A Fixed Point Theory Linked to the Zeros of the Partial Sums of the Riemann Zeta Function

Abstract

For each $n>2$ we consider the corresponding $n\mathrm{th}$-partial sum of the Riemann zeta function $\zeta _{n}(z):=\sum _{j=1}^{n}j^{-z}$ and we introduce two real functions $f_{n}(c)$, $g_{n}(c)$, $c\in \mathbb {R}$, associated with the end-points of the interval of variation of the variable x of the analytic variety $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$, where $\zeta _{n}^{*}(z):=\zeta _{n}(z)-p_{k_{n}}^{-z}$ and $p_{k_{n}}$ is the last prime not exceeding n. The analysis of fixed point properties of $f_{n}$, $g_{n}$ and the behavior of such functions allow us to explain the distribution of the real parts of the zeros of $\zeta _{n}(z)$. Furthermore, the fixed points of $f_{n}$, $g_{n}$ characterize the set $\mathscr {P}^{*}$ of prime numbers greater than 2 and the set $\mathscr {C}^{*}$ of composite numbers greater than 2, proving in this way how close those functions from Arithmetic are. Finally, from the study of the graphs of $f_{n}$, $g_{n}$ we deduce important properties about the set $R_{\zeta _{n}(z)}:=\overline{\{ \mathfrak {R}z:\zeta _{n}(z)=0\} }$ and the bounds $a_{\zeta _{n}(z)}:=\inf \{\mathfrak {R}z:\zeta _{n}(z)=0\}$, $b_{\zeta _{n}(z)}:=\sup \{ \mathfrak {R}z:\zeta _{n}(z)=0\}$ that define the critical strip $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \times \mathbb {R}$ where are located all the zeros of $\zeta _{n}(z)$.

1 Introduction

Since the non-trivial zeros of the Riemann zeta function $\zeta (z)$, until now found, lie on the line $\mathfrak {R}z=1/2$ (the assertion that all them are situated on that line is the Riemann Hypothesis) and the trivial ones are on the real axis (they are the negative even numbers [9, p. 8]), it seems that the zeros of $\zeta (z)$ are situated on those two perpendicular lines. However that is not so for the zeros of the partial sums $\zeta _{n}(z):=\sum _{j=1}^{n}j^{-z}$ of the series $\sum _{j=1}^{\infty }j^{-z}$ that defines the Riemann zeta function $\zeta (z)$ on the half-plane $\mathfrak {R}z>1$. Indeed, except for $\zeta _{2}(z)$ whose zeros all are imaginary (it is immediate to check that the zeros of $\zeta _{2}(z)$ are $z_{2,j}=\frac{(2j+1)\pi i}{\log 2}$, $j\in \mathbb {Z}$), so aligned, the zeros of each $\zeta _{n}(z)$ for any $n>2$ are dispersed in a vertical strip forming a sort of cloud, more or less uniform, that extends up, down and left as n increases, whereas at the right the cloud of zeros is upper bounded (essentially) by the line $\mathfrak {R}z=1$ (see Fig. 1).

Fig. 1

Graphs of the zeros of $\zeta _{n}(z)$ for some values of n, with $\mathfrak {R}z\in [-3,1]$ and $\mathfrak {I}z\in [0,5000]$

An explanation grosso modo why the zeros of the $\zeta _{n}(z)$’s are distributed of such a form is supported by the following facts:

(a) Any exponential polynomial (EP for short) of the form

$$\begin{aligned} P(z):=1+\sum _{j=1}^{N}a_{j}e^{-z\lambda _{j}},\quad z\in \mathbb {C},\quad a_{j}\in \mathbb {C}\setminus \{ 0\} ,\quad 0<\lambda _{1}<\ldots <\lambda _{N},\quad N\ge 1, \end{aligned}$$

(1)

has zeros as a consequence of Hadamard’s Factorization Theorem or from Pólya’s Theorem [13, p. 71]. For $N=1$, it is immediate that an EP of the form (1) has its zeros aligned. For $N>1$, noticing that for any y,

$$ \lim _{x\rightarrow +\infty }P(z)=\lim _{x\rightarrow -\infty }Q(z)=1, $$

where $Q(z):=a_{N}^{-1}e^{z\lambda _{N}}P(z)$ (observe that P(z) and Q(z) have exactly the same zeros), it follows that the zeros of P(z) are situated in a vertical strip. Therefore, for every EP P(z) of the form (1), there exist two real numbers

$$\begin{aligned} a_{P(z)}:=\inf \{ \mathfrak {R}z:P(z)=0\} ,\quad b_{P(z)}:=\sup \{ \mathfrak {R}z:P(z)=0\} , \end{aligned}$$

(2)

that define an interval $[ a_{P(z)},b_{P(z)}]$, called critical interval associated with P(z). Therefore the set $[a_{P(z)},b_{P(z)}] \times \mathbb {R}$, called critical strip associated with P(z), is the minimal vertical strip that contains all the zeros of P(z).

It is immediate that any partial sum $\zeta _{n}(z):=\sum _{j=1}^{n}j^{-z}$, $n\ge 2$, is an EP of the form (1). Therefore the zeros of each $\zeta _{n}(z)$ are situated on its critical strip $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \times \mathbb {R}$ (a detailed proof on the existence of the zeros of $\zeta _{n}(z)$ and their distribution with respect to the imaginary axis can be found in [14, Prop. 1, 2, 3]). Regarding the bounds $a_{\zeta _{n}(z)}$, $b_{\zeta _{n}(z)}$, taking into account that all the zeros of $\zeta _{2}(z)$ lie on the imaginary axis, we get the property

$$\begin{aligned} a_{\zeta _{2}(z)}=b_{\zeta _{2}(z)}=0;\quad a_{\zeta _{n}(z)}<0<b_{\zeta _{n}(z)},\quad n>2, \end{aligned}$$

(3)

that will be proved below in Lemma 2.3, Part (ii). A much more precise estimation of such bounds is given by the formulas:

$$\begin{aligned} b_{\zeta _{n}(z)}=1+\left( \frac{4}{\pi }-1+o(1)\right) \frac{\log \log n}{\log n},\quad n\rightarrow \infty , \end{aligned}$$

(4)

obtained by Montgomery and Vaughan [12] in 2001, by completing a previous work of Montgomery [11] of 1983, and

$$\begin{aligned} a_{\zeta _{n}(z)}=-\frac{\log 2}{\log ( \frac{n-1}{n-2}) }+\varDelta _{n},\quad \lim \sup _{n\rightarrow \infty }| \varDelta _{n}| \le \log 2, \end{aligned}$$

(5)

found by Mora [17] in 2015. Consequently, from (5) and (4), we have

$$ \lim _{n\rightarrow \infty }a_{\zeta _{n}(z)}=-\infty ,\quad \lim _{n\rightarrow \infty }b_{\zeta _{n}(z)}=1, $$

what justifies the fact of the cloud of zeros of $\zeta _{n}(z)$ moves to the left as n increases but not to the right, where the cloud is upper bounded (essentially) by the line $\mathfrak {R}z=1$ (it does not mean that some $\zeta _{n}(z)$ can have zeros with real part greater than 1; in fact, many works prove the existence of such zeros [10, 22, 23, 25], among others).

(b) Since the zeros of an analytic function are isolated, and all the $\zeta _{n}(z)$’s are entire functions, by taking into account the real parts of the zeros of each $\zeta _{n}(z)$ are bounded (the real parts are contained in the critical interval $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$ for every fixed n), their imaginary parts cannot be. Furthermore, as the coefficients of every $\zeta _{n}(z)$ are real, its zeros are conjugate. Consequently the zeros of the $\zeta _{n}(z)$’s are located up and down, symmetrically with respect to the real axis.

(c) From (3) we deduce that, for any $n>2$, $\zeta _{n}(z)$ has zeros with positive and negative real parts.

With the aim to understand what law controls the distribution of the real projections of the zeros of $\zeta _{n}(z)$, we introduce a Fixed Point Theory focused on two real functions, $f_{n}$ and $g_{n}$, for every $n>2$. Firstly, such functions, by virtue of a recent result [19, Theorem 3], allow us to have an easy characterization of the sets

$$\begin{aligned} R_{\zeta _{n}(z)}:=\overline{\{ \mathfrak {R}z:\zeta _{n}(z)=0\} }. \end{aligned}$$

(6)

Secondly, among others relevant results deduced from the fixed point properties of $f_{n}$ and $g_{n}$, we stress those that characterize some notable arithmetic sets such as $\mathscr {P}^{*}$ and $\mathscr {C}^{*}$, the set of primes greater than 2 and the set of composite numbers greater than 2, respectively. In this way, we show how close the arithmetic sets $\mathscr {P}^{*}$ and $\mathscr {C}^{*}$ from the law of the distribution of the zeros of the partial sums of the Riemann zeta function are. Furthermore, our point fixed theory proves the existence of a minimal density interval for each $\zeta _{n}(z)$, that is, a closed interval $[ A_{n},b_{\zeta _{n}(z)}]$, with $a_{\zeta _{n}(z)}\le A_{n}<b_{\zeta _{n}(z)}$ contained in the set $R_{\zeta _{n}(z)}$, for any integer $n>2$, which means that there is no vertical sub-trip contained in $[ A_{n},b_{\zeta _{n}(z)}] \times \mathbb {R}$ zero-free for $\zeta _{n}(z)$. Then, since it is always true that $R_{\zeta _{n}(z)}\subset [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$, when the bound $A_{n}$ coincides with $a_{\zeta _{n}(z)}$ it follows that $R_{\zeta _{n}(z)}=[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$. In this case we will say that $\zeta _{n}(z)$ has a maximum density interval, and it is exactly the critical interval $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$. Finally, we will give a sufficient condition in terms of the quantity of fixed points of $f_{n}$ for $\zeta _{n}(z)$ have a maximum density interval.

