Some remarks on the convergence of the Dirichlet series of L-functions and related questions

Abstract

First we show that the abscissae of uniform and absolute convergence of Dirichlet series coincide in the case of L-functions from the Selberg class ${\mathcal S}$. We also study the latter abscissa inside the extended Selberg class, indicating a different behavior in the two classes. Next we address two questions about majorants of functions in ${\mathcal S}$, showing links with the distribution of the zeros and with independence results.

1 Introduction

Let

$$\begin{aligned} F(s)=\sum _{n=1}^\infty \frac{a(n)}{n^s} \end{aligned}$$

be a Dirichlet series which converges somewhere in the complex plane. It is well known that there are four classical abscissae associated with F(s): the abscissa of convergence $\sigma _c(F)$, of uniform convergence $\sigma _u(F)$, of absolute convergence $\sigma _a(F)$ and of boundedness $\sigma _b(F)$. It may well be, in general, that $\sigma _c(F)=-\infty$, in which case the other three abscissae equal $-\infty$ as well. From the theory of Dirichlet series we know that

$$\begin{aligned} \sigma _c(F)\le \sigma _b(F)= \sigma _u(F)\le \sigma _a(F), \end{aligned}$$

and in general this is best possible, i.e. inequalities cannot be replaced by equalities. We refer to Maurizi and Queffélec [15] for a modern reference for this sort of problems.

Our first result is that $\sigma _b(F)=\sigma _a(F)$ for an important class of Dirichlet series, namely those defining the L-functions of the Selberg class ${\mathcal S}$. We recall that the axiomatic class ${\mathcal S}$ contains, at least conjecturally, most L-functions from number theory and automorphic forms theory, and that $\sigma _b(F)=\sigma _a(F)$ is known in some special cases like the Riemann or the Dedekind zeta functions. The Selberg class ${\mathcal S}$ is defined, roughly, as the class of Dirichlet series absolutely convergent for $\sigma >1$, having analytic continuation to $\mathbb C$ with at most a pole at $s=1$, satisfying a functional equation of Riemann type and having an Euler product representation. Moreover, their coefficients satisfy the Ramanujan condition $a(n)\ll n^\varepsilon$ for any $\varepsilon >0$. We also recall that the extended Selberg class ${\mathcal S}^\sharp$ is the larger class obtained by dropping the Euler product and Ramanujan condition requirements in the definition of ${\mathcal S}$. We refer to our survey papers [7, 9, 17–19] and to the forthcoming book [12] for definitions, examples and the basic theory of the Selberg classes ${\mathcal S}$ and ${\mathcal S}^\sharp$. In particular, we refer to these surveys for the definition of degree $d_F$, conductor $q_F$ and standard twist of F(s).

Theorem 1

Suppose that F(s) belongs to the Selberg class. Then

$$\begin{aligned} \sigma _b(F)= \sigma _u(F) = \sigma _a(F). \end{aligned}$$

(1)

Several months after submitting this result, the note by Brevig and Heap [3] appeared, where the authors prove the same theorem in the much more general framework of Dirichlet series with multiplicative coefficients. Trying to understand Brevig-Heap’s proof, based on Bohr’s theory, we noticed that their result was already known to Bohr himself in 1913 (see [1], Satz XI, p. 480); incidentally, Bohr’s paper [1] appears as item [5] of the reference list in Brevig-Heap [3]. We wish to thank Dr. Mattia Righetti for bringing [1, 3] to our attention and for his advice concerning these papers. We decided to keep Theorem 1 since our proof is different, easier and more direct; moreover, some points in the proof will be useful for the other results in the paper.

We expect that actually $\sigma _a(F)=1$ for all $F\in {\mathcal S}$. This is known for most classical L-functions and, in the general case of the class ${\mathcal S}$, under the assumption of the Selberg orthonormality conjecture; however, an unconditional proof is missing at present. See again the above quoted references for definitions and results about such a conjecture.

Note that the abscissa of convergence $\sigma _c(F)$ can be smaller than 1 for functions in ${\mathcal S}$. For example, the Dirichlet L-functions $L(s,\chi )$ with a primitive non-principal character $\chi$ are convergent in the half-plane $\sigma >0$. Actually, several general results are known about the abscissa $\sigma _c(F)$ for functions F(s) in the extended Selberg class ${\mathcal S}^\sharp$. First of all

$$\begin{aligned} {{ if} F\in {\mathcal S}^\sharp \,\, { is} \,\, { entire}\,\, { with} \,\, { degree} \,\, d\ge 1, { then} \ \frac{1}{2} - \frac{1}{2d} \le \sigma _c(F) \le 1-\frac{2}{d+1}} \end{aligned}$$

(2)

