Some properties of universal Dirichlet series

Abstract

We establish some properties of universal Dirichlet series. In particular we give a new estimate on the growth of their coefficients. As a consequence we obtain an information about the admissible size of coefficients of Dirichlet polynomials that approximate a given entire function on a compact set. Moreover we prove that, for all $\alpha >-1$, the sequence of Riesz means $\left( \left( \sum _{k=1}^nk^{\alpha }\right) ^{-1}\sum _{k=1}^n k^{\alpha }D_k(f)\right)$ of partial sums of an universal Dirichlet series f is automatically universal. Finally we show that the Dirichlet series satisfying the universal approximation property with respect to every compact set K (with connected complement) contained in a strip $\{z\in {\mathbb {C}}:\sigma \le \mathfrak {R}(z)\le 0\}$ are not necessarily universal in the left half-plane $\{z\in {\mathbb {C}}:\mathfrak {R}(z)\le 0\}$.

1 Introduction and notations

For a compact set $K\subset {\mathbb {C}},$A(K) will stand for the space of all continuous functions on K which are holomorphic in its interior. The notion of universal Taylor series was independently introduced by Luh [13] and Chui and Parnes [7], and was strengthened by Nestoridis [17], who showed the existence of holomorphic function f on the unit disc ${\mathbb {D}}=\{z\in {\mathbb {C}}:\vert z\vert <1\}$ whose Taylor series $\sum _{n\ge 0}a_nz^n$ at 0 satisfies the following universal approximation property: for every compact set $K\subset \{z\in {\mathbb {C}};\vert z\vert \ge 1\}$ with connected complement and for every function $h\in A(K),$ there exists a subsequence $(\lambda _n)\subset {\mathbb {N}}$ such that $\sum _{k=0}^{\lambda _n}a_kz^k$ converges to h uniformly on K,  as n tends to infinity. Since Nestoridis’ theorem, many results on universal Taylor series in the complex plane appeared. We refer the reader to [5] and the references therein. Roughly speaking, this curious phenomenon of universality has the following interpretation: the sequence of partial sums of the Taylor series (at 0) of a holomorphic function on ${\mathbb {D}}$ can have the worst possible behavior, even on the unit circle. In 2005, Bayart proved that (ordinary) Dirichlet series share this very strange property [3]. For a complex number z, $\mathfrak {R}(z)$ (resp. $\mathfrak {I}(z)$) denotes its real part (resp. imaginary part). Let ${\mathbb {C}}_+=\{s\in {\mathbb {C}}\ ;\ \mathfrak {R}(s)>0\}$ (resp. ${\mathbb {C}}_-=\{s\in {\mathbb {C}}\ ;\ \mathfrak {R}(s)<0\}$) be the right (resp. left) half-plane. To any Dirichlet series $f=\sum _{n\ge 1}a_nn^{-s}$, one can associate its abscissa of absolute convergence $\sigma _{a}(f)=\inf \{\mathfrak {R}(s):\sum _{n\ge 1}a_nn^{-s}$ is absolutely convergent$\}$ and its n-th partial sum $D_n(f)$, i.e. $D_n(f)(s)=\sum _{k=1}^na_k k^{-s}.$ Notice that for all s with $\mathfrak {R}(s)>\sigma _{a}(f),$ one has $\sum _{n\ge 1}\vert a_n\vert n^{-\mathfrak {R}(s)}<+\infty .$ Let us consider the space ${\mathcal {D}}_a({\mathbb {C}}_+)$ of absolutely convergent Dirichlet series $\sum _{n\ge 1}a_nn^{-s}$ in ${\mathbb {C}}_+$ endowed with its natural topology given by the semi-norms $\Vert \sum _{n\ge 1}a_n n^{-s}\Vert _{\sigma }=\sum _{n\ge 1}\vert a_n\vert n^{-\sigma },$$\sigma >0.$ In [3], the author established the existence of universal Dirichlet series in ${\mathcal {D}}_a({\mathbb {C}}_+)$ in the following sense: for any “admissible” compact set $K\subset \overline{{\mathbb {C}}}_{-}$ with connected complement and for any function $h\in A(K),$ there exists an increasing sequence $(\lambda _n)\subset {\mathbb {N}}$ such that $\sup _{s\in K}\vert D_{\lambda _n}(f)(s)-h(s)\vert \rightarrow 0$ as n tends to infinity. Recently in [2] the authors relaxed the assumption that the compact sets be admissible. In the sequel, we will call ${\mathcal {U}}_d$ the set of such universal Dirichlet series. Several results about these can be found in [2, 5, 8, 9, 15]. Observe that obviously they remain valid for universal Dirichlet series with respect to all compact sets of $\overline{{\mathbb {C}}}_{-}$ with connected complement (not necessarily “admissible” compact sets). In particular in [8] the authors obtain estimates on the growth of coefficients of universal Dirichlet series and, in [15], it is proved that universal Dirichlet series cannot be logarithmically summable at any point of their line of convergence, where the sequence $(\sigma _{L,n}(f))$ of logarithmic means of a Dirichlet series f is given by $\sigma _{L,n}(f)(s)=\frac{1}{\log (n)}\sum _{k=1}^n \frac{1}{k}D_k(f)(s)$. Therefore we deduce that universal Dirichlet series cannot be Cesàro summable at any point of their line of convergence. Finally let us introduce the class of universal Dirichlet series with respect to a specific compact set K.

Definition 1.1

Let $K\subset \overline{{\mathbb {C}}}_{-}$ be a compact set with connected complement. A Dirichlet series $f=\sum _{k\ge 1}a_kk^{-s}\in {\mathcal {D}}_a({\mathbb {C}}_+)$ is said to be universal with respect to K if the set $\{D_n(f)\ :\ n\in {\mathbb {N}}\}$ is dense in A(K) endowed with the topology given by the supremum norm on K. We will call ${\mathcal {U}}_{d,K}$ the set of such universal Dirichlet series.

If $K\cap \{z\in {\mathbb {C}}: \mathfrak {R}(z)=0\}=\emptyset ,$ we know that ${\mathcal {U}}_{d,K}\ne {\mathcal {U}}_d$ [8].