2 The Functions $f_{n}$ and $g_{n}$

The functions $f_{n}$ and $g_{n}$ that we are going to introduce below, are directly linked to the interval of variation of the variable x of the Cartesian equation of an analytic variety associated with the $n\mathrm{th}$-partial sum $\zeta _{n}(z):=\sum _{j=1}^{n}j^{-z}$, $n>2$. First we consider the EP

$$\begin{aligned} \zeta _{n}^{*}(z):=\zeta _{n}(z)-p_{k_{n}}^{-z},\quad n>2, \end{aligned}$$

(7)

where $p_{k_{n}}$ is the last prime not exceeding n. The bounds $a_{\zeta _{n}^{*}(z)}$, $b_{\zeta _{n}^{*}(z)}$ defined in (2) corresponding to $\zeta _{n}^{*}(z)$ satisfy the following crucial property (for details see [16, Theorem 15]) :

$$\begin{aligned} a_{\zeta _{n}^{*}(z)}=b_{\zeta _{n}^{*}(z)}=0,\quad \text {for }n=3,4;\quad a_{\zeta _{n}^{*}(z)}<0<b_{\zeta _{n}^{*}(z)},\quad \text {for all }n>4. \end{aligned}$$

(8)

Now our objective is to analyse the behavior of the end-points of the interval of variation of the variable x of the analytic variety, or level curve [24, p. 121], of equation

$$\begin{aligned} | \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c},\quad n>2,\quad c\in \mathbb {R}. \end{aligned}$$

(9)

To do it, we square (9) and by using elementary formulas of trigonometry we obtain the Cartesian equation of $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$, namely

$$\begin{aligned} \begin{array}{l} \displaystyle \sum _{j=1,j\ne p_{k_{n}}}^{n}j^{-2x}+2\cdot 1^{-x}\sum _{j=2,j\ne p_{k_{n}}}^{n}j^{-x}\cos ( y\log (\frac{j}{1})) + \\ \\ \displaystyle 2\cdot 2^{-x}\sum _{j=3,j\ne p_{k_{n}}}^{n}j^{-x}\cos ( y\log (\frac{j}{2})) +\cdots + \\ \\ \displaystyle 2(n-1)^{-x}\sum _{j=n,j\ne p_{k_{n}}}^{n}j^{-x}\cos ( y\log (\frac{j}{n-1})) =p_{k_{n}}^{-2c}. \end{array} \end{aligned}$$

(10)

It is immediate to see that for any value of y, the left-hand side of (10) tends to $+\infty$ as $x\rightarrow -\infty$. Then, as the right-hand side of (10) is a constant, the variation of x is always lower bounded by a number denoted by $a_{n,c}$. On the other hand, the limit of the left-hand side of (10) is 1 when $x\rightarrow +\infty$. Then, if $c\ne 0$, the variation of x is upper bounded by a number denoted by $b_{n,c}$. Therefore, fixed an integer $n>2$, we have:

If $c\ne 0$, the variable x in the Eq. (10) varies on an open interval $( a_{n,c},b_{n,c})$ satisfying the properties: (a) Given $x\in ( a_{n,c},b_{n,c})$, there is at least a point of the level curve $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$ with abscissa x. Exceptionally $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$ could have points of abscissas $a_{n,c}$, $b_{n,c}$. In this case we will say that $a_{n,c}$, $b_{n,c}$ are accessible. Otherwise the lines $x=a_{n,c}$, $x=b_{n,c}$ are asymptotes of the variety. (b) For $x<a_{n,c}$ or $x>b_{n,c}$ there is no point of the variety $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$.

If $c=0$, x varies on $( a_{n,0},+\infty )$, so $b_{n,0}$ can be defined as $+\infty$, satisfying: (c) Given $x\in ( a_{n,0},+\infty )$, there is at least a point of the variety $| \zeta _{n}^{*}(z)| =1$ with abscissa x. If there is a point of $| \zeta _{n}^{*}(z)| =1$ with abscissa $a_{n,0}$, we will say that $a_{n,0}$ is accessible. Otherwise the line $x=a_{n,0}$ is an asymptote of the variety. (d) For $x<a_{n,0}$ there is no point of $| \zeta _{n}^{*}(z)| =1$.

We show in Fig. 2 the varieties $| \zeta _{n}^{*}(z)|=p_{k_{n}}^{-c}$ for $n=3$ and some values of c.

Fig. 2

Graphs of the varieties $| \zeta _{3}^{*}(z)|=3^{-c}$ for some values of c

The end-points $a_{3,c}$, $b_{3,c}$ corresponding to the variety $|\zeta _{3}^{*}(z)| =p_{k_{3}}^{-c}$ can be easily determined by a completely similar way to those of the variety $| \zeta _{3}^{*}(-z)| =p_{k_{3}}^{c}$ (see [8, p. 49]). Each bound $a_{3,c}$, $b_{3,c}$ as a function of c is given by the formulas

$$\begin{aligned} a_{3,c}=-\frac{\log (1+3^{-c})}{\log 2},\quad c\in \mathbb {R}; \quad b_{3,c}=\left\{ \begin{array}{c} -\frac{\log (3^{-c}-1)}{\log 2},\quad \text {if }c<0 \\ \\ -\frac{\log (1-3^{-c})}{\log 2},\quad \text {if }c>0 \end{array} \right. . \end{aligned}$$

(11)

By virtue of above considerations (a), (b), (c), (d), and by using an elementary geometric reasoning, similar to that it was used to find the graphs of $| \zeta _{n}^{*}(-z)| =p_{k_{n}}^{c}$ (see [16, Proposition 8]), the graphs of the varieties $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$ are described in the next result.

Proposition 2.1

Fixed an integer $n>2$, we have:

• (i) If $c>0$, $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$ has infinitely many arc-connected components which are closed curves and x varies on a finite interval $( a_{n,c},b_{n,c})$, where $a_{n,c},b_{n,c}$ could be accessible.

• (ii) If $c=0$, $| \zeta _{n}^{*}(z)| =1$ has infinitely many arc-connected components which are open curves with horizontal asymptotes of equations $y=( 2j+1) \frac{\pi }{2\log 2}$, $j\in \mathbb {Z}$, and x varies on the infinite interval $( a_{n,0},+\infty )$, where $a_{n,0}$ could be accessible.

• (iii) If $c<0$, $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$ has only one arc-connected component which is an open curve; x varies on a finite interval $(a_{n,c},b_{n,c})$, where $a_{n,c},b_{n,c}$ could be accessible. The variable y takes all real values. Furthermore, $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$ intersects the real axis at a unique point of abscissa $b_{n,c}$, so $b_{n,c}$ is always accessible when $c<0$.

In Fig. 3 we show the graph of $| \zeta _{n}^{*}(z)|=p_{k_{n}}^{-c}$ for some values of $n>3$ and c.

Fig. 3

Graphs of the varieties $| \zeta _{7}^{*}(z)|=7^{-c}$ and $| \zeta _{12}^{*}(z)|=11^{-c}$ for some values of c

From Proposition 2.1, a simple consequence is deduced:

Corollary 2.1

Fixed an integer $n>2$, if $u\in \mathbb {C}$ satisfies $| \zeta _{n}^{*}(u)| <p_{k_{n}}^{-c}$ (in this case we will say that u is an interior point of the variety $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$), then there exists a point w of $| \zeta _{n}^{*}(z)|=p_{k_{n}}^{-c}$, so $a_{n,c}\le \mathfrak {R}w\le$ $b_{n,c}$, such that $\mathfrak {R}w<\mathfrak {R}u$.

Definition 2.1

Given an integer $n>2$, we define the real functions

$$\begin{aligned} f_{n}(c):=a_{n,c},\quad c\in \mathbb {R};\quad \quad g_{n}(c):=b_{n,c},\quad c\in \mathbb {R}\setminus \{ 0\} , \end{aligned}$$

(12)

where $a_{n,c}$, $b_{n,c}$ are the end-points of the interval of variation of the variable x in the Eq. (10).

We show in Fig. 4 the graph of the functions $f_{3}(c)$ and $g_{3}(c)$, defined by the Eq. (11), and the function $f_{4}(c)$.

Fig. 4

Left: Graph of the functions $f_{3}(c)$ (blue), $g_{3}(c)$ (red) and $y=x$ (plotted). Right: Graph of the function $f_{4}(c)$ (blue) and $y=x$ (plotted)

Since $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-d}$ tends to $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$ as d tends to c, it is immediate that $f_{n}$, $g_{n}$ are both continuous on $\mathbb {R}\setminus \{ 0\}$, and $f_{n}$ is continuous on whole of $\mathbb {R}$. For $c=0$, by Part (ii) of Proposition 1 we can agree $b_{n,0}=+\infty$, and then we should define $g_{n}(0):=$ $+\infty$.

Now we are ready to give a characterization of the set $R_{\zeta _{n}(z)}$, defined in (6), by using the functions $f_{n}$ and $g_{n}$.

Theorem 2.1

Let $n>2$ be a fixed integer. A real number $c\in R_{\zeta _{n}(z)}$ if and only if

$$\begin{aligned} f_{n}(c)\le c\le g_{n}(c). \end{aligned}$$

(13)

Proof

If c $\in$ $R_{\zeta _{n}(z)}$, there exists a sequence $(z_{m}) _{m=1,2,\ldots }$ of zeros of $\zeta _{n}(z)$ such that $\lim _{m\rightarrow \infty }\mathfrak {R}z_{m}=c$. From (7), $\zeta _{n}^{*}(z_{m})=-p_{k_{n}}^{-z_{m}}$ for each $m=1,2,\ldots$. By taking the modulus, we have $| \zeta _{n}^{*}(z_{m})| =p_{k_{n}}^{-x_{m}}$, where $x_{m}:=\mathfrak {R}z_{m}$. This means that each $z_{m}$ is a point of the variety $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-x_{m}}$, so $x_{m}\in [ a_{n,x_{m}},b_{n,x_{m}}]$ and then we get

$$ f_{n}(x_{m})=a_{n,x_{m}}\le x_{m}\le b_{n,x_{m}}=g_{n}(x_{m}),\quad \text {for all }m. $$

Now by taking the limit when $m\rightarrow \infty$, noticing that $\lim _{m\rightarrow \infty }x_{m}=c$, because of the continuity of $f_{n}$ and $g_{n}$, the inequalities (13) follow. Conversely, if $f_{n}(c)<c<g_{n}(c)$, by taking into account the definitions of $f_{n}$, $g_{n}$, the value c is in the interval of variation of x of the variety $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$ and then the line $x=c$ intersects the variety. Therefore, by applying [16, Theorem 3], $c\in$ $R_{\zeta _{n}(z)}$. If $f_{n}(c)=c$ or $g_{n}(c)=c$, the line $x=c$ intersects the variety $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$ provided that $a_{n,c}$ or $b_{n,c}$ be accessible. Otherwise the line $x=c$ is an asymptote of $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$. Therefore, in both cases, again by [19, Theorem 3], the point $c\in$ $R_{\zeta _{n}(z)}$.       $\square$

As we can easily check, the function $f_{3}(c):=a_{3,c}$, with $a_{3,c}$ given in (11), is strictly increasing; this property is true for all the functions $f_{n}(c)$, $n>2$, defined in (12), as we prove below.

Lemma 2.1

For every integer $n>2$, $f_{n}$ is a strictly increasing function on $\mathbb {R}$.