(recall that there exist no functions $F\in {\mathcal S}^\sharp$ with degree $0<d<1$, see [8] and Conrey and Ghosh [5]). Indeed, the first inequality in (2) is Corollary 3 in [11] and is based on the properties of the standard twist, while the second inequality follows from a well known theorem of Landau [13]. Moreover, in accordance with classical degree 2 conjectures and with the general $\Omega$-theorem in Corollary 2 of [11], we expect that equality holds in the left inequality in (2). Further

$$\begin{aligned} \sigma _c(F)=-\infty \quad \,{ if} \,\,{ and} \,\,{ only} \,\,{ if} \quad \,d_F=0, \end{aligned}$$

since the degree 0 functions of ${\mathcal S}^\sharp$ are Dirichlet polynomials (see [8]). From (2) we also deduce that

$$\begin{aligned} \sigma _c(F)=1\,\, \textit{if}\, \textit{and} \,\textit{only} \,\textit{if} \, F(s) \, \textit{has} \,\,a \,\,\textit{pole} \,\,\textit{at} \,s=1. \end{aligned}$$

We also remark that if the Lindelöf Hypothesis holds for $F\in {\mathcal S}^\sharp$, then $\sigma _c(F)\le 1/2$.

The behavior of $\sigma _a(F)$ in the extended class ${\mathcal S}^\sharp$ is different from the expected behavior in ${\mathcal S}$. Indeed, in the next section, which is also of independent interest, we show that

$$\begin{aligned} \textit{there} \, \textit{exist} \,\textit{functions} \,F\in {\mathcal S}^\sharp \quad \textit{with}\, \sigma _a(F) \, \textit{arbitrarily}\, \textit{close} \,\textit{to} \, 1/2. \end{aligned}$$

We conclude this section with the following

Question

Does (1) hold for the functions in the extended Selberg class ? $\square$

A variant of the question is: does (1) hold for linear combinations

$$\begin{aligned} F(s) = \sum _{j=1}^N c_jF_j(s) \end{aligned}$$

with $F_j\in {\mathcal S}$ and $c_j\in \mathbb C$? If needed, one may assume that F(s) belongs to ${\mathcal S}^\sharp$.

Since $\sigma _a(F)=1$ for most classical L-functions F(s), Theorem 1 prevents the possibility of getting information on the non-trivial zeros exploiting the properties of the abscissa of uniform convergence. On the other hand, if $F\in {\mathcal S}$ is bounded for $\sigma >1-\delta$ for some $\delta >0$, then its Dirichlet series is absolutely convergent for $\sigma >1-\delta$ and hence $F(s)\ne 0$ by Euler’s identity. In the next theorems we replace boundedness by more general majorants and deduce some consequences.

Let $F\in {\mathcal S}$ be of degree d, $N_F(\sigma ,T)$ be the number of zeros $\rho =\beta +i\gamma$ with $\beta >\sigma$ and $|\gamma |\le T$, and denote the density abscissa $\sigma _D(F)$ by

$$\begin{aligned} \sigma _D(F) = \inf \{\sigma : N_F(\sigma ,T)=o(T)\}. \end{aligned}$$

An inspection of the proof of Lemma 3 in [10], obtained by a rudimentary version of Montgomery’s zero-detecting method, shows that

$$\begin{aligned} N_F(\sigma ,T) \ll T^{4(d+3)(1-\sigma )+\varepsilon }. \end{aligned}$$

Hence in general

$$\begin{aligned} \frac{1}{2} \le \sigma _D(F) \le 1-\frac{1}{4(d+3)}, \end{aligned}$$

although it is well known that the classical L-functions F(s) of degree 1 and 2 have $\sigma _D(F)=1/2$, see e.g. Luo [14]. Actually, one can prove that $\sigma _D(F)=1/2$ for all $F\in {\mathcal S}$ with degree $0<d\le 2$. Further, let f(s) be holomorphic in $\sigma >1-\delta$ for some $\delta >0$ and almost periodic on the line $\sigma =A$ for some $A>1$. We say that f(s) is a $\delta$-almost periodic majorant of F(s) if

$$\begin{aligned} |F(s)|\le c(\sigma )|f(s)| \end{aligned}$$

(3)

in the half-plane $\sigma >1-\delta$, where $c(\sigma )>0$ is a continuous function for $\sigma >1-\delta$.

Theorem 2

Let $F\in {\mathcal S}$ and f(s) be a $\delta$-almost periodic majorant of F(s). Then F(s) and f(s) have the same zeros, with the same multiplicity, in the half-plane $\sigma >\max (1-\delta ,\sigma _D(F))$.