In the present short paper, we are going to establish some properties of universal Dirichlet series. First we are interested in the notion of restricted universality. As has been pointed out, an universal Dirichlet series with respect to a specific compact set $K\subset {\mathbb {C}}_{-}$ with connected complement is not necessarily an element of ${\mathcal {U}}_d$. A natural question that arises is whether the same conclusion holds for Dirichlet series that are universal with respect to a compact set $K\subset \overline{{\mathbb {C}}}_-$ (with connected complement) such that $K\cap \{z\in {\mathbb {C}}: \mathfrak {R}(z)=0\}\ne \emptyset$. Actually we are going to establish a stronger result: there exists a Dirichlet series which satisfies the universal approximation property with respect to every compact set K (with connected complement) contained in a strip $\{z\in {\mathbb {C}}:\sigma < \mathfrak {R}(z)\le 0\}$ ($\sigma <0$) whose partial sums tend to infinity for almost every z (in the sense of Lebesgue measure) in the left half-plane $\{z\in {\mathbb {C}}:\mathfrak {R}(z)<\sigma -\frac{1}{2}\}$. It is a Dirichlet version of a theorem of Kahane and Melas for universal Taylor series with respect to compact sets that are contained in an annulus [11]. Then we study some boundary behaviors of universal Dirichlet series. First we study the effect of Cesàro summability methods on universal Dirichlet series. The same problem has been studied in the case of universal Taylor series ( [4, 12, 16]) and in particular it is showed that a Taylor series is universal if and only if the sequence of Cesàro means of its partial sums is universal. In the case of Dirichlet series, we already know that a universal element cannot be Cesàro summable on the boundary. It is easy to prove that the sequence of Cesàro means $\left( n^{-1}\sum _{k=1}^n D_k(f)(s)\right)$ of an element $f\in {\mathcal {U}}_d$ satisfies the universal approximation property with respect to all compact subsets of $\overline{{\mathbb {C}}}_{-}$ with connected complement. Furthermore we show that the same result remains true for the sequence of weighted Riesz means of the form $\left( \left( \sum _{k=1}^n k^{\alpha }\right) ^{-1}\sum _{k=1}^n k^{\alpha } D_k(f)(s)\right)$ for $\alpha >-1.$ Notice that the case $\alpha =1$ corresponds to logarithmic means, but we don’t know if the universal Dirichlet series remain automatically universal under this summability method. As mentioned above, we only know that they cannot be logarithmically summable on $i{\mathbb {R}}$. Next we improve the result of [8] on the admissible growth of coefficients of such series. As application, we are interested in the following result of the set of Dirichlet polynomials: let $K\subset \overline{{\mathbb {C}}}_{-}$ be a compact set with connected complement and $\delta >0$, then for every integer N the set $\left\{ \sum _{n=N}^Ma_n n^{-(1-\delta )}n^{-s}; M\ge N,\vert a_n\vert \le 1\right\}$ is dense in A(K) [2] (see Lemma 4.5 below). We show that this result does not hold anymore if the arbitrarily small real number $\delta$ is replaced by the sequence $\delta _n=1/\log ^{1+\varepsilon }(\log (n))$ ($\varepsilon >0$) for instance. Finally we deal with an extension of a result of Gauthier [10]: we exhibit universal Dirichlet series whose coefficients are generated by the Riemann zeta-function. To do this, we combine the famous Voronin’s theorem with the lemma of approximation by Dirichlet polynomials.

The paper is organized as follows: in Sect. 2 we deal with the results on restricted universality. In Sect. 3 we are interested in Riesz summability methods preserving the universality of Dirichlet series. Section 4 is devoted to the study of the growth of coefficients of universal Dirichlet series and its applications for the approximation by Dirichlet polynomials with a control on the size of coefficients, whereas in Sect. 5 we build universal Dirichlet series thanks to Riemann zeta-function. Section 6 concludes the paper with some open problems.

2 Restricted universality

For power series, the notion of universality with respect to a specific compact set does not coincide with that of universality with respect to all suitable compact sets. We know that a Taylor series of $H({\mathbb {D}})$ that is universal with respect to any compact set K with connected complement contained in a closed annulus $\{z\in {\mathbb {C}}:1\le \vert z\vert \le d\}$ is not necessarily universal in the sense of Nestoridis. We refer the reader to [6, Section 4] for an elementary proof using the fact that all universal Taylor series possess Ostrowski-gaps. But more than this is true: Kahane and Melas showed that there exists a power series $\sum _{k\ge 0} a_kz^k$ having radius of convergence equal to 1, that is universal with respect to any compact set K with connected complement contained in the closed annulus $\{z\in {\mathbb {C}}:1\le \vert z\vert \le d\}$ and satisfies $\sum _{k=0}^n a_kz^k\rightarrow \infty$ for almost every (in the sense of Lebesgue measure) z in $\{z\in {\mathbb {C}}:\vert z\vert > d\}$, as n tends to infinity [11]. Thus in the context of Dirichlet series, one can wonder if any Dirichlet series which satisfies the universal approximation property in a given strip is necessarily universal. We give a negative answer. To do this, we establish a stronger result in the spirit of Kahane-Melas theorem: following their main ideas, we show that there exists a Dirichlet series which satisfies the universal approximation property in a given strip but which has a regular behavior in a left half-plane.

Theorem 2.1

Let $\sigma <0.$ There exists a Dirichlet series $\sum _{n\ge 1}a_nn^{-s}$ in ${\mathcal {D}}_a({\mathbb {C}}_+)$ satisfying the universal approximation property in the strip $\{s\in {\mathbb {C}}:\sigma <\mathfrak {R}(s)\le 0\}$ such that $\sum _{j=1}^na_j j^{-s}\rightarrow \infty ,$ as n tends to infinity, for almost every s in the half-plane $\{s\in {\mathbb {C}}: \mathfrak {R}(s)<\sigma -\frac{1}{2}\}$ (in the sense of the Lebesgue measure).