Proof

Firstly, for each fixed c $\in \mathbb {R}$, we claim that $f_{n}$ satisfies

$$\begin{aligned} \inf \{ | \zeta _{n}^{*}(f_{n}(c)+iy)| :y\in \mathbb {R}\} =p_{k_{n}}^{-c}. \end{aligned}$$

(14)

Indeed, we put $\lambda _{n,c}:=\inf \{ | \zeta _{n}^{*}(f_{n}(c)+iy)| :y\in \mathbb {R}\}$. By assuming $\lambda _{n,c}<p_{k_{n}}^{-c}$, there exists a point $z_{c}:=f_{n}(c)+iy_{c}$ such that

$$ \lambda _{n,c}\le | \zeta _{n}^{*}(z_{c})|<p_{k_{n}}^{-c} , $$

and then it means that $z_{c}$ is an interior point of $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$. By Corollary 2.1 there exists w belonging to the variety $| \zeta _{n}^{*}(z)|=p_{k_{n}}^{-c}$, so $a_{n,c}\le \mathfrak {R}w\le$ $b_{n,c}$, such that $\mathfrak {R}w<\mathfrak {R}z_{c}=f_{n}(c)=a_{n,c}$. But this is a contradiction, and then necessarily

$$\begin{aligned} \lambda _{n,c}\ge p_{k_{n}}^{-c}. \end{aligned}$$

(15)

For $\varepsilon >0$ sufficiently small, we consider the strip

$$ S_{\varepsilon }:=\{ z\in \mathbb {C}:a_{n,c}\le \mathfrak {R}z<a_{n,c}+\varepsilon \} , $$

and put

$$ \lambda _{n,c,\varepsilon }:=\inf \{ | \zeta _{n}^{*}(z)| :z\in S_{\varepsilon }\} . $$

From the definition of $a_{n,c}$, the set $S_{\varepsilon }$ contains infinitely many points of the variety $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$. Then $\lambda _{n,c,\varepsilon }$ $\le p_{k_{n}}^{-c}$ for all $\varepsilon >0$, so $\lambda _{n,c}\le p_{k_{n}}^{-c}$. Therefore, according to (15), $\lambda _{n,c}=p_{k_{n}}^{-c}$ and then (14) follows. Let d be a real number such that $d<c$, so $p_{k_{n}}^{-d}>p_{k_{n}}^{-c}$. Let $\eta$ be such that $0<\eta <p_{k_{n}}^{-d}-p_{k_{n}}^{-c}$. From (14), there exists some point $z_{\eta }:=f_{n}(c)+iy_{\eta }$ such that

$$ p_{k_{n}}^{-c}\le | \zeta _{n}^{*}(z_{\eta })|<p_{k_{n}}^{-c}+\eta <p_{k_{n}}^{-d}, $$

so $z_{\eta }$ is interior of $| \zeta _{n}^{*}(z)|=p_{k_{n}}^{-d}$. By Corollary 2.1, there exists a point $w_{\eta }$ of $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-d}$, so $a_{n,d}\le \mathfrak {R}w_{\eta }\le b_{n,d}$, such that $\mathfrak {R}w_{\eta }<\mathfrak {R}z_{\eta }$. Then

$$ f_{n}(d)=a_{n,d}\le \mathfrak {R}w_{\eta }<\mathfrak {R}z_{\eta }=f_{n}(c), $$

which definitely proves the lemma.       $\square$

In the next result we prove that $f_{n}$ is upper bounded by the number $a_{\zeta _{n}^{*}(z)}$ defined in (2) corresponding to the EP $\zeta _{n}^{*}(z)$, defined in (7).

Lemma 2.2

For every $n>2$, the function $f_{n}$ satisfies

$$ f_{n}(c)<a_{\zeta _{n}^{*}(z)}\text { for any }c\in \mathbb {R}. $$

Proof

Let c be an arbitrary real number. By taking into account the definition of $a_{\zeta _{n}^{*}(z)}$, there exists a sequence $( z_{m})_{m=1,2,\ldots }$of zeros of $\zeta _{n}^{*}(z)$, with $\mathfrak {R}z_{m}\ge a_{\zeta _{n}^{*}(z)}$, such that

$$\begin{aligned} \lim _{m\rightarrow \infty }\mathfrak {R}z_{m}=a_{\zeta _{n}^{*}(z)}. \end{aligned}$$

(16)

Since $\zeta _{n}^{*}(z_{m})=0$, we get $| \zeta _{n}^{*}(z_{m})| <p_{k_{n}}^{-c}$, for all m. Then, from Corollary 2.1, there exists a sequence $( w_{m}) _{m=1,2,\ldots }$ of points of $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$, so $a_{n,c}\le \mathfrak {R}w_{m}\le b_{n,c}$, such that $\mathfrak {R}w_{m}<\mathfrak {R}z_{m}$, for all m. Therefore, since $f_{n}(c)=a_{n,c}$, we have

$$ f_{n}(c)\le \mathfrak {R}w_{m}<\mathfrak {R}z_{m},\quad \text {for all }m. $$

Now, by taking the limit in the above inequality when $m\rightarrow \infty$, by (16), we get

$$ f_{n}(c)\le a_{\zeta _{n}^{*}(z)}\quad \text {for any }c\in \mathbb {R}, $$

implying, noticing that by Lemma 2.1 $f_{n}$ is strictly increasing, that $f_{n}(c)<a_{\zeta _{n}^{*}(z)}$ for any $c\in \mathbb {R}$.       $\square$

For every $n>2$, let $a_{\zeta _{n}(z)}$, $b_{\zeta _{n}(z)}$ be the bounds, defined in (2), corresponding to the EP $\zeta _{n}(z)$. The function $g_{n}$, defined in (12), has the following properties.

Lemma 2.3

For every $n>2$, the function $g_{n}$ satisfies:

• (i) $g_{n}$ is strictly increasing on $( -\infty ,0)$ and decreasing on $(0,+\infty )$.

• (ii) If n is composite, then $c\le g_{n}(c)$ for any $c\in ( -\infty ,b_{\zeta _{n}(z)}] \setminus \{ 0\}$ and the inequality is strict for all $c\in ( -\infty ,b_{\zeta _{n}(z)}) \setminus \{ 0\}$; if $c\in ( b_{\zeta _{n}(z)},+\infty )$, then $c>g_{n}(c)$.

• (iii) If n is prime, then $c\le g_{n}(c)$ for any $c\in [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \setminus \{ 0\}$ and the inequality is strict for all $c\in ( a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}) \setminus \{0\}$; if $c\in ( -\infty ,a_{\zeta _{n}(z)}) \cup (b_{\zeta _{n}(z)},+\infty )$, then $c>g_{n}(c)$.

Proof

Part (i). Let c, d be real numbers such that $c<d<0$. From Proposition 2.1, $b_{n,c}$ and $b_{n,d}$ are the unique points of $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$ and $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-d}$ that intersect the real axis, respectively. Therefore $b_{n,c}$ and $b_{n,d}$ satisfy the equations

$$\begin{aligned} \sum _{\begin{array}{c} m=1\\ m\ne p_{k_{n}} \end{array}}^{n}m^{-x}=p_{k_{n}}^{-c},\quad \sum _{\begin{array}{c} m=1\\ m\ne p_{k_{n}} \end{array}}^{n}m^{-x}=p_{k_{n}}^{-d}, \end{aligned}$$

(17)

respectively. Each equation of (17) has only one real solution by virtue of [20, p. 46] and then, since $p_{k_{n}}^{-c}>p_{k_{n}}^{-d}$, the real solution of the first equation is obviously greater than the second one. Therefore $-b_{n,c}>-b_{n,d}$, equivalently, $b_{n,c}<b_{n,d}$. Consequently, $g_{n}(c)<g_{n}(d)$ and then $g_{n}$ is strictly increasing in $( -\infty ,0)$. Let c, d be such that $c>d>0$. From Proposition 2.1, $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$ and $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-d}$ have infinitely many arc-connected components which are closed curves. Since $p_{k_{n}}^{-c}<p_{k_{n}}^{-d}$, any point of $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$ is interior of $| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-d}$, so $b_{n,c}\le b_{n,d}$. That is, $g_{n}(c)\le g_{n}(d)$, which means that $g_{n}$ is decreasing on $( 0,+\infty )$.

Part (ii). We firstly demonstrate that the bounds $a_{\zeta _{n}(z)}$, $b_{\zeta _{n}(z)}$ defined in (2) corresponding to $\zeta _{n}(z)$ satisfy the second inequality of (3), that is

$$\begin{aligned} a_{\zeta _{n}(z)}<0<b_{\zeta _{n}(z)}\quad \text {for all }n>2. \end{aligned}$$

(18)

Indeed, we introduce the EP

$$\begin{aligned} G_{n}(z):=\zeta _{n}(-z). \end{aligned}$$

(19)

In [7, Chap. 3, Theorem 3.20] was shown that

$$ b_{G_{n}(z)}:=\sup \{ \mathfrak {R}z:G_{n}(z)=0\}>0\quad \text {for all }n>2, $$

now we claim that

$$ a_{G_{n}(z)}:=\inf \{ \mathfrak {R}z:G_{n}(z)=0\} <0\quad \text {for all }n>2. $$

Otherwise, if all the zeros of $G_{n}(z)$, say $(z_{n,k})_{k=1,2,\ldots }$, satisfy $\mathfrak {R}z_{n,k}\ge 0$, since $b_{G_{n}(z)}>0$, there is at least a zero $z_{n,k_{0}}$ with $\mathfrak {R}z_{n,k_{0}}>0$. Then, as $G_{n}(z)$ is almost-periodic (see for instance [4, 5] and [10, Chap. VI]), $G_{n}(z)$ has infinitely many zeros in the strip

$$ S_{\varepsilon }:=\{ z:\mathfrak {R}z_{n,k_{0}}-\varepsilon<\mathfrak {R}z<\mathfrak {R}z_{n,k_{0}}+\varepsilon \} ,\quad 0<\varepsilon <\mathfrak {R}z_{n,k_{0}}, $$

and consequently

$$\begin{aligned} \sum _{k=1}^{\infty }\mathfrak {R}z_{n,k}=+\infty . \end{aligned}$$

(20)

However, as all the coefficients of $G_{n}(z)$ are equal to 1, [21, formula (9)] applies and then we get $\sum _{k=1}^{\infty }\mathfrak {R}z_{n,k}=O(1)$, contradicting (20). Therefore the claim follows, that is, $a_{G_{n}(s)}<0$ for all $n>2$. By (19) we have $a_{\zeta _{n}(z)}=-b_{G_{n}(z)}$ and $b_{\zeta _{n}(z)}=-a_{G_{n}(z)}$, so (18) follows.

We now consider the point $b_{\zeta _{n}(z)}$. It is immediate that $b_{\zeta _{n}(z)}$ belongs to the set $R_{\zeta _{n}(z)}$ defined in (6). Then from Theorem 2.1 we have $b_{\zeta _{n}(z)}\le g_{n}(b_{\zeta _{n}(z)})$, so the property $c\le g_{n}(c)$ is true for $c=b_{\zeta _{n}(z)}$. From (18) and by using that $g_{n}$ is decreasing on $( 0,\infty )$ by virtue of Part (i), for any c $\in ( 0,b_{\zeta _{n}(z)})$ we have

$$\begin{aligned} 0<c<b_{\zeta _{n}(z)}\le g_{n}(b_{\zeta _{n}(z)})\le g_{n}(c). \end{aligned}$$

(21)

Consequently, Part (ii) follows for $c\in ( 0,b_{\zeta _{n}(z)}]$. We now assume $c<0$ and n composite, so $p_{k_{n}}<n$. If $b_{n,c}\ge 0$, then $c<b_{n,c}=g_{n}(c)$ and again Part (ii) is true. Finally, we suppose $b_{n,c}<0$. Since $c<0$, $b_{n,c}$ satisfies the first equation of (17) and then $p_{k_{n}}^{-c}>n^{-b_{n,c}}$. Consequently $-c>-b_{n,c}$, so $c<b_{n,c}$ and then Part (ii) follows for $c\in ( -\infty ,b_{\zeta _{n}(z)}] \setminus \{ 0\}$. Finally, we claim that $c>g_{n}(c)$ when $c>b_{\zeta _{n}(z)}$. Indeed, because of Lemma 2.2 and (8), we have $f_{n}(c)<a_{\zeta _{n}^{*}(z)}\le 0$ for any c. Therefore, since $c>b_{\zeta _{n}(z)}$, by (18) c is positive and then $f_{n}(c)<c$. Assume $c>g_{n}(c)$ is not true. Then we would have $f_{n}(c)<c\le g_{n}(c)$ and by Theorem 2.1, $c\in R_{\zeta _{n}(z)}\subset [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$ which means that $c\le b_{\zeta _{n}(z)}$. This is a contradiction because $c>b_{\zeta _{n}(z)}$, so the claim follows. This definitely proves Part (ii).