Remark

Clearly, in view of (3) each zero of f(s) is also a zero of F(s); the non-trivial part of Theorem 2 says that the opposite assertion holds true as well. Note that we do not require that f(s) is almost periodic for $\sigma >1-\delta$, but only on some vertical line far on the right. We already noticed that, as a consequence of Theorem 1, $F(s)\ne 0$ in every right half-plane where it is bounded. An immediate consequence of Theorem 2 is that $F(s)\ne 0$ for $\sigma >\max (1-\delta ,\sigma _D(F))$ if f(s) is a non-vanishing $\delta$-almost periodic majorant. In particular, from the density estimates reported above when $d\le 2$, if $\delta =1/2$ then the Riemann Hypothesis holds for such F(s). $\square$

Our final result is a kind of new independence statement for L-functions from the Selberg class. Several forms of independence are known in ${\mathcal S}$, such as the linear independence, the multiplicity one property and the orthogonality conjecture and some of its consequences; see our above quoted surveys on the Selberg class. The new independence result is expressed in terms of majorants as follows.

Theorem 3

Let $F,G\in {\mathcal S}$ be such that $F(s)\ll |G(s)|$ for $\sigma >1/2$. Then $F(s)=G(s)$.

The special nature of the majorant is very important here. Indeed, suppose that G(s) is entire; then Theorem 2 gives only that F(s) and G(s) have the same zeros for $\sigma >\sigma _D(F)$. Instead, exploiting the information that $G\in {\mathcal S}$, Theorem 3 shows that actually $F(s)=G(s)$. In other words, no function from ${\mathcal S}$ can dominate in $\sigma >1/2$ another function from ${\mathcal S}$. We may regard this as a weak form of a well known result obtained, under stronger assumptions, by Selberg [20] and Bombieri and Hejhal [2] about the statistical independence of the values of L-functions.

2 The lift operator

Let $Q>0$, ${\varvec{\lambda }}=(\lambda _1,\ldots ,\lambda _r)$ with $\lambda _j>0$, ${\varvec{\mu }}=(\mu _1,\ldots ,\mu _r)$ with $\mu _j\in \mathbb C$ and $|\omega |=1$. We denote by $W(Q,{\varvec{\lambda }},{\varvec{\mu }},\omega )$ the $\mathbb R$-linear space of the Dirichlet series F(s) solutions of the functional equation

$$\begin{aligned} Q^s \prod _{j=1}^r\Gamma (\lambda _js+\mu _j)F(s) = \omega Q^{1-s} \prod _{j=1}^r\Gamma (\lambda _j(1-s)+\overline{\mu _j})\overline{F(1-\overline{s})}. \end{aligned}$$

(4)

Given an integer $k\ge 1$, we define the k-lift operator by

$$\begin{aligned} F(s)\longmapsto F_k(s)=F\left( ks+\frac{1-k}{2}\right) ; \end{aligned}$$

clearly, the operator is trivial for $k=1$. A simple computation shows that

$$\begin{aligned} if F\in W(Q,{\varvec{\lambda }},{\varvec{\mu }},\omega ) \quad then \quad F_k\in W\left( Q^k,k{\varvec{\lambda }},{\varvec{\mu }}+ \frac{1-k}{2}\mathbf {\varvec{\lambda }},\omega \right) . \end{aligned}$$

(5)

In particular, from (5) we have that degree $d_{F_k}$ and conductor $q_{F_k}$ of $F_k(s)$ satisfy

$$\begin{aligned} d_{F_k} = kd_F \quad q_{F_k} = q_F^k k^{kd_F}. \end{aligned}$$

(6)

We recall (see the above references) that the class ${\mathcal S}^\sharp$ consists of the Dirichlet series satisfying a functional equation of type (4), where now $\mathfrak {R}{\mu _j}\ge 0$, with the following properties: F(s) is absolutely convergent for $\sigma >1$ and $(s-1)^mF(s)$ is entire of finite order for some integer $m\ge 0$. Therefore we consider

$$\begin{aligned} B_F= 2\min _{1\le j\le r} \frac{\mathfrak {R}{\mu _j}}{\lambda _j}+1, \end{aligned}$$

which is an invariant of ${\mathcal S}^\sharp$ (see again the above references) since a simple computation shows that

$$\begin{aligned} B_F=-2\max _{\rho } \mathfrak {R}{\rho } + 1, \end{aligned}$$

where $\rho$ runs over the trivial zeros of F(s). From the definition of the k-lift operator and (5) we see that, given $F\in {\mathcal S}^\sharp$, the lifted function $F_k(s)$ also belongs to ${\mathcal S}^\sharp$ provided $1\le k\le B_F$ and, if $B_F\ge 2$, F(s) is entire. Indeed, if $k\ge 2$, F(s) has to be holomorphic at $s=1$ otherwise the pole of $F_k(s)$ is not at $s=1$, and the bound $k\le B_F$ is needed to have non-negative real part of the $\mu$’s data of $F_k(s)$. Therefore, defining $V(Q,{\varvec{\lambda }},{\varvec{\mu }},\omega )$ to be the $\mathbb R$-linear space of the entire functions $F\in {\mathcal S}^\sharp$ satisfying (4) (again with $\mathfrak {R}{\mu _j}\ge 0$), we have that