Proof

Let us consider an universal Dirichlet series $f(s)=\sum _{n\ge 1} b_n n^{-s}$ in ${\mathcal {D}}_a({\mathbb {C}}_+).$ We are going to build inductively the sequence of coefficients $(a_n).$ Set $a_1=b_1$ and assume that we constructed $a_2,\dots ,a_{n-1}.$ For every complex number d,  we define the sets $E_n(d)$ as follows

$$\begin{aligned} E_n(d)=\left\{ s\in B_n:\left| \sum _{k=1}^{n-1}a_k k^{-s}+b_n n^{-s}+dn^{-s}\right| \le \log (\log (n+2))\right\} , \end{aligned}$$

where $B_n$ is the compact rectangle

$$\begin{aligned} B_n= & {} \left\{ s\in {\mathbb {C}} : \sigma -\frac{1}{2}-\log (\log (n+2))\right. \nonumber \\\le & {} \left. \mathfrak {R}(s)\le \sigma -\frac{1}{2}-\frac{\log (\log (n+2))}{\log (n)}\hbox { and }\vert \mathfrak {I}(s)\vert \le \log (\log (n+2))\right\} . \end{aligned}$$

Clearly the fact $E_n(d)\cap E_n(d')\ne \emptyset$ implies that

$$\begin{aligned} \vert d-d'\vert n^{-\sigma }\sqrt{n}\log (n)\le 2\log (\log (n+2)). \end{aligned}$$

Let us choose $\{w_1,\dots ,w_{q_n}\}$ in the closed disc $\{w\in {\mathbb {C}} : \vert w+\sum _{k=1}^{n-1}(a_k-b_k)k^{-\sigma }\vert \le 1/\log (\log (n+2))\}$ maximal under the requirement

$$\begin{aligned} \vert w_j-w_k\vert \ge 2\frac{ \log (\log (n+2))}{\sqrt{n}\log (n)} \end{aligned}$$

for $j\ne k.$ A simple argument of recovering gives

$$\begin{aligned} q_n\ge C\frac{n \log ^2(n)}{\log ^4(\log (n+2))}, \end{aligned}$$

where C is a constant. We set $d_k=w_kn^{\sigma }$, for $k=1,\dots ,q_n.$ We conclude that, for $i=1,\dots ,q_n$, the sets $E_n(d_i)$ are pairwise disjoint. Hence there exists $d_{i_n}$ such that

$$\begin{aligned} \lambda (E_n(d_{i_n}))\le \frac{\lambda (B_n)}{q_n}\le C'\frac{\log ^6(\log (n+2))}{n \log ^2(n)}, \end{aligned}$$

where $C'$ depends only on C. We let $a_n=b_n+d_{i_n}$ and $g(s)=\sum _{n\ge 1}a_nn^{-s}.$ Then, for every $s\in B_n{\setminus } E_n(d_{i_n}),$ we have $\vert \sum _{k=1}^na_kk^{-s}\vert \ge \log (\log (n+2)),$ which implies $\sum _{k=1}^na_kk^{-s}\rightarrow \infty$ as $n\rightarrow +\infty ,$ unless $s\in E_n(d_{i_n})$ for infinitely many values of n. Since $\sum _{n\ge 2}\frac{\log ^6(\log (n+2))}{n \log ^2(n)}<+\infty ,$ we have $\sum _{n\ge 1}\lambda (E_n(d_{i_n})) <+\infty ,$ and the Borel-Cantelli lemma applied to any fixed set $B_M$ ensures that $\sum _{j=1}^na_j j^{-s}\rightarrow \infty ,$ as $n\rightarrow +\infty ,$ for almost every s in the half-plane $\{s\in {\mathbb {C}}: \mathfrak {R}(s)<\sigma -\frac{1}{2}\}.$ On the other hand we have, for every $n\ge 1,$

$$\begin{aligned} \left| \sum _{k=1}^{n}(a_k-b_k)k^{-\sigma }\right| \le 1/\log (\log (n+2)). \end{aligned}$$

Hence the abscissa of convergence $\sigma _c(h)$ of the Dirichlet series $h(s)=\sum _{k\ge 1}(a_k-b_k)k^{-s}$ satisfies $\sigma _c(h)\le \sigma .$ Therefore a classical result (see for instance [1, Theorem 11.11]) ensures that $\sum _{k\ge 1}(a_k-b_k)k^{-s}$ converges uniformly on every compact subset lying interior to the half-plane $\{s\in {\mathbb {C}}: \mathfrak {R}(s)>\sigma \},$ which implies that $\sum _{k\ge N}(a_k-b_k)k^{-s}$ converges uniformly to 0,  as N tends to infinity, on such compact sets. Now, let $K\subset \{s\in {\mathbb {C}}:\sigma <\mathfrak {R}(s)\le 0\}$ be a compact set such that $K^c$ is connected and let P be any holomorphic polynomial. Since f is an universal Dirichlet series, there exists an increasing sequence of natural numbers $(\lambda _n)$ such that

$$\begin{aligned}&\sup _{s\in K}\left| \sum _{k=1}^{\lambda _n}b_k k^{-s}- P(s)+\sum _{k=1}^{+\infty }(a_k-b_k) k^{-s}\right| \\&\qquad = \sup _{s\in K}\left| \sum _{k=1}^{\lambda _n}a_k k^{-s}- P(s)+\sum _{k=1+\lambda _n}^{+\infty }(a_k-b_k) k^{-s}\right| \rightarrow 0 \end{aligned}$$

as n tends to infinity. Thus by the triangle inequality we get

$$\begin{aligned}&\sup _{s\in K}\vert \sum _{k=1}^{\lambda _n}a_k k^{-s}- P(s)\vert \le \sup _{s\in K}\vert \sum _{k=1}^{\lambda _n}a_k k^{-s}- P(s)\\&\quad +\sum _{k=1+\lambda _n}^{+\infty }(a_k-b_k) k^{-s}\vert + \sup _{s\in K}\vert \sum _{k=1+\lambda _n}^{+\infty }(a_k-b_k) k^{-s}\vert , \end{aligned}$$

and combining the last two inequalities with the fact $\sup _{s\in K}\vert \sum _{k=N}^{+\infty }(a_k-b_k) k^{-s}\vert \rightarrow 0,$ as $N\rightarrow +\infty ,$ we conclude that the Dirichlet series g satisfies the universal approximation property in the strip $\{s\in {\mathbb {C}}:\sigma <\mathfrak {R}(s)\le 0\}.$ This finishes the proof. $\square$

Remark 2.2

• (1) In the above construction, there is a grey zone. What is the behavior of the Dirichlet series g in the strip $\{s\in {\mathbb {C}}: \sigma -\frac{1}{2}\le \mathfrak {R}(s)\le \sigma \}$?

• (2) Contrary to the case of Taylor series, we do not know an elementary proof of the simple fact that a Dirichlet series which satisfies the universal approximation property in a given strip is not necessarily universal.