Part (iii). We first note that, since n is prime, $p_{k_{n}}=n$. Therefore the first equation in (17) becomes $\sum _{m=1}^{n-1}m^{-x}=n^{-c}$. By assuming $c<0$, $b_{n,c}$ satisfies the above equation and then we have

$$\begin{aligned} \sum _{m=1}^{n-1}m^{-b_{n,c}}=n^{-c}. \end{aligned}$$

(22)

For every $n\ge 2$, we consider the number $\beta _{G_{n}(z)}$, defined as the unique real solution of the equation $\sum _{m=1}^{n-1}m^{x}=n^{x}$ (see [20, p. 46]). By [6, Proposition 5], $\beta _{G_{n}(z)}\ge b_{G_{n}(z)}$ and the equality is attained for n prime. Therefore the set $\mathbb {R}$ of real numbers is partitioned in two sets:

$$\begin{aligned} ( -\infty ,\beta _{G_{n}(z)}] =\{ x\in \mathbb {R}:\sum _{m=1}^{n-1}m^{x}\ge n^{x}\} , \end{aligned}$$

(23)

and

$$\begin{aligned} ( \beta _{G_{n}(z)},\infty ) =\{ x\in \mathbb {R}:\sum _{m=1}^{n-1}m^{x}<n^{x}\} . \end{aligned}$$

(24)

Now we claim that $c\le g_{n}(c)$ when $a_{\zeta _{n}(z)}\le c<0$. Indeed, by (19), $b_{G_{n}(z)}=-a_{\zeta _{n}(z)}$, so c is such that $0<-c\le b_{G_{n}(z)}=\beta _{G_{n}(z)}$. Then, according to (23), we have

$$\begin{aligned} \sum _{m=1}^{n-1}m^{-c}\ge n^{-c}. \end{aligned}$$

(25)

Therefore, if we assume $c>g_{n}(c)=b_{n,c}$, by applying (25) and taking into account (22), we get

$$ n^{-c}\le \sum _{m=1}^{n-1}m^{-c}<\sum _{m=1}^{n-1}m^{-b_{n,c}}=n^{-c}, $$

which is a contradiction. Therefore $c\le g_{n}(c)$ is true for c such that $a_{\zeta _{n}(z)}\le c<0$. Consequently, taking into account (21), it follows

$$ c\le g_{n}(c)\text {, for any }c\in [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \setminus \{ 0\} , $$

where the inequality is strict for all c of $( a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}) \setminus \{ 0\}$. Now suppose $c\in ( -\infty ,a_{\zeta _{n}(z)})$. Then, since $-c >-a_{\zeta _{n}(z)}=b_{G_{n}(z)}=\beta _{G_{n}(z)}$, by applying (24) we have

$$\begin{aligned} \sum _{m=1}^{n-1}m^{-c}<n^{-c}. \end{aligned}$$

(26)

It implies that $c>g_{n}(c)$. Indeed, by supposing $c\le g_{n}(c)=b_{n,c}$, from (22) and (26) we are led to the following contradiction:

$$ n^{-c}=\sum _{m=1}^{n-1}m^{-b_{n,c}}\le \sum _{m=1}^{n-1}m^{-c}<n^{-c}. $$

Therefore $c>g_{n}(c)$ if $c\in ( -\infty ,a_{\zeta _{n}(z)})$. Finally, if $c\in ( b_{\zeta _{n}(z)},+\infty )$, the reasoning used to demonstrate the end of Part (ii) of the lemma proves that $c>g_{n}(c)$.       $\square$

As a consequence of Lemma 2.3 we find the fixed points of the function $g_{n}$.

Corollary 2.2

For every composite number $n>2$, $b_{\zeta _{n}(z)}$ is the fixed point of the function $g_{n}$. If $n>2$ is prime, $a_{\zeta _{n}(z)}$, $b_{\zeta _{n}(z)}$ are the fixed points of $g_{n}$.

Proof

Fixed an integer $n>2$, by (18) $a_{\zeta _{n}(z)}$, $b_{\zeta _{n}(z)}\ne 0$, so $g_{n}$ is well defined at $a_{\zeta _{n}(z)}$ and $b_{\zeta _{n}(z)}$. By applying Part (ii) of Lemma 2.3 for $n>2$ composite, it is immediate, by the continuity of $g_{n}$, that the unique fixed point of $g_{n}$ is $b_{\zeta _{n}(z)}$. If $n>2$ is prime, by Part (iii) of Lemma 2.3, we get $g_{n}(a_{\zeta _{n}(z)})=a_{\zeta _{n}(z)}$ and $g_{n}(b_{\zeta _{n}(z)})=b_{\zeta _{n}(z)}$. Furthermore, Part (iii) of Lemma 2.3 also proves that $a_{\zeta _{n}(z)}$, $b_{\zeta _{n}(z)}$ are the unique fixed points of $g_{n}$.       $\square$

In the next result we obtain a characterization of $\mathscr {P}^{*}$, the set of prime numbers greater than 2.

Theorem 2.2

An integer $n>2$ belongs to $\mathscr {P}^{*}$ if and only if $a_{\zeta _{n}(z)}$ is a fixed point of the function $g_{n}$.

Proof

Assume $n>2$ is prime, from Corollary 2.2, $a_{\zeta _{n}(z)}$ is a fixed point of $g_{n}$. Conversely, if

$$\begin{aligned} g_{n}(a_{\zeta _{n}(z)})=a_{\zeta _{n}(z)}, \end{aligned}$$

(27)

by supposing n composite, from Part (ii) of Lemma 2.3, we have $c<g_{n}(c)$ for all $c\in ( -\infty ,b_{\zeta _{n}(z)}) \setminus \{0\}$. From (18), $a_{\zeta _{n}(z)}\in ( -\infty ,b_{\zeta _{n}(z)}) \setminus \{ 0\}$. Then, $a_{\zeta _{n}(z)}<g_{n}(a_{\zeta _{n}(z)})$. This contradicts (27). Consequently n is a prime number and then the theorem follows.       $\square$

3 The Fixed Points of $f_{n}$ and the Sets $R_{\zeta _{n}(z)}$

For every integer $n>2$, the function $f_{n}$ defined in (12) allows us to give a sufficient condition to have points of the set $R_{\zeta _{n}(z)}$, defined in (6).

Theorem 3.1

For every integer $n>2$, if a point c $\in$ $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$ satisfies $f_{n}(c)\le c$, then $c\in R_{\zeta _{n}(z)}$.

Proof

We first claim that

$$\begin{aligned} a_{\zeta _{n}(z)}, 0, b_{\zeta _{n}(z)}\in R_{\zeta _{n}(z)}\quad \text {for every }n\ge 2. \end{aligned}$$

(28)

Indeed, for $n=2$, the claim trivially follows because as we have seen in Introduction all the zeros of $\zeta _{2}(z)$ are imaginary, so $a_{\zeta _{2}(z)}=$ $b_{\zeta _{2}(z)}=0$ and then $R_{\zeta _{2}(z)}=\{0\}$. Therefore we assume $n>2$. By taking into account the definitions of $a_{\zeta _{n}(z)}$, $b_{\zeta _{n}(z)}$, both numbers obviously belong to $R_{\zeta _{n}(z)}$. Regarding the fact that $0\in R_{\zeta _{n}(z)}$for all $n>2$, it was proved in [18, (3.7)]. Then (28) is true. Hence it only remains to prove the theorem for c $\in$ $(a_{G_{n}(z)},b_{G_{n}(z)})\setminus \{ 0\}$. But in this case,since by hypothesis $f_{n}(c)\le c$, by using Parts (ii) and (iii) of Lemma 2.3 we are lead to $f_{n}(c)\le c<g_{n}(c)$ and then, by Theorem 2.1, $c\in R_{\zeta _{n}(z)}$.       $\square$

An important conclusion is deduced from the above theorem.

Theorem 3.2

For every integer $n>2$, if c belongs to $R_{\zeta _{n}(z)}$ then

$$\begin{aligned}{}[ f_{n}(c),c] \cap [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \subset R_{\zeta _{n}(z)}. \end{aligned}$$

(29)

If $n>2$ is composite and c belongs to $R_{\zeta _{n}(z)}$, then

$$\begin{aligned}{}[ f_{n}(c),c] \subset R_{\zeta _{n}(z)}. \end{aligned}$$

(30)

Proof

Assume $c\in$ $R_{\zeta _{n}(z)}$. Then, by Theorem 2.1, $f_{n}(c)\le c\le g_{n}(c)$. Therefore the interval $[ f_{n}(c),c]$ is well defined. If $f_{n}(c)=c$ the theorem trivially follows. Suppose $f_{n}(c)<c$. Let t be a point of $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$ such that $f_{n}(c)<t<c$. By Lemma 2.1, $f_{n}(t)<f_{n}(c)$. Therefore we have

$$ f_{n}(t)<f_{n}(c)<t<c, $$

and then, by applying Theorem 3.1, $t\in R_{\zeta _{n}(z)}$. Consequently

$$ (f_{n}(c),c)\cap [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \subset R_{\zeta _{n}(z)}, $$

and from the closedness of $R_{\zeta _{n}(z)}$, (29) follows.

Assume $n>2$ is composite. Since $c\in$ $R_{\zeta _{n}(z)}$ and

$$ R_{\zeta _{n}(z)}\subset [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] , $$

we have $c\le b_{\zeta _{n}(z)}$. Furthermore, from Theorem 2.1, $f_{n}(c)\le c\le g_{n}(c)$. Then, if $f_{n}(c)$ $=c$, (30) is obviously true. Suppose $f_{n}(c)<c$. Consider a number t such that $f_{n}(c)\le t<c$. Then, we get

$$\begin{aligned} f_{n}(c)\le t<c\le b_{\zeta _{n}(z)}. \end{aligned}$$

(31)

If $t=0$, by virtue of (28), $t\in R_{\zeta _{n}(z)}$. If $t\ne 0$, from (31), $t\in ( -\infty ,b_{\zeta _{n}(z)}) \setminus \{0\}$. Then, as n is composite, by Part (ii) of Lemma 2.3, $t<g_{n}(t)$. On the other hand, since $t<c$, from Lemma 2.1, $f_{n}(t)<f_{n}(c)$ and then, again by (31), we have

$$ f_{n}(t)<f_{n}(c)\le t<g_{n}(t). $$

Now, by applying Theorem 2.1, $t\in R_{\zeta _{n}(z)}$. Consequently $[f_{n}(c),c)$ $\subset R_{\zeta _{n}(z)}$ and then, since by hypothesis $c\in$ $R_{\zeta _{n}(z)}$, we get $[ f_{n}(c),c]\subset R_{\zeta _{n}(z)}$. The proof is now completed.       $\square$

As a consequence of the two preceding results we characterize the set $\mathscr {C}^{*}$ of composite numbers $n>2$.