$$\begin{aligned} for\,\, 1\le & {} k\le B_F, the\,\,k-lift \,\,operator \,\,maps\,\, V(Q,{\varvec{\lambda }},{\varvec{\mu }},\omega )\,\, into \,\, \\&\quad V\left( Q^k,k{\varvec{\lambda }},{\varvec{\mu }}+\frac{1-k}{2}\mathbf {\varvec{\lambda }},\omega \right) . \end{aligned}$$

Note that $B_F$ depends only on ${\varvec{\lambda }}$ and ${\varvec{\mu }}$, so it is the same for all functions in $V(Q,{\varvec{\lambda }},{\varvec{\mu }},\omega )$. Note also that the Selberg class ${\mathcal S}$ is not preserved under the above mappings since the Ramanujan condition is not (necessarily) satisfied by $F_k(s)$ even if F(s) does; see the examples below. Further, a simple computation shows that the k-lift operator commutes with the map sending F(s) to its standard twist. We also remark that the requirement $\mathfrak {R}{\mu _j}\ge 0$ in the definition of ${\mathcal S}^\sharp$, which is responsible for the limitation $k\le B_F$ in (6), is apparently not of primary importance in the theory of the Selberg class. Hence, although formally not belonging to ${\mathcal S}^\sharp$, the lifts $F_k(s)$ of entire $F\in {\mathcal S}^\sharp$ with $k>B_F$ are further examples of Dirichlet series with continuation over $\mathbb C$ and functional equation. A similar remark applies to the other condition in the definition of $V(Q,{\varvec{\lambda }},{\varvec{\mu }},\omega )$, namely the holomorphy at $s=1$.

Example

The Riemann zeta function $\zeta (s)$ cannot be lifted inside ${\mathcal S}^\sharp$ since it has $B_\zeta =1$. The same holds for the Dirichlet L-functions with even primitive characters, while those associated with odd primitive characters may be lifted inside ${\mathcal S}^\sharp$ for $k=2$ and $k=3$. However, after lifting their Dirichlet coefficients do not satisfy the Ramanujan condition, hence the lifted Dirichlet L-functions do not belong to ${\mathcal S}$. Note that, once suitably normalized, the lifts with $k=2$ become the L-functions associated with half-integral weight modular forms; see the books by Hecke [6] and Ogg [16]. Concerning degree 2, we consider the L-functions associated with holomorphic eigenforms of level N and integral weight K; see Ogg [16]. Denoting by F(s) their normalization satisfying a functional equation reflecting $s\mapsto 1-s$ (instead of the original $s\mapsto K-s$), we have that

$$\begin{aligned} B_F=K. \end{aligned}$$

Hence the normalized L-functions of eigenforms of weight K may be lifted inside ${\mathcal S}^\sharp$ with k up to their weight. Here we consider only eigenforms since in general the L-functions of modular forms of level N satisfy a slightly different functional equation, not of ${\mathcal S}^\sharp$ type.

$\square$

We finally turn to the problem of the absolute convergence abscissa in ${\mathcal S}^\sharp$. Let $F\in {\mathcal S}^\sharp$ be of degree $d\ge 1$. Then, thanks again to the properties of the standard twist, we know that

$$\begin{aligned} \sigma _a(F) \ge \frac{1}{2} + \frac{1}{2d}; \end{aligned}$$

(7)

this folows from Theorem 1 of [11]. On the other hand, if $F\in {\mathcal S}^\sharp$ we have that the series

$$\begin{aligned} \sum _{n=1}^\infty \frac{|a(n)|}{n^{k\sigma +(1-k)/2}} \end{aligned}$$

converges for $\sigma >1/2 + 1/(2k)$. Hence from (6) and (7) we obtain that if both F(s) of degree $d\ge 1$ and $F_k(s)$ belong to ${\mathcal S}^\sharp$, then

$$\begin{aligned} \frac{1}{2} + \frac{1}{2kd} \le \sigma _a(F_k) \le \frac{1}{2} +\frac{1}{2k}. \end{aligned}$$

(8)

Since the above examples show that there exist functions $F\in {\mathcal S}^\sharp$ with arbitrarily large $B_F$ (e.g. the holomorphic eigenforms with arbitrarily large weight K), (8) shows that $\sigma _a(F)$ can be arbitrarily close to 1 / 2 inside ${\mathcal S}^\sharp$. Hence the behavior of $\sigma _a(F)$ in the extended class ${\mathcal S}^\sharp$ is definitely different from its expected behavior in the class ${\mathcal S}$.