3 Riesz means of universal Dirichlet series

For a Dirichlet series $f(s)=\sum _{n\ge 1}a_nn^{-s},$ let us consider the Cesàro means of its partial sums:

$$\begin{aligned} \sigma _{n}(f)(s)=\frac{D_1(f)(s)+D_2(f)(s)+\dots +D_n(f)(s)}{n}. \end{aligned}$$

Let $K\subset \{s\in {\mathbb {C}}\ ;\ \mathfrak {R}(s)\le 0\}$ be a compact set with connected complement and h be an entire function. Let L be a compact set with connected complement such that $K\subset L$ and $\{s-1,s\in K\}\subset L.$ Assume that f is an universal Dirichlet series. Therefore there exists a sequence $(\lambda _n)$ such that

$$\begin{aligned} \sup _{s\in L}\vert D_{\lambda _n}(f)(s)-h(s)\vert \rightarrow 0,\hbox { as }n\rightarrow +\infty . \end{aligned}$$

Observe that we have $\sigma _{\lambda _n}(f)(s)=\left( 1+\frac{1}{\lambda _n}\right) D_{\lambda _n}(f)(s)- \frac{1}{\lambda _n}D_{\lambda _n}(f)(s-1).$ Thus the triangle inequality gives

$$\begin{aligned} \begin{array}{rcl} \displaystyle \sup _{s\in K}\vert \sigma _{\lambda _n}(f)(s)-h(s)\vert &{}\le &{}\displaystyle \sup _{s\in K}\vert D_{\lambda _n}(f)(s)-h(s)\vert + \frac{1}{\lambda _n}\sup _{s\in K}\vert D_{\lambda _n}(f)(s)-h(s)\vert \\ &{}&{}+\,\displaystyle \frac{1}{\lambda _n}\sup _{s\in K}\vert D_{\lambda _n}(f)(s-1)-h(s-1)\vert \\ &{}&{}+\,\frac{1}{\lambda _{n}}(\sup _{s\in K}\vert h(s)\vert +\sup _{s\in K}\vert h(s-1)\vert )\\ &{}\le &{}\displaystyle \sup _{s\in L}\vert D_{\lambda _n}(f)(s)-h(s)\vert + \frac{2}{\lambda _n}\sup _{s\in L}\vert D_{\lambda _n}(f)(s)-h(s)\vert \\ &{}&{}+\frac{2}{\lambda _{n}}\sup _{s\in L}\vert h(s)\vert \end{array} \end{aligned}$$

and we deduce that $\sup _{s\in K}\vert \sigma _{\lambda _n}(f)(s)-h(s)\vert \rightarrow 0,$ as n tends to infinity. Therefore we proved the following Proposition:

Proposition 3.1

Let f be a Dirichlet series in ${\mathcal {D}}_a({\mathbb {C}}_+).$ Assume that f is a universal Dirichlet series, then for every compact set $K\subset \overline{{\mathbb {C}}}_{-}$ with connected complement and every entire function h,  there exists an increasing sequence $(\lambda _n)\subset {\mathbb {N}}$ such that $\sigma _{\lambda _n}(f)$ converges to h uniformly on K as n tends to infinity, i.e. the sequence of the Cesàro means of partial sums of an universal Dirichlet series is universal.

More generally we are interested in the Riesz means related to the summability matrix $A_{\alpha }=(a_{n,k}(\alpha ))$ given by

$$\begin{aligned}a_{n,k}(\alpha )=\left\{ \begin{array}{ll}\displaystyle \frac{k^{-\alpha }}{\sum _{j=1}^nj^{-\alpha }}&{} \quad \hbox { for }1\le k\le n,\\ 0 &{} \quad \hbox { for }k\ge n+1,\end{array}\right. \end{aligned}$$

with $0<\alpha <1.$ For a Dirichlet series $f(s)=\sum _{n\ge 1}a_nn^{-s},$ we define the $A_{\alpha }$-Riesz means as follows:

$$\begin{aligned} \sigma _{A_{\alpha },n}(f)(s)=\displaystyle \frac{1}{\sum _{k=1}^n k^{-\alpha }} \left( \sum _{k=1}^n k^{-\alpha }D_k(f)(s)\right) . \end{aligned}$$

The case $\alpha =0$ corresponds to the Cesàro means.

We will need the following lemma, which is an exercise left to the reader.

Lemma 3.2

Let $0<\alpha <1.$ For any integer $l\ge 1,$ we have

$$\begin{aligned} \sum _{k=1}^nk^{-\alpha }=\frac{n^{1-\alpha }}{1-\alpha }+l_{\alpha }+\sum _{j=0}^l\frac{C_{\alpha ,j}}{n^{\alpha +j}}+ \frac{\varepsilon _{\alpha }(n)}{n^{\alpha +l}}, \end{aligned}$$

where the numbers $l_{\alpha }$ and $C_{\alpha ,j},$$j=0,\dots ,l,$ don’t depend on n ($C_{\alpha ,0}=\frac{1}{2},$$C_{\alpha ,1}=\frac{-\alpha }{12},$...) and $\varepsilon _{\alpha }(n)\rightarrow 0$ as $n\rightarrow +\,\infty .$

Theorem 3.3

Let f be a Dirichlet series in ${\mathcal {D}}_a({\mathbb {C}}_+).$ Assume that f is a universal Dirichlet series, then for every compact set $K\subset \overline{{\mathbb {C}}}_{-}$ with connected complement and every entire function h,  there exists an increasing sequence $(\lambda _n)\subset {\mathbb {N}}$ such that $\sigma _{A_{\alpha },\lambda _n}(f)$ converges to h uniformly on K as n tends to infinity, i.e. the sequence of the Riesz means $(\sigma _{A_{\alpha },n}(f))$ is universal.