Corollary 3.1

For every $n\in \mathscr {C}^{*}$, $a_{\zeta _{n}(z)}$ is a fixed point of the function $f_{n}$.

Proof

Assume $n\in \mathscr {C}^{*}$. From (28), $a_{\zeta _{n}(z)}\in R_{\zeta _{n}(z)}$. Since n is composite and greater than 2, by (30) we have $[ f_{n}(a_{\zeta _{n}(z)}),a_{\zeta _{n}(z)}] \subset R_{\zeta _{n}(z)}$. Noticing $R_{\zeta _{n}(z)}$ $\subset$ $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$, necessarily $f_{n}(a_{\zeta _{n}(z)})=a_{\zeta _{n}(z)}$.       $\square$

In the next result we prove that $a_{\zeta _{n}(z)}$ is not a fixed point of $f_{n}$ for any $n\in \mathscr {P}^{*}$.

Corollary 3.2

For every $n\in \mathscr {P}^{*}$, $f_{n}(a_{\zeta _{n}(z)})<a_{\zeta _{n}(z)}$.

Proof

For every $n>2$, the variety $| \zeta _{n}^{*}(z)|=p_{k_{n}\text { }}^{-c}$, for arbitrary $c\in \mathbb {R}$, by virtue of equation (10) is not contained in a vertical line, so the interval of the variation of the variable x in the variety $|\zeta _{n}^{*}(z)| =p_{k_{n}\text { }}^{-c}$ is not degenerate. Therefore, taking into account (12), we have

$$\begin{aligned} f_{n}(c)<g_{n}(c)\text { for every integer }n>2,\quad \text {for all }c\in \mathbb {R} . \end{aligned}$$

(32)

Assume $n>2$ prime. By Corollary 2.2, $g_{n}(a_{\zeta _{n}(z)})=a_{\zeta _{n}(z)}$. Then, by taking $c=a_{\zeta _{n}(z)}$ in (32), the corollary follows.       $\square$

As a simple consequence from Corollary 3.2 we obtain a characterization of $\mathscr {C}^{*}$.

Theorem 3.3

An integer $n>2$ belongs to $\mathscr {C}^{*}$ if and only if $a_{\zeta _{n}(z)}$ is a fixed point of the function $f_{n}$.

Proof

From Corollary 3.1, if $n>2$ is composite, $a_{\zeta _{n}(z)}$ is a fixed point of $f_{n}$. Reciprocally, if $a_{\zeta _{n}(z)}$ is a fixed point of $f_{n}$, by assuming $n>2$ is not composite, by applying Corollary 3.2 we are led to a contradiction. Therefore, the theorem follows.       $\square$

The bounds $a_{\zeta _{n}(z)}$, $a_{\zeta _{n}^{*}(z)}$ satisfy the following inequality.

Proposition 3.1

For every integer $n>2$, $a_{\zeta _{n}(z)}<a_{\zeta _{n}^{*}(z)}$.

Proof

By taking $c=a_{\zeta _{n}^{*}(z)}$ in Lemma 2.2 we have

$$\begin{aligned} f_{n}(a_{\zeta _{n}(z)}^{*})<a_{\zeta _{n}(z)}^{*}\text { for all }n>2 . \end{aligned}$$

(33)

Again from Lemma 2.2, for $c=a_{\zeta _{n}(z)}$, we get $f_{n}(a_{\zeta _{n}(z)})<a_{\zeta _{n}(z)}^{*}$. If n is composite, by Corollary 3.1 $f_{n}(a_{\zeta _{n}(z)})=a_{\zeta _{n}(z)}$ and from (33) we then deduce that $a_{\zeta _{n}(z)}<a_{\zeta _{n}(z)}^{*}$. This proves the proposition for n composite.

Assume n is prime. Then $p_{k_{n}}=n$ and, from (7), $\zeta _{n}^{*}(z)=\zeta _{n-1}(z)$, so $a_{\zeta _{n}^{*}(z)}=a_{\zeta _{n-1}(z)}$. Now we consider the function $G_{n}(z)$ defined in (19). As we have seen in the proof of Lemma 2.3, because of [6, Proposition 5] we have $b_{G_{n}(z)}\le \beta _{G_{n}(z)}$ for all $n\ge 2$ and the equality is attained for n prime. Noticing [17, Lemma 1], $\beta _{G_{n-1}(z)}<\beta _{G_{n}(z)}$ for all $n>2$. Then we get

$$\begin{aligned} b_{G_{n-1}(z)}\le \beta _{G_{n-1}(z)}<\beta _{G_{n}(z)}=b_{G_{n}(z)},\quad \text {for all prime }n>2, \end{aligned}$$

(34)

or equivalently

$$ -b_{G_{n-1}(z)}\ge -\beta _{G_{n-1}(z)}>-\beta _{G_{n}(z)}=-b_{G_{n}(z)},\quad \text {for all prime }n>2. $$

Now, since from (19) $a_{\zeta _{n}(z)}=-b_{G_{n}(z)}$ for all $n\ge 2$, from the above chain of inequalities we deduce

$$ a_{\zeta _{n}^{*}(z)}=a_{\zeta _{n-1}(z)}=-b_{G_{n-1}(z)}>-b_{G_{n}(z)}=a_{\zeta _{n}(z)},\quad \text {for all prime }n>2. $$

The proof is now completed.       $\square$

Corollary 3.3

For every integer $n>2$, $a_{\zeta _{n}(z)}^{*}\in R_{\zeta _{n}(z)}$.

Proof

For $n=3,4$, because of (8) we have $a_{\zeta _{n}(z)}^{*}=0$. Therefore, from (28), $a_{\zeta _{n}(z)}^{*}\in R_{\zeta _{n}(z)}$ for $n=3$, 4. Assume $n>4$. By Proposition 3.1, $a_{\zeta _{n}(z)}<a_{\zeta _{n}^{*}(z)}$ for all $n>2$. Then, from (8) and (18), $a_{\zeta _{n}(z)}^{*}\in [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$ for all $n>4$. Therefore, by using (33) and applying Theorem 3.1, $a_{\zeta _{n}(z)}^{*}\in R_{\zeta _{n}(z)}$ for all $n>4$. This proves the corollary.       $\square$

In the next result we prove the existence of a minimal density interval for every $\zeta _{n}(z)$, $n>2$.

Theorem 3.4

For every integer $n>2$ there exists a number $A_{n}\in [ a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)})$ such that $[ A_{n},b_{\zeta _{n}(z)}] \subset R_{\zeta _{n}(z)}$.

Proof

Firstly we note that, by Proposition 3.1, the interval $[ a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)})$ is well defined. On the other hand, by (18) $b_{\zeta _{n}(z)}>0$ and, by (8) $a_{\zeta _{n}^{*}(z)}\le 0$ for all $n>2$, so by Proposition 3.1 we have

$$\begin{aligned} a_{\zeta _{n}(z)}<a_{\zeta _{n}^{*}(z)}\le 0<b_{\zeta _{n}(z)},\quad \text {for all }n>2. \end{aligned}$$

(35)

This means that $[ a_{\zeta _{n}^{*}(z)},b_{\zeta _{n}(z)}]$ is a non-degenerate sub-interval of $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$ for any $n>2$. By Lemma 2.2, we have $f_{n}(b_{\zeta _{n}(z)})<a_{\zeta _{n}^{*}(z)}$. Then, according to (35), we get

$$ f_{n}(b_{\zeta _{n}(z)})\le a_{\zeta _{n}^{*}(z)}<b_{\zeta _{n}(z)}, $$

so

$$ [ a_{\zeta _{n}^{*}(z)},b_{\zeta _{n}(z)}] \subset [f_{n}(b_{\zeta _{n}(z)}),b_{\zeta _{n}(z)}] . $$

Now, since $b_{\zeta _{n}(z)}\in R_{\zeta _{n}(z)}$, because of Theorem 3.2 we obtain

$$\begin{aligned}{}[ a_{\zeta _{n}^{*}(z)},b_{\zeta _{n}(z)}] \subset [f_{n}(b_{\zeta _{n}(z)}),b_{\zeta _{n}(z)}] \cap [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \subset R_{\zeta _{n}(z)} . \end{aligned}$$

(36)

This implies that $a_{\zeta _{n}^{*}(z)}\in R_{\zeta _{n}(z)}$ (observe that from Corollary 3.3 we already knew that $a_{\zeta _{n}^{*}(z)}\in R_{\zeta _{n}(z)})$) so, again by Theorem 3.2, we have

$$\begin{aligned}{}[ f_{n}(a_{\zeta _{n}^{*}(z)}),a_{\zeta _{n}^{*}(z)} \cap [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \subset R_{\zeta _{n}(z)}. \end{aligned}$$

(37)

If $f_{n}(a_{\zeta _{n}^{*}(z)})\le a_{\zeta _{n}(z)}$, from (37) we deduce that $[ a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)}] \subset R_{\zeta _{n}(z)}$ and then, by (36) we get $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \subset R_{\zeta _{n}(z)}$. In this case by taking $A_{n}=a_{\zeta _{n}(z)}$, the theorem follows. Moreover, $\zeta _{n}(z)$ has a maximum density interval and it coincides with its critical interval $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$.

If $f_{n}(a_{\zeta _{n}^{*}(z)})>a_{\zeta _{n}(z)}$, from (37) we deduce

$$\begin{aligned}{}[ f_{n}(a_{\zeta _{n}^{*}(z)}),a_{\zeta _{n}^{*}(z)}]\subset R_{\zeta _{n}(z)}. \end{aligned}$$

(38)

Therefore $f_{n}(a_{\zeta _{n}^{*}(z)})$ $\in R_{\zeta _{n}(z)}$ and, again by Theorem 3.2, we have

$$\begin{aligned}{}[ f_{n}^{(2)}(a_{\zeta _{n}^{*}(z)}),f_{n}(a_{\zeta _{n}^{*}(z)})] \cap [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \subset R_{\zeta _{n}(z)}, \end{aligned}$$

(39)

where $f_{n}^{(2)}$ denotes $f_{n}$ composed with itself. Then, if $f_{n}^{(2)}(a_{\zeta _{n}^{*}(z)})\le a_{\zeta _{n}(z)}$, from (39), we have $[ a_{\zeta _{n}(z)},f_{n}(a_{\zeta _{n}^{*}(z)})]\subset R_{\zeta _{n}(z)}$ and by (38), we get $[ a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)}] \subset R_{\zeta _{n}(z)}$. Therefore taking into account (36) we obtain $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \subset R_{\zeta _{n}(z)}$. Consequently, by taking $A_{n}=a_{\zeta _{n}(z)}$, the theorem follows and $\zeta _{n}(z)$ has a maximum density interval that coincides with its critical interval $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$. If $f_{n}^{(2)}(a_{\zeta _{n}^{*}(z)})>a_{\zeta _{n}(z)}$, from (39), we get

$$ [ f_{n}^{(2)}(a_{\zeta _{n}^{*}(z)}),f_{n}(a_{\zeta _{n}^{*}(z)})] \subset R_{\zeta _{n}(z)} . $$

Therefore $f_{n}^{(2)}(a_{\zeta _{n}^{*}(z)})\in R_{\zeta _{n}(z)}$ and, again by Theorem 3.2, we have

$$ [ f_{n}^{(3)}(a_{\zeta _{n}^{*}(z)}),f_{n}^{(2)}(a_{\zeta _{n}^{*}(z)})] \cap [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \subset R_{\zeta _{n}(z)}, $$

and so on. Therefore, by denoting $f_{n}^{(k)}=f_{n}^{(k-1)}\circ f_{n}$ for $k\ge 2$ and repeating the process above, we are led to one of the two cases:

(i) There is some $k\ge 1$ such that $f_{n}^{(k)}(a_{\zeta _{n}^{*}(z)})\le a_{\zeta _{n}(z)}$. In this case, as we have seen $A_{n}=a_{\zeta _{n}(z)}$ and then $\zeta _{n}(z)$ has a maximum density interval that coincides with its critical interval $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$.