3 Proof of Theorem 1

Observe that the case $d=0$ is trivial, since F(s) is identically 1; see Conrey and Ghosh [5]. For d positive we have $\sigma _b(F)\ge 1/2$, since F(s) is unbounded for $\sigma <1/2$ by the functional equation and the properties of the $\Gamma$ function. Therefore, to prove the assertion it suffices to show the following fact: if for a certain $1/2<\sigma _0\le 1$ the function F(s) is bounded for $\sigma >\sigma _0$, then $\sigma _a(F)\le \sigma _0$.

Let us fix an $\varepsilon \in (0,\sigma _0-1/2)$, and let $c_0=c_0(\varepsilon )$ be such that $|a(n)|\le c_0n^{\varepsilon \slash 2}$ for all $n\ge 1$. Without loss of generality we may assume that $c_0\ge 3$. Consider the finite set of primes

$$\begin{aligned} S_{\varepsilon }=\{p: |a(p)|>p^{\varepsilon \slash 2} \ \ \text {or}\ \ p< c_0^{2\slash \varepsilon }\}. \end{aligned}$$

Let

$$\begin{aligned} F_p(s) = \sum _{m=0}^\infty \frac{a(p^m)}{p^{ms}} \end{aligned}$$

(9)

denote the pth Euler factor of F(s). We split the Euler product as

$$\begin{aligned} \begin{aligned} F(s)&=\prod _{p\not \in S_{\varepsilon }}\left( 1+\frac{a_F(p)}{p^s}\right) \prod _{p\in S_{\varepsilon }}F_p(s) \prod _{p\not \in S_{\varepsilon }}\left( F_p(s)\left( 1+\frac{a_F(p)}{p^s}\right) ^{-1}\right) \\&=P_1(s)P_2(s)P_3(s), \end{aligned} \end{aligned}$$

(10)

say. Both $P_2(s)$ and its inverse $1\slash P_2(s)$ have Dirichlet series representations which converge absolutely for $\sigma >\theta$ for some $\theta <1\slash 2$. This is a simple consequence of the definition of the Selberg class; see the above quoted references. Therefore, $P_2(s)$ and $1\slash P_2(s)$ are bounded for $\sigma >\sigma _0$.

In view of (9) we have

$$\begin{aligned} P_3(s)=\prod _{p\not \in S_{\varepsilon }}\left( 1+\sum _{m=2}^{\infty }\frac{b(p^m)}{p^{ms}}\right) \end{aligned}$$

with

$$\begin{aligned} b(p^m)=\sum _{l=0}^m(-1)^la(p)^la(p^{m-l}). \end{aligned}$$

Hence, recalling that $p\not \in S_{\varepsilon }$, $m\ge 2$ and $c_0\ge 3$, we have

$$\begin{aligned} |b(p^m)|\le \sum _{l=0}^m|a(p)|^l|a(p^{m-l})|\le c_0mp^{m\varepsilon \slash 2}\le p^{m\varepsilon }. \end{aligned}$$

Thus for $\sigma >1/2+\varepsilon$ and $p\not \in S_{\varepsilon }$ we have

$$\begin{aligned} \sum _{m=2}^{\infty }\frac{|b(p^m)|}{p^{m\sigma }} <1 \quad \text {and} \quad \sum _{p\not \in S_{\varepsilon }} \sum _{m=2}^{\infty }\frac{|b(p^m)|}{p^{m\sigma }}\ll 1. \end{aligned}$$

Hence both $P_3(s)$ and $1\slash P_3(s)$ are bounded and have Dirichlet series representations which converge absolutely for $\sigma >\sigma _0$ (recall that $\sigma _0>1/2+\varepsilon$).

We therefore see that $P_1(s)=F(s)\slash (P_2(s)P_3(s))$ is bounded for $\sigma >\sigma _0$. Let us write

$$\begin{aligned} P_1(s)=\sum _{n=1}^{\infty }\frac{c(n)}{n^s}. \end{aligned}$$

The coefficients c(n) are completely multiplicative, and the series converges for $\sigma >\sigma _0$. Fix such a $\sigma$, and a positive $\delta <\sigma -\sigma _0$. Consider the following familiar Mellin’s transform

$$\begin{aligned} \sum _{n=1}^{\infty }\frac{c(n)}{n^{\sigma +it}}e^{-n\slash Y} = \frac{1}{2\pi i}\int _{1-i\infty }^{1+i\infty } \frac{F(w+\sigma +it)}{P_2(w+\sigma +it)P_3(w+\sigma +it)} \Gamma (w) Y^w \text {d}w. \end{aligned}$$