Proof

Let $K\subset \overline{{\mathbb {C}}}_{-}$ be a compact set with connected complement and h be an entire function. We set $\sigma _K=\inf \{\mathfrak {R}(s);s\in K\},$$t_K=\sup \{\vert \mathfrak {I}(s)\vert ;s\in K\}$ and $l_K=\min \{l\in {\mathbb {N}};\sigma _K+\alpha +l>0\}.$ Let us define the compact sets $K_{\alpha },$$K_{\alpha }^{-}$ and $K_{\alpha }^{+}$ as follows:

$$\begin{aligned}&K_{\alpha }=\{s\in {\mathbb {C}}; \sigma _K+\alpha -1\le \mathfrak {R}(s)\le \alpha +l_K\hbox { and }\vert \mathfrak {I}(s)\vert \le t_K\},\hbox { }\\&K_{\alpha }^{-}=K_{\alpha }\cap \overline{{\mathbb {C}}}_{-}\hbox { and }K_{\alpha }^{+}=K_{\alpha }\cap \overline{{\mathbb {C}}}_{+}. \end{aligned}$$

Observe that $K_{\alpha }^{-}$ has connected complement and $K\subset K_{\alpha }^{-}.$ Since $f=\sum _{n\ge 1}a_nn^{-s}$ is an universal Dirichlet series, there exists an increasing sequence $(\lambda _n)\subset {\mathbb {N}}$ such that

$$\begin{aligned} \sup _{s\in K_{\alpha }^{-}}\vert D_{\lambda _n}(f)(s)-h(s)\vert \rightarrow 0,\hbox { as }n\rightarrow +\infty . \end{aligned}$$

(1)

We are going to prove that $\sigma _{A_{\alpha },\lambda _n}(f)$ converges to h uniformly on K as n tends to infinity. Let us write

$$\begin{aligned} \begin{array}{rcl}\displaystyle \sigma _{A_{\alpha },n}(f)(s)&{}=&{}\displaystyle \frac{\sum _{k=1}^n k^{-\alpha }D_k(f)(s)}{\sum _{k=1}^n k^{-\alpha }}\\ &{}=&{}\displaystyle \frac{\sum _{j=1}^na_j\left( \sum _{k=1}^nk^{-\alpha }-(\sum _{k=1}^j k^{-\alpha }-j^{-\alpha })\right) j^{-s}}{\sum _{k=1}^n k^{-\alpha }}\\ &{}=&{}\displaystyle D_n(f)(s)+\frac{D_n(f)(s+\alpha )}{\sum _{k=1}^n k^{-\alpha }}- \frac{\sum _{j=1}^na_j(\sum _{k=1}^jk^{-\alpha })j^{-s}}{\sum _{k=1}^n k^{-\alpha }}\end{array} \end{aligned}$$

(2)

Lemma 3.2 ensures that we can find $N\ge 1$ such that for every $n\ge N,$ we have

$$\begin{aligned} \left| \sum _{k=N+1}^nk^{-\alpha }-\frac{n^{1-\alpha }}{1-\alpha }-l_{\alpha }-\sum _{j=0}^{l_K}\frac{C_{\alpha ,j}}{n^{\alpha +j}}\right| <\frac{1}{n^{\alpha +l_K}}. \end{aligned}$$

(3)

Then we write, for all n with $\lambda _n\ge N,$

$$\begin{aligned}&\sum _{j=1}^{\lambda _n}a_j\left( \sum _{k=1}^jk^{-\alpha }\right) j^{-s}=\sum _{j=1}^Na_j\left( \sum _{k=1}^jk^{-\alpha }\right) j^{-s}\nonumber \\&\quad + \sum _{j=N+1}^{\lambda _n}a_j\left( \sum _{k=1}^Nk^{-\alpha }\right) j^{-s}+\sum _{j=N+1}^{\lambda _n}a_j(\sum _{k=N+1}^jk^{-\alpha })j^{-s}. \end{aligned}$$

(4)

Clearly, we have

$$\begin{aligned} \sup _{s\in K}\left| \frac{\sum _{j=1}^Na_j(\sum _{k=1}^jk^{-\alpha })j^{-s}}{\sum _{k=1}^{\lambda _n} k^{-\alpha }}\right| \rightarrow 0 \hbox { as }n\rightarrow +\infty . \end{aligned}$$

By the triangle inequality, we get

$$\begin{aligned} \left| \sum _{j=N+1}^{\lambda _n}a_j(\sum _{k=1}^Nk^{-\alpha })j^{-s}\right| \le \left| \sum _{k=1}^Nk^{-\alpha }\right| \left( \left| D_{\lambda _n}(f)(s)-h(s)\right| + \left| \sum _{j=1}^{N}a_jj^{-s}\right| +\vert h(s)\vert \right) .\nonumber \\ \end{aligned}$$

(5)

Therefore we deduce, using both (1) and $K\subset K_{\alpha }^{-},$

$$\begin{aligned} \sup _{s\in K}\left| \frac{\sum _{j=N+1}^{\lambda _n}a_j(\sum _{k=1}^Nk^{-\alpha })j^{-s}}{\sum _{k=1}^{\lambda _n} k^{-\alpha }}\right| \rightarrow 0 \hbox { as }n\rightarrow +\infty . \end{aligned}$$

(6)

Let us now consider the last term of (4). Using Lemma 3.2 and the estimate (3) we can write, for all positive integer n with $\lambda _n\ge N,$

$$\begin{aligned} \begin{array}{rcl}\displaystyle \sum _{j=N+1}^{\lambda _n}a_j(\sum _{k=N+1}^jk^{-\alpha })j^{-s}&{}=&{} \displaystyle \frac{1}{1-\alpha }\sum _{j=N+1}^{\lambda _n}a_j j^{-(s+{\alpha }-1)}+ l_{\alpha }\sum _{j=N+1}^{\lambda _n}a_j j^{-s}\\ &{}&{}+\displaystyle \sum _{i=0}^{l_K}C_{\alpha ,i} \sum _{j=N+1}^{\lambda _n}a_j j^{-(s +\alpha +i)}\\ &{}&{}+\sum \limits _{j=N+1}^{\lambda _n}a_j j^{-(s+\alpha +l_K)}\varepsilon _{\alpha }(j),\end{array} \end{aligned}$$

(7)

with $\vert \varepsilon _{\alpha }(j)\vert \le 1,$ for all $j\ge N.$ Clearly we have

$$\begin{aligned} \sup _{s\in K}\left| \sum _{j=N+1}^{\lambda _n}a_j j^{-s}\right| \le \sup _{s\in K}\vert D_{\lambda _n}(f)(s)-h(s)\vert +\sup _{s\in K}\vert h(s)\vert +\sup _{s\in K}\left| D_N(f)(s)\right| .\nonumber \\ \end{aligned}$$

(8)

Combining (1) with (8), we get

$$\begin{aligned} \frac{\sup _{s\in K}\vert \sum _{j=N+1}^{\lambda _n}a_j j^{-s}\vert }{\sum _{k=1}^{\lambda _n} k^{-\alpha }} \rightarrow 0 \hbox { as }n\rightarrow +\infty . \end{aligned}$$