(ii) For all k, $f_{n}^{(k)}(a_{\zeta _{n}^{*}(z)})>a_{\zeta _{n}(z)}$ and then, by virtue of Lemma 2.1 and (33), we have

$$ a_{\zeta _{n}(z)}<\cdots<f_{n}^{(k)}(a_{\zeta _{n}^{*}(z)})<\cdots<f_{n}^{(2)}(a_{\zeta _{n}^{*}(z)})<f_{n}(a_{\zeta _{n}^{*}(z)})<a_{\zeta _{n}^{*}(z)}. $$

Consequently there exists $\lim _{k\rightarrow \infty }f_{n}^{(k)}(a_{\zeta _{n}^{*}(z)})$ and then, by defining

$$ A_{n}:=\lim _{k\rightarrow \infty }f_{n}^{(k)}(a_{\zeta _{n}^{*}(z)}), $$

we have $a_{\zeta _{n}(z)}\le A_{n}<a_{\zeta _{n}^{*}(z)}$. On the other hand, by reiterating Theorem 3.2, we get

$$\begin{aligned}{}[ f_{n}^{(k)}(a_{\zeta _{n}^{*}(z)}),f_{n}^{(k-1)}(a_{\zeta _{n}^{*}(z)})] \subset R_{\zeta _{n}(z)}\text {, for all }k\ge 2 . \end{aligned}$$

(40)

Then taking into account (36) and (38), by (40) we deduce that $[A_{n},b_{\zeta _{n}(z)}] \subset R_{\zeta _{n}(z)}$. This definitely proves the theorem.       $\square$

Remark 3.5

Observe that if the case (ii) of above theorem holds, $A_{n}$ will be a fixed point of $f_{n}$ by virtue of the continuity of $f_{n}$. Then if $n\in \mathscr {C}^{*}$, by Theorem 14, the point $A_{n}$ could be $a_{\zeta _{n}(z)}$. But if $n\in \mathscr {P}^{*}$, from Corollary 3.2, $A_{n}$ can not be equal to $a_{\zeta _{n}(z)}$.

In the next result we prove that the number of fixed points of $f_{n}$ influences on the existence of a maximum density interval of $\zeta _{n}(z)$.

Theorem 3.6

For every integer $n>2$, if $f_{n}$ has at most a fixed point in the interval $(a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)})$ then $\zeta _{n}(z)$ has a maximum density interval that coincides with the critical interval $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$ associated with $\zeta _{n}(z)$.

Proof

We first assume $f_{n}$ has no fixed point in $(a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)})$. Then we claim that $f_{n}(c)<c$ for all $c\in (a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)}]$. Indeed, we define the function $h_{n}(c):=f_{n}(c)-c$. Then $h_{n}$ is continuous on $\mathbb {R}$, and by virtue of Lemma 2.2 and (33), $h_{n}$ is negative on $[a_{\zeta _{n}^{*}(z)},\infty )$. Then, since $f_{n}$ by hypothesis has no fixed point on $(a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)})$, $h_{n}(c)$ has no zero on $( a_{\zeta _{n}(z)},\infty )$. Consequently, $h_{n}(c)<0$ for any $c\in ( a_{\zeta _{n}(z)},\infty )$ and in particular we have

$$\begin{aligned} f_{n}(c)<c\text { for all }c\in ( a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)}] . \end{aligned}$$

(41)

Hence the claim follows. On the other hand, by Corollary 3.3 $a_{\zeta _{n}^{*}(z)}\in$ $R_{\zeta _{n}(z)}\subset [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$, so

$$ ( a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)}] \subset [a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] . $$

Consequently, by taking into account (41) and by applying Theorem 3.1 we have

$$ ( a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)}] \subset R_{\zeta _{n}(z)} . $$

Therefore, since from (28) $a_{\zeta _{n}(z)}\in$ $R_{\zeta _{n}(z)}$, we get $[ a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)}] \subset R_{\zeta _{n}(z)}$ and then by (36) it follows that $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \subset R_{\zeta _{n}(z)}$. As always is true that $R_{\zeta _{n}(z)}\subset [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$ we deduce that $R_{\zeta _{n}(z)}=[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$, i.e. $\zeta _{n}(z)$ has a maximum density interval. Then the theorem follows in this case.

We now suppose $f_{n}$ has only one fixed point, say $c_{1}$, in $(a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)})$. Then the function $h_{n}(c):=f_{n}(c)-c$, continuous on $\mathbb {R}$, is non-positive on $[ c_{1},+\infty )$ by virtue of Lemma 2.2. Therefore, in particular, $f_{n}(c)\le c$ for all $c\in [c_{1},a_{\zeta _{n}^{*}(z)}]$. Since $[ c_{1},a_{\zeta _{n}^{*}(z)}] \subset [ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$, by applying the Theorem 3.1 at any $c\in [ c_{1},a_{\zeta _{n}^{*}(z)}]$ we have

$$\begin{aligned}{}[ c_{1},a_{\zeta _{n}^{*}(z)}] \subset R_{\zeta _{n}(z)}. \end{aligned}$$

(42)

Now we claim that $h_{n}$ is negative on $( a_{\zeta _{n}(z)},c_{1})$. Indeed, if we assume that $h_{n}$ is non-negative on $( a_{\zeta _{n}(z)},c_{1})$, since $c_{1}$ is the unique fixed point of $f_{n}$ in $(a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)})$, then $f_{n}(c)>c$ for all $c\in ( a_{\zeta _{n}(z)},c_{1})$. Then, by Theorem 2.1, $c\notin R_{\zeta _{n}(z)}$ for all $c\in$ $(a_{\zeta _{n}(z)},c_{1})$. This means that $\zeta _{n}(z)$ has no zero on the strip $( a_{\zeta _{n}(z)},c_{1}) \times \mathbb {R}$. But, taking into account that $a_{\zeta _{n}(z)}\in R_{\zeta _{n}(z)}$, $a_{\zeta _{n}(z)}$ would be an isolated point of $R_{\zeta _{n}(z)}$ and it contradicts [2, Corollary 3.2]. Therefore the claim follows. Consequently, $f_{n}(c)<c$ for all $c\in ( a_{\zeta _{n}(z)},c_{1})$ and then, by Theorem 3.1, $( a_{\zeta _{n}(z)},c_{1}) \subset R_{\zeta _{n}(z)}$. From the closedness of $R_{\zeta _{n}(z)}$, we have

$$\begin{aligned}{}[ a_{\zeta _{n}(z)},c_{1}] \subset R_{\zeta _{n}(z)}. \end{aligned}$$

(43)

Then, from (43), (42) and (36) we deduce that $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \subset R_{\zeta _{n}(z)}$. Consequently, $\zeta _{n}(z)$ has a maximum density interval and it coincides with its critical interval $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$.       $\square$

As a first application of the usefulness of Theorem 3.6 we prove a result on $\zeta _{3}(z)$ (the same result can be also deduced from others methods as we can see in [13, 15]).

Corollary 3.4

$\zeta _{3}(z)$ has a maximum density interval and it coincides with its critical interval $[ a_{\zeta _{3}(z)},b_{\zeta _{3}(z)}]$.

Proof

The function $f_{3}(c):=a_{3,c}$ is explicitly given by the formula (11). Then it is immediate to check that $f_{3}(c)<c$ for all $c\in \mathbb {R}$. Therefore $f_{3}(c)$ has no fixed point and then, from Theorem 3.6, $\zeta _{3}(z)$ has a maximum density interval and it coincides with $[a_{\zeta _{3}(z)},b_{\zeta _{3}(z)}]$.       $\square$

4 The Fixed Point Theory and the Maximum Density Interval for $\zeta _{n}(z)$

In this section our aim is to give a very useful result (see below Lemma 4.1) based on Kronecker Theorem [8, Theorem 444] that allows us to apply our fixed point theory to prove the existence of a maximum density interval.

Let $\mathscr {P}:=$ $\{ p_{j}:j=1,2,3,\ldots \}$ be the set of prime numbers and $U:=\{ 1,-1\}$. For every map $\delta :\mathscr {P}\rightarrow U$, we define the function $\omega _{\delta }: \mathbb {N}\rightarrow U$ as

$$\begin{aligned} \omega _{\delta }(1):=1,\quad \omega _{\delta }(m):=(\delta (p_{k_{1}}))^{\alpha _{1}}\ldots (\delta (p_{k_{l(m)}}))^{\alpha _{l(m)}},\quad m>1, \end{aligned}$$

(44)

where $(p_{k_{1}})^{\alpha _{1}}\ldots (p_{k_{l(m)}})^{\alpha _{l(m)}}$, with $\alpha _{1}$, ..., $\alpha _{l(m)}\in \mathbb {N}$, is the decomposition of m in prime factors. Let $\varOmega$ be the set of all the $\omega _{\delta }$’s defined in (44). Observe that all functions of $\varOmega$ are completely multiplicative (see for instance [1, p. 138]).

Lemma 4.1

Let $n>2$ a fixed integer, $p_{k_{n}}$ the last prime not exceeding n and $f_{n}$ defined in (12). Given an arbitrary $\omega _{\delta }\in \varOmega$, the inequality

$$\begin{aligned} p_{k_{n}}^{-c}\le | \sum _{ \begin{array}{c} m=1\\ m\ne p_{k_{n}} \end{array}}^{n}\omega _{\delta }(m)m^{-f_{n}(c)}| , \end{aligned}$$

(45)

holds for all $c\in \mathbb {R}$.