We shift the line of integration to $\mathfrak {R}(w)=-\delta$ and obtain

$$\begin{aligned} \sum _{n=1}^{\infty }\frac{c(n)}{n^{\sigma +it}}e^{-n\slash Y} = \frac{F(\sigma +it)}{P_2(\sigma +it)P_3(\sigma +it)} + O(Y^{-\delta })\ll 1 \end{aligned}$$

uniformly in $t\in {\mathbb R}$ and $Y\ge 1$. Since $|c(n)|\le n^{\varepsilon /2}$, due to the decay of the exponential we may cut the sum on the left hand side to $n\le 3Y\log Y$, say, producing an extra error term of size $O(1\slash Y)$. Thus

$$\begin{aligned} \sum _{n\le 3Y\log Y}\frac{c(n)}{n^{\sigma +it}}e^{-n\slash Y} \ll 1 \end{aligned}$$

(11)

uniformly in $t\in \mathbb R$ and $Y\ge 1$.

Now we apply Kronecker’s theorem in the following form, see Theorem 8 of Ch. VIII of Chandrasekharan [4]. If $\theta _1,\ldots ,\theta _k\in \mathbb R$ are linearly independent over $\mathbb Z$, $\beta _1,\ldots ,\beta _k\in \mathbb R$ and $T,\eta >0$, then there exist $t>T$ and $n_1,\ldots ,n_k\in \mathbb Z$ such that

$$\begin{aligned} |t\theta _\ell - n_\ell -\beta _\ell |<\eta \qquad \ell =1,\ldots ,k. \end{aligned}$$

(12)

We choose the $\theta$’s as $-\frac{1}{2\pi }\log p$ with the primes $p\le 3Y\log Y$ not in $S_{\varepsilon }$ and, correspondingly, the $\beta$’s such that $|c(p)|=c(p)e^{2\pi i\beta _p}$ for each such p. Hence by (12) there exists a sequence of real numbers $t_\nu \rightarrow +\infty$ such that

$$\begin{aligned} c(p)p^{-it_\nu } \rightarrow |c(p)| \qquad \nu \rightarrow \infty \end{aligned}$$

uniformly for the primes $p\le 3Y\log Y$ not in $S_\varepsilon$. By the complete multiplicativity of c(n) we infer that

$$\begin{aligned} c(n)n^{-it_{\nu }} \rightarrow |c(n)| \quad \quad \nu \rightarrow \infty \end{aligned}$$

uniformly for $n\le 3Y\log Y$. Thus putting $t=t_{\nu }$ in (11) and making $\nu \rightarrow \infty$ we obtain

$$\begin{aligned} \sum _{n\le Y}\frac{|c(n)|}{n^{\sigma }}\le e\sum _{n\le 3Y\log Y}\frac{|c(n)|}{n^{\sigma }}e^{-n\slash Y} = e\lim _{\nu \rightarrow \infty } \sum _{n\le 3Y\log Y}\frac{c(n)}{n^{\sigma +it_{\nu }}}e^{-n\slash Y} \ll 1 \end{aligned}$$

uniformly for $Y\ge 1$. Letting $Y\rightarrow \infty$, we see that the Dirichlet series of $P_1(s)$ converges absolutely for $\sigma >\sigma _0$.

Summarizing, we have shown that the Dirichlet series of $P_1(s)$, $P_2(s)$ and $P_3(s)$ are absolutely convergent for $\sigma >\sigma _0$, hence the Dirichlet series of F(s) is also absolutely convergent for $\sigma >\sigma _0$ thanks to (10), and the result follows.

4 Proof of Theorem 2

As in Theorem 1 the case $d=0$ is trivial, hence we assume $d>0$ and consider the function

$$\begin{aligned} h(s) = \frac{F(s)}{f(s)} \end{aligned}$$

for $\sigma >1-\delta$. From (3) we have that h(s) is holomorphic for $\sigma >1-\delta$, bounded on every closed vertical strip inside $\sigma >1-\delta$ and almost periodic on the line $\sigma =A$. For a given $\varepsilon >0$, let $\tau$ be an $\varepsilon$-almost period of $h(A+it)$, namely for every $t\in \mathbb R$

$$\begin{aligned} |h(A+i(t+\tau )) - h(A+it)| < \varepsilon . \end{aligned}$$

Then, by the convexity following from Phragmén-Lindelöf’s theorem applied to $h(s+i\tau )-h(s)$, given $\eta >1-\delta$ and any $\eta<\sigma <A$ we have

$$\begin{aligned} \begin{aligned} \sup _{t\in \mathbb R} |h(\sigma +i(t+\tau )) - h(\sigma +it)|&\le \big (\sup _{t\in \mathbb R} |h(\eta +i(t+\tau )) - h(\eta +it)|\big )^{\frac{A-\sigma }{A-\eta }} \\&\quad \times \,\big (\sup _{t\in \mathbb R} |h(A+i(t+\tau )) - h(A+it)|\big )^{\frac{\sigma -\eta }{A-\eta }}. \end{aligned} \end{aligned}$$