(9)

In the same way, observe that for $s\in K,$ we necessarily have $s+\alpha -1\in K_{\alpha }^-$ and we have

$$\begin{aligned} \sup _{s\in K}\left| \sum _{j=N+1}^{\lambda _n}a_j j^{-(s+{\alpha }-1)}\right|&\le \sup _{s\in K_{\alpha }^{-}}\vert D_{\lambda _n}(f)(s)-h(s)\vert +\sup _{s\in K_{\alpha }^{-}}\vert h(s)\vert \nonumber \\&+\sup _{s\in K_{\alpha }^{-}}\left| D_N(f)(s)\right| , \end{aligned}$$

(10)

which gives, using (1),

$$\begin{aligned} \frac{\sup _{s\in K}\vert \sum _{j=N+1}^{\lambda _n}a_j j^{-(s+{\alpha }-1)}\vert }{\sum _{k=1}^{\lambda _n} k^{-\alpha }} \rightarrow 0 \hbox { as }n\rightarrow +\infty . \end{aligned}$$

(11)

Let us now estimate the sums $\sum _{j=N+1}^{\lambda _n}a_j j^{-(s +\alpha +i)},$ for $i=0,\dots ,l_K-1$ and $s\in K.$ By construction the property $s\in K$ implies the following property $s+\alpha +i\in K_{\alpha },$ for all $i=0,\dots ,l_K-1.$ For any integer $i\in \{0,\dots ,l_K-1\},$ we have to consider two cases:

Case$s+\alpha +i\in K_{\alpha }^{-}$: we have, for any complex number $s\in K$ with $s+\alpha +i\in K_{\alpha }^{-},$

$$\begin{aligned} \left| \sum _{j=N+1}^{\lambda _n}a_j j^{-(s+{\alpha }+i)}\right| \!\le \! \sup _{s\in K_{\alpha }^{-}}\vert D_{\lambda _n}(f)(s)\!-\!h(s)\vert \!+\!\sup _{s\in K_{\alpha }^{-}}\vert h(s)\vert +\!\sup _{s\in K_{\alpha }^{-}}\left| D_N(f)(s)\right| .\nonumber \\ \end{aligned}$$

(12)

Case$s+\alpha +i\in K_{\alpha }^{+}$: let us choose $\eta >0$ with $1-\alpha -\eta >0.$ Thus, for any complex number $s\in K$ with $s+\alpha +i\in K_{\alpha }^{+},$ the following estimate holds

$$\begin{aligned} \begin{array}{rcl}\displaystyle \left| \sum _{j=N+1}^{\lambda _n}a_j j^{-(s+{\alpha }+i)}\right| &{}\le &{}\displaystyle \left| \sum _{j=N+1}^{\lambda _n}a_j j^{-(s+{\alpha }+i+\eta )}j^{\eta }\right| \\ &{}\le &{} \displaystyle \lambda _n^{\eta }\sum _{j=1}^{+\infty }\vert a_j\vert j^{-\eta } \end{array} \end{aligned}$$

(13)

Thus, combining (12) with (13), we get

$$\begin{aligned} \begin{array}{rcl}\displaystyle \frac{\sup _{s\in K}\vert \sum _{j=N+1}^{\lambda _n}a_j j^{-(s+{\alpha }+i)}\vert }{\sum _{k=1}^{\lambda _n} k^{-\alpha }}&{}\le &{}\displaystyle \frac{\sup _{s\in K_{\alpha }^{-}}\vert D_{\lambda _n}(f)(s)-h(s)\vert }{\sum _{k=1}^{\lambda _n} k^{-\alpha }}+\frac{\sup _{s\in K_{\alpha }^{-}}\vert h(s)\vert }{\sum _{k=1}^{\lambda _n} k^{-\alpha }} \\ &{}&{}+\displaystyle \frac{\sup _{s\in K_{\alpha }^{-}}\left| D_N(f)(s)\right| }{\sum _{k=1}^{\lambda _n} k^{-\alpha }} +\frac{\lambda _n^{\eta }}{\sum _{k=1}^{\lambda _n} k^{-\alpha }}\sum _{j=1}^{+\infty }\vert a_j\vert j^{-\eta } \end{array} \end{aligned}$$

(14)

Therefore taking into consideration (1), the estimate $\sum _{k=1}^{\lambda _n} k^{-\alpha }\sim \frac{\lambda _n^{1-\alpha }}{1-\alpha }$ as $n\rightarrow +\infty$ and the property $1-\alpha -\eta >0,$ we get

$$\begin{aligned} \frac{\sup _{s\in K}\vert \sum _{j=1}^{\lambda _n}a_j j^{-(s+{\alpha }+i)}\vert }{\sum _{k=1}^{\lambda _n} k^{-\alpha }}\rightarrow 0, \hbox { as }n\rightarrow +\infty . \end{aligned}$$

(15)

In the same way, we have, for all $\lambda _n\ge N,$

$$\begin{aligned} \frac{\sup _{s\in K}\left| \sum _{j=N+1}^{\lambda _n}a_j j^{-(s+\alpha +l_K)}\varepsilon _{\alpha }(j)\right| }{\sum _{k=1}^{\lambda _n} k^{-\alpha }}\le \frac{\lambda _n^\eta }{\sum _{k=1}^{\lambda _n} k^{-\alpha }} \sum _{j=N+1}^{+\infty }\vert a_j\vert j^{-\eta }\rightarrow 0\hbox { as }n\rightarrow +\infty .\nonumber \\ \end{aligned}$$

(16)

Let us recall that $K\subset K_{\alpha }^{-}.$ Applying (1), we have also $\sup _{s\in K}\vert D_{\lambda _n}(f)(s)-h(s)\vert \rightarrow 0$ as $n\rightarrow +\infty .$ Hence combining (4), (6), (7), (9), (11), (15), (16) with (2), we obtain

$$\begin{aligned} \sup _{s\in K}\vert \sigma _{A_{\alpha },\lambda _n}(f)(s)-h(s)\vert \rightarrow 0\hbox { as }n\rightarrow +\infty . \end{aligned}$$

This finishes the proof. $\square$

Remark 3.4

• (1) On the other hand, the same method allows to prove that Theorem 3.3 remains true in the case $\alpha <0$. Therefore if f is an universal Dirichlet series, then for all $\alpha <1$, the sequence of the Riesz means $(\sigma _{A_{\alpha },n}(f))$ is universal.