Proof

Because of (7), $\zeta _{n}^{*}(z):=$ $\sum _{m=1,m\ne p_{k_{n}}}^{n}m^{-z}$. Therefore, given $c\in \mathbb {R}$ we have

$$ \zeta _{n}^{*}(f_{n}(c)+iy)=\sum _{\begin{array}{c} m=1\\ m\ne p_{k_{n}} \end{array}}^{n}m^{-f_{n}(c)}(\cos (y\log m)-i\sin (y\log m)) . $$

Then taking into account (14),

$$\begin{aligned} p_{k_{n}}^{-c}\le | \sum _{\begin{array}{c} m=1\\ m\ne p_{k_{n}} \end{array}}^{n}m^{-f_{n}(c)}(\cos (y\log m)-i\sin (y\log m))| \text {, for all }y\in \mathbb {R}. \end{aligned}$$

(46)

Given $n>2$, we define $J_{n}:=\{ 1,2,3,\ldots \pi (n)\}$, where $\pi (n)$ denotes the number of prime numbers not exceeding n. As the set $\{ \log p_{j}:j\in J_{n}\}$ is rationally independent, the set $\{ \frac{\log p_{j}}{2\pi }:j\in J_{n}\}$ is also rationally independent. Then by Kronecker Theorem [8, Theorem 444] fixed an arbitrary set of real numbers $\{ \gamma _{j}:j\in J_{n}\}$ and given an integer $N\ge 1$, there exists a real number $y_{N}>N$ and integers $m_{j,N}$, such that

$$\begin{aligned} | y_{N}\frac{\log p_{j}}{2\pi }-m_{j,N}-\gamma _{j}| <\frac{1}{N},\quad \text { for all }j\in J_{n}. \end{aligned}$$

(47)

For each $n>2$, we define the set $\mathscr {P}_{n}:=\{ p_{j}\in \mathscr {P}:p_{j}\le n\}$. Then, given a mapping $\delta :=\mathscr {P}_{n}\rightarrow U$, we consider the set $\{ \gamma _{j}:j\in J_{n}\}$ where $\gamma _{j}=1$ for those j such that $\delta (p_{j})=1$ and $\gamma _{j}=1/2$ for those j such that $\delta (p_{j})=-1$. Then by applying the aforementioned Kronecker Theorem for $N=1,2 \ldots$, we can determine a sequence $(y_{N})_{N}$ satisfying, by virtue of (47), that

$$ \cos (y_{N}\log p_{j})\rightarrow 1,\quad \sin (y_{N}\log p_{j})\rightarrow 0\text { as }N\rightarrow \infty \text {, for }p_{j}\text { with }\delta (p_{j})=1 , $$

and

$$\begin{aligned} \cos (y_{N}\log p_{j})\rightarrow -1,\quad \sin (y_{N}\log p_{j})\rightarrow 0\text { as }N\rightarrow \infty \text {, for }p_{j}\text { with }\delta (p_{j})=-1. \end{aligned}$$

Therefore for each m such that $1\le m\le n$ we get

$$\begin{aligned} \cos (y_{N}\log m)\rightarrow \omega _{\delta }(m),\quad \sin (y_{N}\log m)\rightarrow 0\quad \text {as }N\rightarrow \infty . \end{aligned}$$

(48)

Now, we substitute y by $y_{N}$ in (46) and we take the limit as $N\rightarrow \infty$. Then, according to (48), the inequality (45) follows.       $\square$

Theorem 4.1

For all prime numbers $n>2$ except at most for a finite quantity, $f_{n}$ has no fixed point in the interval $(a_{\zeta _{n}(z)},a_{\zeta _{n}^{*}(z)})$.

Proof

Corollary 3.4 proves the theorem for $n=3$. Assume $n>3$ prime. The numbers $n-2$ and $n-1$ are relatively primes and both cannot be perfect squares, so there exists $\omega _{\delta }\in \varOmega$ such that $\omega _{\delta }(n-2)\omega _{\delta }(n-1)=-1$. Since n is prime, $a_{\zeta _{n}^{*}(z)}=a_{\zeta _{n-1}(z)}$ and $p_{k_{n}}=n$. By supposing the existence of a fixed point $c_{n}\in (a_{\zeta _{n}(z)},a_{\zeta _{n-1}(z)})$ for the function $f_{n}$ for infinitely many prime $n>3$, we are led to the following contradiction:

By (45) we have

$$\begin{aligned} n^{-c_{n}}\le | \pm ((n-1)^{-c_{n}}-(n-2)^{-c_{n}})+\sum _{m\in P_{n-3,\omega _{\delta }}}m^{-c_{n}}-\sum _{m\notin P_{n-3,\omega _{\delta }}}m^{-c_{n}}| , \end{aligned}$$

(49)

where, for a fixed integer $n>2$ and $\omega _{\delta }\in \varOmega$, the set $P_{n,\omega _{\delta }}$ is defined as

$$ P_{n,\omega _{\delta }}:=\{ m:1\le m\le n\text { such that }\omega _{\delta }(m)=1\} . $$

On the other hand, $\lim _{n\rightarrow \infty }\frac{a_{\zeta _{n}(z)}}{n}=-\log 2$ (see [3, Theorem 1] and [17, Theorem 2]). Then noticing that $a_{\zeta _{n}(z)}<c_{n}<a_{\zeta _{n-1}(z)}$, we get

$$ \lim _{\begin{array}{c} n\text { prime} \\ n\rightarrow \infty \end{array}} \frac{c_{n}}{n-1}=-\log 2. $$

Therefore, for each fixed $j\ge 0$, it follows

$$\begin{aligned} \lim _{\begin{array}{c} n\text { prime} \\ n\rightarrow \infty \end{array}} \left( \frac{n-j}{n-1}\right) ^{-c_{n}}=2^{-j+1}. \end{aligned}$$

(50)

Now, dividing by $(n-1)^{-c_{n}}$ the inequality (49), we have

$$\begin{aligned} \left( \frac{n}{n-1}\right) ^{-c_{n}}&\le \Bigg | \pm \Bigg ( 1-\left( \frac{n-2}{n-1}\right) ^{-c_{n}}\Bigg ) \nonumber \\&\qquad +\sum _{m\in P_{n-3,\omega _{\delta }}}\left( \frac{m}{n-1}\right) ^{-c_{n}} -\sum _{m\notin P_{n-3,\omega _{\delta }}}\left( \frac{m}{n-1}\right) ^{-c_{n}}\Bigg | \nonumber \\&\le \Bigg | \pm \Bigg ( 1-\left( \frac{n-2}{n-1}\right) ^{-c_{n}}\Bigg )\Bigg | \\&\qquad + \Bigg | \sum _{m\in P_{n-3,\omega _{\delta }}} \left( \frac{m}{n-1}\right) ^{-c_{n}} -\sum _{m\notin P_{n-3,\omega _{\delta }}}\left( \frac{m}{n-1}\right) ^{-c_{n}} \Bigg |\nonumber \\&\le \left( 1-\left( \frac{n-2}{n-1}\right) ^{-c_{n}}\right) +\sum _{j=3}^{n-1}\left( \frac{n-j}{n-1}\right) ^{-c_{n}}.\nonumber \end{aligned}$$

(51)

According to (50), by taking the limit in (51) for n prime, $n\rightarrow \infty$, it follows that the limit of the left-hand side of (51) is 2 whereas the limit of the right-hand side one is $1/2+\sum _{j=3}^{\infty }2^{-j+1}=1$. This is the contradiction desired. Hence the theorem follows.       $\square$

As a consequence from Theorem 4.1, an important property of the partial sums of order n prime can be deduced.

Theorem 4.2

For all prime numbers $n>2$ except at most for a finite quantity, $\zeta _{n}(z)$ has a maximum density interval and it coincides with its critical interval $[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}]$.

Proof

It is enough to apply Theorems 3.6 and 4.1.       $\square$

5 Numerical Experiences

Simple numerical experiences carried out for some values of n in inequality (45) joint with the application of Theorem 3.6 and Lemma 4.1, allows us to prove the existence of a maximum density interval of $\zeta _{n}(z)$ for all $2\le n\le 8$. Indeed: For $n=2$, we have already seen in the Introduction section that the zeros of $\zeta _{2}(z)$ are all imaginary, so the set $R_{\zeta _{2}(z)}=\{ 0\}$ and then $a_{\zeta _{2}(z)}=b_{G_{2}(z)}=0$ which means that we trivially have

$$\begin{aligned} R_{\zeta _{2}(z)}=[ a_{\zeta _{2}(z)},b_{\zeta _{2}(z)}] . \end{aligned}$$

Therefore $\zeta _{2}(z)$ has a maximum density interval (in this case degenerate).

For $n=3$, Corollary 3.4 proves that

$$\begin{aligned} R_{\zeta _{3}(z)}=[ a_{\zeta _{3}(z)},b_{\zeta _{3}(z)}] \end{aligned}$$

and then $\zeta _{3}(z)$ has a maximum density interval. In this case the end-points $a_{\zeta _{3}(z)}$, $b_{\zeta _{3}(z)}$ can be easily computed, being $a_{\zeta _{3}(z)}=-1$ and $b_{\zeta _{3}(z)}\approx 0.79$. Thus, $R_{\zeta _{3}(z)}\approx [ -1,0.79]$.

For $n=4$, we firstly claim that $f_{4}$ has no fixed point in the interval $(a_{\zeta _{4}(z)},a_{\zeta _{4}^{*}(z)})$. Indeed, by (8), $a_{\zeta _{4}^{*}(z)}=0$ and from (18), $a_{\zeta _{4}(z)}<0$. Therefore we only study the behavior of $f_{4}(c)$ for $c<0$. We recall that from (12) $f_{4}(c)=a_{4,c}$, where $a_{4,c}$ is the left end-point of the interval of variation of the variable x in the Cartesian equation of the variety $| \zeta _{4}^{*}(z)| =p_{k_{4}}^{-c}$. By taking into account formula (10) for $n=4$, the equation of that variety is

$$\begin{aligned} 1+2^{-2x}+4^{-2x}+2\cdot 2^{-x}(1+4^{-x})\cos (y\log 2)+2\cdot 4^{-x}\cos ( y\log 4)) =3^{-2c}. \end{aligned}$$

(52)

By putting $\cos ( y\log 4)) =2\cos ^{2}(y\log 2)-1$ in (52) and solving it for $\cos (y\log 2)$ we have

$$\begin{aligned} \cos (y\log 2)=\frac{-(1+4^{-x})\pm \sqrt{(2\cdot 3^{-c})^{2}-(\sqrt{3}(4^{-x}-1))^{2}}}{4\cdot 2^{-x}}. \end{aligned}$$

Then the variable x must satisfy the inequality $(\sqrt{3}(4^{-x}-1))^{2}\le (2\cdot 3^{-c})^{2}$ which is equivalent to say that

$$\begin{aligned} 4^{-x}\in [ 1-2\cdot 3^{-c-\frac{1}{2}},1+2\cdot 3^{-c-\frac{1}{2}}]. \end{aligned}$$

(53)

Since $1-2\cdot 3^{-c-\frac{1}{2}}<0$ for all $c<0$, by noting that $4^{-x}>0$ for any x, (53) is in turn equivalent to

$$\begin{aligned} -\frac{\log ( 1+2\cdot 3^{-c-\frac{1}{2}}) }{\log 4}\le x. \end{aligned}$$