Hence we obtain that

$$\begin{aligned} \sup _{t\in \mathbb R} |h(\sigma +i(t+\tau )) - h(\sigma +it)| \ll \varepsilon ^c \end{aligned}$$

uniformly in any closed strip contained in $\eta<\sigma <A$, where $c>0$ depends on the strip. Since $\varepsilon$ is arbitrarily small, h(s) is uniformly almost periodic in such strips. Suppose now that $h(\rho )=0$ for some $\rho$ with $\mathfrak {R}{\rho }>1-\delta$. Then by a well known argument based on Rouché’s theorem we have that for any $1-\delta<\eta <\mathfrak {R}{\rho }$

$$\begin{aligned} T \ll N_h(\eta ,T) \le N_F(\eta ,T) =o(T) \end{aligned}$$

if $\eta >\sigma _D(F)$, a contradiction. Thus $h(s)\ne 0$ for $\sigma >\max (1-\delta ,\sigma _D(F))$, hence every zero of F(s) in this half-plane is a zero of f(s). Theorem 2 is therefore proved, since the opposite implication is a trivial consequence of (3).

5 Proof of Theorem 3

Again the case $d=0$ is trivial, since in this case $F(s)\equiv 1$ and so G(s) does not vanish inside the critical strip, thus its degree is 0 and hence $G(s)\equiv 1$ as well. Let $F,G\in {\mathcal S}$ be with positive degrees and coefficients $a_F(n)$ and $a_G(n)$, respectively, and consider the function

$$\begin{aligned} H(s)=\frac{F(s)}{G(s)} = \sum _{n=1}^\infty \frac{h(n)}{n^s}, \end{aligned}$$

say. By our hypothesis H(s) is bounded, and hence holomorphic, for $\sigma >1/2$. We modify the proof of Theorem 1 at several points. By Lemma 1 of [10] we have that for every $\varepsilon >0$ there exists an integer $K=K(\varepsilon )$ such that the coefficients $a^{-1}_G(n)$ of 1 / G(s) satisfy

$$\begin{aligned} a^{-1}_G(n) \ll n^\varepsilon \quad \quad \quad (n,K)=1, \end{aligned}$$

and hence

$$\begin{aligned} h(n)\ll n^{2\varepsilon } \quad \quad \quad (n,K)=1. \end{aligned}$$

Therefore the set

$$\begin{aligned} S = \{p: |h(p^m)|>p^{m/10} \quad \text {for some} \quad m\ge 1 \text { or } p\le 10^4\} \end{aligned}$$

is finite and we write

$$\begin{aligned} \begin{aligned} H(s)&= \prod _p \frac{F_p(s)}{G_p(s)} = \prod _p H_p(s) \\&=\prod _{p\not \in S}\left( 1+\frac{h(p)}{p^s} + \frac{h(p^2)}{p^{2s}}\right) \prod _{p\in S}H_p(s) \prod _{p\not \in S}\left( H_p(s)\left( 1+\frac{h(p)}{p^s} + \frac{h(p^2)}{p^{2s}}\right) ^{-1}\right) \\&=Q_1(s)Q_2(s)Q_3(s), \end{aligned} \end{aligned}$$

(13)

say. As in the prof of Theorem 1, $Q_2(s)$ and $1/Q_2(s)$ are holomorphic and bounded for $\sigma \ge 1/2$. Moreover we have

$$\begin{aligned} Q_3(s) = \prod _{p\not \in S}\left( 1 + \frac{\sum _{m=3}^\infty \frac{h(p^m)}{p^{ms}}}{1+\frac{h(p)}{p^s} + \frac{h(p^2)}{p^{2s}}}\right) = \prod _{p\not \in S}\left( 1+ \sum _{m=3}^\infty \frac{k(p^m)}{p^{ms}}\right) , \end{aligned}$$

say, and a computation shows that for $\sigma \ge 1/2$

$$\begin{aligned} \sum _{m=3}^\infty \frac{|k(p^m)|}{p^{m\sigma }} \le \frac{1}{3} \ \ \quad \text {for every }\quad p\not \in S \quad {\text{ and }} \quad \sum _{p\not \in S} \sum _{m=3}^\infty \frac{|k(p^m)|}{p^{m\sigma }} \ll 1. \end{aligned}$$

Therefore, no factor of the product vanishes, and $Q_3(s)$ and $1/Q_3(s)$ are holomorphic and bounded for $\sigma \ge 1/2$ as well.

In order the treat $Q_1(s)$ we need the following elementary lemma.