• (2) Let $k\ge 1$ be an integer. Let us consider the Cesàro means of order k

  $$\begin{aligned} \sigma _{(C,k),n}(f)(s)=\sum _{j=1}^{n}\left( 1-\frac{j-1}{n}\right) \left( 1-\frac{j-1}{n+1}\right) \dots \left( 1-\frac{j-1}{n+k-1}\right) a_j j^{-s} \end{aligned}$$

  of a Dirichlet series $f=\sum _{j\ge 1}a_j j^{-s}.$ As in Proposition 3.1, it is easy to check that the universality of f implies the universality of the sequence $(\sigma _{(C,k),n}(f))$ for all integer $k\ge 1$.

4 Growth of the coefficients of universal Dirichlet series and applications

We begin by giving a slight improvement of the estimate of the growth of coefficients of universal Dirichlet series. In [8], the authors showed that for an universal Dirichlet series $\sum _{n\ge 1}a_n n^{-s}$ with respect to a compact set $K\subset i{\mathbb {R}}$ we have $\limsup (n\vert a_n\vert e^{-\sqrt{b_n\log (n)}})=+\infty ,$ provided that $(b_n)$ is a decreasing sequence satisfying $\sum _{n\ge 2}b_n/(n\log (n))<+\infty$. The proof was inspired by that employed by [14] to handle the case of coefficients of universal Taylor series. Using similar ideas, we slightly strengthen this result. In the sequel, for $j\ge 1,$ we denote by $\log _j$ the j-th iterated of the function $\log$, i.e. for every positive integer n sufficiently large, we have $\log _1(n)=\log (n),$$\log _2(n)= \log (\log (n)),...$. We set $q_j=\min (n\in {\mathbb {N}}:\log _j(n)>1)$. To proceed further, it is convenient to state the following key-lemma.

Lemma 4.1

Let $\eta >0,$$k\ge 1$ be an integer and let $\sum _{n\ge 1} a_n n^{-s}$ be a Dirichlet series in ${\mathcal {D}}_a({\mathbb {C}}_+)$ that satisfies the universal approximation property on $\{it: -\eta \le t\le \eta \}$. Assume that $(\varepsilon _n)$ is a decreasing sequence of positive real numbers satisfying $\sum \varepsilon _n/(n\log _1(n)\dots \log _k(n))<+\infty$ and $\varepsilon _n \log (n)\rightarrow +\infty ,$ as n tends to infinity. Then we have

$$\begin{aligned} \sum _{} \vert a_n\vert e^{-\frac{\varepsilon _n\log (n)}{\log _2(n)\dots \log _{k+1}(n)}}=+\infty . \end{aligned}$$

Proof

We follow the main ideas of the proof of [8, Lemma 2.1]. We set $\delta _n=e\varepsilon _n$ for $n\in {\mathbb {N}}.$ There exists $N_0\ge q_{k+1}$ such that, for every $n\ge N_0,$ we have

$$\begin{aligned} \log _k(\varepsilon _n\log (n))>1,\quad \frac{\varepsilon _n\log (n)}{\log _1(\varepsilon _n\log (n))\dots \log _k(\varepsilon _n\log (n))}\ge q_{k+1} \end{aligned}$$

(17)

and

$$\begin{aligned} \sum _{n=N_0}^\infty \frac{\delta _n}{n\log _1(n)\dots \log _k(n)} <\eta . \end{aligned}$$

We define the functions from $i{\mathbb {R}}$ which are $2i\pi$-periodic letting

$$\begin{aligned} \left\{ \begin{array}{ll} H_n(it)=\displaystyle \frac{n\log (n)}{\delta _n}\pi &{}\hbox { for } \vert t\vert <\displaystyle \frac{\delta _n}{n\log _1(n)\dots \log _k(n)},\\ \\ H_n(it)=0&{}\hbox { for } \displaystyle \frac{\delta _n}{n \log _1(n)\dots \log _k(n)}\le \vert t\vert \le \pi . \end{array} \right. \end{aligned}$$

For $f\in L^{1}([-\pi ,\pi ]),$ let us define its Fourier coefficients ${\hat{f}}(\log (m))=\displaystyle \frac{1}{2\pi }\displaystyle \int _{-\pi }^{\pi }f(it)m^{it}dt.$ We get

$$\begin{aligned} {\hat{H}}_n(\log (m))=\left\{ \begin{array}{ll}1&{}\hbox { for }m=1,\\ \displaystyle \frac{n \log _1(n)\dots \log _k(n)}{\delta _n\log (m)} \sin \left( \displaystyle \frac{\delta _n\log (m)}{n \log _1(n)\dots \log _k(n)}\right) &{}\hbox { for }m\ne 1.\end{array}\right. \end{aligned}$$

Let $N\ge N_0$ be an integer. By hypothesis we can approximate the Dirichlet polynomial $1 + \sum \nolimits _{{m = 1}}^{{N - 1}} {a_{m} } m^{{ - z}}$ by a subsequence of partial sums of f uniformly on the compact set $\{it:-\eta \le t\le \eta \}.$ Therefore there exists an integer $M>N$ such that we have, for all $t\in [-\eta ,\eta ]$, $\left| {1 - \sum \nolimits _{{m = N}}^{M} {a_{m} } m^{{ - it}} } \right| < \frac{1}{2}$ which implies $\frac{1}{2}\le \mathfrak {R}\left( \sum _{m=N}^{M}a_mm^{-it}\right)$. Thus we proceed as in the proof of Lemma 2.1 of [8] to obtain

$$\begin{aligned} \frac{1}{2}\le \sum _{m=N}^{M}\vert a_m\vert \times \vert {\hat{f}}(\log (m))\vert . \end{aligned}$$

(18)

with

$$\begin{aligned} {\hat{f}}(\log (m))=\displaystyle \prod _{n=N_0}^{M}\displaystyle \frac{\sin \left( \displaystyle \frac{\delta _n\log (m)}{n \log _1(n)\dots \log _k(n)}\right) }{\displaystyle \frac{\delta _n\log (m)}{n \log _1(n)\dots \log _k(n)}}. \end{aligned}$$