Hence the minimum value for x is $-\frac{\log ( 1+2\cdot 3^{-c-\frac{1}{2}}) }{\log 4}$, so $a_{4,c}=-\frac{\log ( 1+2\cdot 3^{-c-\frac{1}{2}}) }{\log 4}$ and consequently for $c<0$ the function $f_{4}(c)$ is given by the formula

$$ f_{4}(c)=-\frac{\log ( 1+2\cdot 3^{-c-\frac{1}{2}}) }{\log 4}. $$

Then the fixed points of $f_{4}(c)$ are the solutions of the equation $f_{4}(c)=c$, that is

$$\begin{aligned} 1+2\cdot 3^{-c-1/2}=4^{-c}. \end{aligned}$$

(54)

According to [20, p. 46] Eq. (54) has a unique real solution, say $c_{0}$, whose approached value is $-1.21$. On the other hand, since $n=4$ belongs to $\mathscr {C}^{*}$, by Theorem 3.3 $a_{\zeta _{4}(z)}$ is a fixed point of the function $f_{4}$. Since $c_{0}$ is the unique solution of $f_{4}(c)=c$, necessarily $a_{\zeta _{4}(z)}=c_{0}\approx -1.21$ and then $f_{4}$ has no fixed point in $(a_{\zeta _{4}(z)},a_{\zeta _{4}^{*}(z)})$. Hence the claim follows. Then, by applying Theorem 3.6, $\zeta _{4}(z)$ has a maximum density interval and consequently

$$ R_{\zeta _{4}(z)}=[ a_{\zeta _{4}(z)},b_{\zeta _{4}(z)}] . $$

For $n=5$ we take a mapping $\delta :\mathscr {P}$ $\rightarrow U$ satisfying $\delta (2)=\delta (3)=-1$ and consider its corresponding $\omega _{\delta }:\mathbb {N}\rightarrow U$ defined in (44). Assume $f_{5}$ has some fixed point, say $c_{0}$, in the interval $(a_{\zeta _{5}(z)},a_{\zeta _{5}^{*}(z)})$. By (8) $a_{\zeta _{5}^{*}(z)}<0$ and then $(a_{\zeta _{5}(z)},a_{\zeta _{5}^{*}(z)})$ contains only negative numbers, so $c_{0}<0$. By applying (45) for $n=5$, $f_{5}$ and the above defined $\omega _{\delta }$, under the assumption $f_{5}(c_{0})=c_{0}$, we have

$$ 5^{-c_{0}}\le | 1-2^{-c_{0}}-3^{-c_{0}}+4^{-c_{0}}| . $$

But this inequality is clearly impossible for any $c_{0}<0$. Hence $f_{5}$ has no fixed point in $(a_{\zeta _{5}(z)},a_{\zeta _{5}^{*}(z)})$. Then, by applying Theorem 3.6, $\zeta _{5}(z)$ has a maximum density interval and consequently

$$ R_{\zeta _{5}(z)}=[ a_{\zeta _{5}(z)},b_{\zeta _{5}(z)}] . $$

For $n=6$, we take a mapping $\delta :\mathscr {P}$ $\rightarrow U$ satisfying $\delta (2)=-1$, $\delta (3)=1$ and consider its corresponding $\omega _{\delta }:\mathbb {N}\rightarrow U$ defined in (44). Assume $f_{6}$ has some fixed point, say $c_{0}$, in the interval $(a_{\zeta _{6}(z)},a_{\zeta _{6}^{*}(z)})$. By (8) $a_{\zeta _{6}^{*}(z)}<0$ and then $(a_{\zeta _{6}(z)},a_{\zeta _{6}^{*}(z)})$ contains only negative numbers, so $c_{0}<0$. By applying (45) for $n=6$, $f_{6}$ and the above defined $\omega _{\delta }$, under the assumption $f_{6}(c_{0})=c_{0}$, we have

$$\begin{aligned} 5^{-c_{0}}\le | 1-2^{-c_{0}}+3^{-c_{0}}+4^{-c_{0}}-6^{-c_{0}}| . \end{aligned}$$

(55)

Regarding inequality (55) we consider the two possible cases: (a) $1-2^{-c_{0}}+3^{-c_{0}}+4^{-c_{0}}-6^{-c_{0}}\ge 0$, (b) $1-2^{-c_{0}}+3^{-c_{0}}+4^{-c_{0}}-6^{-c_{0}}<0$. In (a), according to (55), we have the inequality

$$ 1+3^{-c_{0}}+4^{-c_{0}}\ge 2^{-c_{0}}+5^{-c_{0}}+6^{-c_{0}}, $$

that as we easily can check is not possible for any $c_{0}<0$. In (b), because of (55), we get

$$\begin{aligned} 1+3^{-c_{0}}+4^{-c_{0}}+5^{-c_{0}}\le 2^{-c_{0}}+6^{-c_{0}}. \end{aligned}$$

(56)

By a direct computation we see that (56) is only true for $c_{0}\le a_{\zeta _{6}(z)}\approx -2.8$ (observe that for $c_{0}\approx -2.8$, inequality (56) becomes an equality and since $n=6$ belongs to $C^{*}$, by Theorem 3.3, $a_{\zeta _{6}(z)}$ is a fixed point of the function $f_{6}$). Therefore for $c_{0}>a_{\zeta _{6}(z)}$, (56) is not possible. Hence $f_{6}$ has no fixed point in $(a_{\zeta _{6}(z)},a_{\zeta _{6}^{*}(z)})$. Then, by applying Theorem 3.6, $\zeta _{6}(z)$ has a maximum density interval and consequently

$$ R_{\zeta _{6}(z)}=[ a_{\zeta _{6}(z)},b_{\zeta _{6}(z)}] . $$

For $n=7$, we take a mapping $\delta :\mathscr {P}$ $\rightarrow U$ satisfying $\delta (2)=$ $\delta (3)=\delta (5)=-1$ and consider its corresponding $\omega _{\delta }:\mathbb {N} \rightarrow U$ defined in (44). Assume $f_{7}$ has some fixed point, say $c_{0}$, in the interval $(a_{\zeta _{7}(z)},a_{\zeta _{7}^{*}(z)})$. By (8) $a_{\zeta _{7}^{*}(z)}<0$ and then $(a_{\zeta _{7}(z)},a_{\zeta _{7}^{*}(z)})$ contains only negative numbers, so $c_{0}<0$. By applying (45) for $n=7$, $f_{7}$ and the above defined $\omega _{\delta }$, under the assumption $f_{7}(c_{0})=c_{0}$, we have

$$\begin{aligned} 7^{-c_{0}}\le |1-2^{-c_{0}}-3^{-c_{0}}+4^{-c_{0}}-5^{-c_{0}}+6^{-c_{0}}| . \end{aligned}$$

(57)

We consider the two possible cases: (a) $1-2^{-c_{0}}-3^{-c_{0}}+4^{-c_{0}}-5^{-c_{0}}+6^{-c_{0}}\ge 0$, (b) $1-2^{-c_{0}}-3^{-c_{0}}+4^{-c_{0}}-5^{-c_{0}}+6^{-c_{0}}<0$. In (a), according to (57), we have the inequality

$$ 1+4^{-c_{0}}+6^{-c_{0}}\ge 2^{-c_{0}}+3^{-c_{0}}+5^{-c_{0}}+7^{-c_{0}}, $$

that is clearly impossible for any $c_{0}<0$. In (b), because of (57), we get

$$\begin{aligned} 1+4^{-c_{0}}+6^{-c_{0}}+7^{-c_{0}}\le 2^{-c_{0}}+3^{-c_{0}}+5^{-c_{0}}. \end{aligned}$$

(58)

It is immediate to check that inequality (58) is false for any $c_{0}<0$. Hence $f_{7}$ has no fixed point in $(a_{\zeta _{7}(z)},a_{\zeta _{7}^{*}(z)})$. Then, by applying Theorem 3.6, $\zeta _{7}(z)$ has a maximum density interval and consequently

$$ R_{\zeta _{7}(z)}=[ a_{\zeta _{7}(z)},b_{\zeta _{7}(z)}] . $$

For $n=8$, we take a mapping $\delta :\mathscr {P}$ $\rightarrow U$ satisfying $\delta (2)=1$, $\delta (3)=\delta (5)=-1$ and consider its corresponding $\omega _{\delta }:=\mathbb {N}\rightarrow U$ defined in (44). Assume $f_{8}$ has some fixed point, say $c_{0}$, in the interval $(a_{\zeta _{8}(z)},a_{\zeta _{8}^{*}(z)})$. By (8) $a_{\zeta _{8}^{*}(z)}<0$ and then $(a_{\zeta _{8}(z)},a_{\zeta _{8}^{*}(z)})$ contains only negative numbers, so $c_{0}<0$. By applying (45) for $n=8$, $f_{8}$ and the above defined $\omega _{\delta }$, under the assumption $f_{8}(c_{0})=c_{0}$, we have

$$\begin{aligned} 7^{-c_{0}}\le |1+2^{-c_{0}}-3^{-c_{0}}+4^{-c_{0}}-5^{-c_{0}}-6^{-c_{0}}+8^{-c_{0}}|. \end{aligned}$$

(59)

Regarding inequality (59) we consider the two possible cases: (a) $1+2^{-c_{0}}-3^{-c_{0}}+4^{-c_{0}}-5^{-c_{0}}-6^{-c_{0}}+8^{-c_{0}}<0$, (b) $1+2^{-c_{0}}-3^{-c_{0}}+4^{-c_{0}}-5^{-c_{0}}-6^{-c_{0}}+8^{-c_{0}}\ge 0$. In case (a), according to (59), we have the inequality

$$ 3^{-c_{0}}+5^{-c_{0}}+6^{-c_{0}}\ge 1+2^{-c_{0}}+4^{-c_{0}}+7^{-c_{0}}+8^{-c_{0}}, $$

which is clearly impossible for any $c_{0}<0$. In case (b), because of (59), we get

$$\begin{aligned} 1+2^{-c_{0}}+4^{-c_{0}}+8^{-c_{0}}\ge 3^{-c_{0}}+5^{-c_{0}}+6^{-c_{0}}+7^{-c_{0}}. \end{aligned}$$

(60)

By an elementary analysis we can see that (60) is only true for $c_{0}\le a_{\zeta _{8}(z)}\approx -4.1$ (observe that for $c_{0}\approx -4.1$ inequality (60) becomes an equality and since $n=8$ belongs to $C^{*}$, by Theorem 3.3, $a_{\zeta _{8}(z)}\approx -4.1$ is a fixed point of the function $f_{8}$). Therefore for $c_{0}\in ( a_{\zeta _{8}(z)},0)$, (60) is not possible. Then, since by (8) $a_{\zeta _{8}^{*}(z)}<0$, in particular (60) is not possible in $(a_{\zeta _{8}(z)},a_{\zeta _{8}^{*}(z)})$. Hence $f_{8}$ has no fixed point in the interval $(a_{\zeta _{8}(z)},a_{\zeta _{8}^{*}(z)})$. Then, by applying Theorem 3.6, $\zeta _{8}(z)$ has a maximum density interval and consequently

$$ R_{\zeta _{8}(z)}=[ a_{\zeta _{8}(z)},b_{\zeta _{8}(z)}] . $$