Lemma

For every $a,b\in \mathbb C$ there exists $\theta \in \mathbb C$ with $|\theta |=1$ such that

$$\begin{aligned} |1+\theta a + \theta ^2b| \ge 1 + \frac{1}{24}(|a|+|b|). \end{aligned}$$

Proof

Suppose first that $|a|\le |b|/2$. Then

$$\begin{aligned} \max _{|\theta |=1} |1+\theta a + \theta ^2b| \ge 1+|b|-|a| \ge 1+\frac{1}{2}|b| \ge 1+\frac{1}{3}(|a|+|b|), \end{aligned}$$

and the result follows in this case. In the opposite case $|a|> |b|/2$ we apply the maximum modulus principle to the function $f(z) = 1 +az+bz^2$, thus obtaining

$$\begin{aligned} \begin{aligned} \max _{|\theta |=1} |1+\theta a + \theta ^2b|&\ge \max _{|\theta |=1} \left| 1+\frac{1}{4}\theta a +\frac{1}{16} \theta ^2b\right| \\&\ge 1+\frac{1}{4}|a| -\frac{1}{16}|b| \ge 1+ \frac{1}{24} (|a|+|b|), \end{aligned} \end{aligned}$$

and the Lemma follows. Note that the constant 1 / 24 is neither optimal nor important in what follows; moreover, in general it cannot be made arbitrarily close to 1. $\square$

From (13), our hypothesis and the above information on $Q_2(s)$ and $Q_3(s)$ we deduce that there exists $M>0$ such that for $\sigma >1/2$

$$\begin{aligned} |Q_1(s)| = \prod _{p\not \in S}\left| 1+p^{-it}\frac{h(p)}{p^\sigma } +p^{-2it} \frac{h(p^2)}{p^{2\sigma }}\right| \le M. \end{aligned}$$

By the Lemma, for every $\sigma$ and p there exists $|\theta _{p,\sigma }|=1$ such that

$$\begin{aligned} \left| 1+\theta _{p,\sigma }\frac{h(p)}{p^\sigma } +\theta _{p,\sigma }^2\frac{h(p^2)}{p^{2\sigma }}\right| \ge 1 + \frac{1}{24}\left( \frac{|h(p)|}{p^\sigma } +\frac{|h(p^2)|}{p^{2\sigma }}\right) . \end{aligned}$$

Assuming that $\sigma >1/2$ and $p\not \in S$, applying Kronecker’s theorem as in the last part of the proof of Theorem 1 we find that

$$\begin{aligned} \prod _{p\not \in S} \left( 1 + \frac{1}{24}\left( \frac{|h(p)|}{p^\sigma } +\frac{|h(p^2)|}{p^{2\sigma }} \right) \right) \le M. \end{aligned}$$

Then, letting $\sigma \rightarrow 1/2^+$, we deduce that the product

$$\begin{aligned} \prod _{p\not \in S} \left( 1 + \frac{1}{24}\left( \frac{|h(p)|}{p^{1/2}} +\frac{|h(p^2)|}{p} \right) \right) \end{aligned}$$

is convergent. Thus the series

$$\begin{aligned} \sum _{p\not \in S} \left( \frac{|h(p)|}{p^{1/2}} +\frac{|h(p^2)|}{p} \right) \end{aligned}$$

is convergent as well and, in turn, the product

$$\begin{aligned} \prod _{p\not \in S} \left( 1 + \left( \frac{|h(p)|}{p^{1/2}} +\frac{|h(p^2)|}{p} \right) \right) \end{aligned}$$

converges. Hence $Q_1(s)$ and $Q_1(s)^{-1}$ are non-vanishing for $\sigma \ge 1/2$.

From (13) and the above properties of $Q_j(s)$, $j=1,2,3$, we immediately see that H(s) is holomorphic and non-vanishing for $\sigma \ge 1/2$. Denoting by $\gamma _F(s)$ and $\gamma _G(s)$ the $\gamma$-factors of F(s) and G(s), thanks to the functional equation we deduce that

$$\begin{aligned} \frac{\gamma _F(s)}{\gamma _G(s)}H(s) \end{aligned}$$

is a non-vanishing entire function of order $\le 1$, and hence by Hadamard’s theory we have

$$\begin{aligned} H(s) = \frac{\gamma _G(s)}{\gamma _F(s)} e^{as+b} \end{aligned}$$

(14)

with some $a,b\in \mathbb C$. Now we can conclude by means of the almost periodicity argument that we used in our proof of the multiplicity one property of ${\mathcal S}$. For this we refer to Lemma 2.1 of [9] and to Theorem 2.3.2 of [7]; in particular, (14) is exactly the last displayed formula of p. 167 of [7]. This way we get that $H(s)\equiv 1$, hence Theorem 3 is proved.