As $(\delta _n)$ is a decreasing sequence and the series $\displaystyle \sum \displaystyle \frac{\delta _n}{n \log _1(n)\dots \log _k(n)}$ converges, we get $\delta _n\rightarrow 0,$ as $n\rightarrow +\infty .$ Therefore, there exists an integer N such that we have the following two inequalities $\displaystyle \frac{\delta _{N_0}}{N_0 l_1(N_0)\dots l_k(N_0)}\log (N)>e$ and $\delta _N<e.$ For every $m\in \{N,\dots ,M\},$ we have

$$\begin{aligned} \displaystyle \frac{\delta _{N_0}}{N_0 \log _1(N_0)\dots \log _k(N_0)}\log (m)\!>\!e \hbox { and } \displaystyle \frac{\delta _{m}}{m \log _1(m)\dots \log _k(m)}\log (m)\!<\!\delta _m\!\le \!\delta _N\!<\!e. \end{aligned}$$

Then there exists an integer $l\in \{N_0,\cdots ,m-1\}$ satisfying

$$\begin{aligned} \displaystyle \frac{\delta _{l}}{l \log _1(l)\dots \log _k(l)}\log (m)\ge e \quad \hbox {and}\quad \displaystyle \frac{\delta _{l+1}}{(l+1)\log _1(l+1)\dots \log _{k}(l+1)}\log (m)<e. \end{aligned}$$

Since the sequence $(\delta _n)$ is decreasing, we get

$$\begin{aligned} \vert {\hat{f}}(\log (m))\vert \le \displaystyle \prod _{n=N_0}^{l}\displaystyle \frac{n\log _1(n)\dots \log _k(n)}{\delta _n\log (m)}\le \Big (\displaystyle \frac{l\log _1(l)\dots \log _k(l)}{\delta _l\log (m)}\Big )^{l+1-N_0}\le \left( \frac{1}{e}\right) ^{l+1-n_0}.\nonumber \\ \end{aligned}$$

(19)

Set $u_m=\frac{\varepsilon _m\log (m)}{\log _1(\varepsilon _m\log (m))\dots \log _k(\varepsilon _m\log (m))}.$ Taking into account (17), observe that we have

$$\begin{aligned} u_m\log _1(u_m)\dots \log _k(u_m)\le u_m\log _1(\varepsilon _m\log (m))\dots \log _k(\varepsilon _m\log (m))\le \varepsilon _m\log (m). \end{aligned}$$

We deduce

$$\begin{aligned} l+1\ge u_m\ge \frac{\varepsilon _m\log (m)}{\log _2(m)\dots \log _{k+1}(m)}. \end{aligned}$$

(20)

Combining (18) with (19) and (20) we obtain

$$\begin{aligned} \sum _{m=N}^M\vert a_m\vert e^{n_0}e^{-\frac{\varepsilon _m\log (m)}{\log _2(m)\dots \log _{k+1}(m)}}\ge \frac{1}{2}. \end{aligned}$$

Since this holds for infinitely many pairs (N, M),  we have the conclusion. $\square$

Remark 4.2

The estimate given by (20) seems to be optimal since the unique solution $x_n$ of the equation $x\log (x)\log _2(x)\dots \log _k(x)=n$ has the following behavior: $x_n\sim \frac{n}{\log _2(n)\dots \log _{k+1}(n)}$ as n tends to infinity.

Next Lemma 4.1 leads to the following statement.

Theorem 4.3

Let $\eta >0,$$k\ge 1$ be an integer and let $\sum _{n\ge 1} a_n n^{-s}$ be a Dirichlet series in ${\mathcal {D}}_a({\mathbb {C}}_+)$ satisfying the universal approximation property on $\{it: -\eta \le t\le \eta \}$. Let $(b_n)$ be a decreasing sequence satisfying $\sum _{}b_n/(n\log _1(n)\dots \log _k(n))<+\infty$ and $b_n \log (n)\rightarrow +\infty ,$ as n tends to infinity. Then, we have

$$\begin{aligned} \limsup _{n\rightarrow +\infty }\left( n\vert a_n\vert e^{-\frac{b_n\log (n)}{\log _2(n)\dots \log _{k+1}(n)}}\right) =+\infty . \end{aligned}$$

Proof

Assume that there exists a real number M such that, for all integer n large enough,

$$\begin{aligned} \vert a_n\vert \le \displaystyle \frac{M}{n}e^{\frac{b_n\log (n)}{\log _2(n)\dots \log _{k+1}(n)}}. \end{aligned}$$

(21)

Let $\varepsilon >0.$ We set $\varepsilon _n=\max \left( b_n,\displaystyle \frac{\log _{k+1}(n)}{\log _k^{\varepsilon }(n)}\right) +b_n$ for all integer n large enough. Obviously the sequence $(\varepsilon _n)$ is decreasing, $\displaystyle \sum _{}\displaystyle \frac{\varepsilon _n}{n\log _1(n)\dots \log _{k}(n)}$ converges and $\varepsilon _n\log (n)\rightarrow +\infty$, as n tends to infinity. Thus Lemma 4.1 ensures that $\displaystyle \sum \vert a_n\vert e^{-\frac{\varepsilon _n\log (n)}{\log _2(n)\dots \log _{k+1}(n)}}=+\infty$. On the other hand, combining (21) with the equality $b_n=\varepsilon _n-\max \left( b_n,\displaystyle \frac{\log _{k+1}(n)}{\log _k^{\varepsilon }(n)}\right)$ and the estimate $\max \left( b_n,\displaystyle \frac{\log _{k+1}(n)}{\log _k^{\varepsilon }(n)}\right) \ge \displaystyle \frac{\log _{k+1}(n)}{\log _k^{\varepsilon }(n)},$ we have for every positive integer B, with $B>A$ and A fixed large enough,

$$\begin{aligned} \displaystyle \sum _{n=A}^B \vert a_n\vert e^{-\frac{\varepsilon _n\log (n)}{\log _2(n)\dots \log _{k+1}(n)}}\le M\displaystyle \sum _{n=A}^B\displaystyle \frac{1}{n}e^{-\frac{\log (n)}{\log _2(n)\dots \log _{k-1}(n)\log _{k}^{1+\varepsilon }(n)}}. \end{aligned}$$

However it is easy to check that $\displaystyle \sum \displaystyle \frac{1}{n}e^{-\frac{\log (n)}{\log _2(n)\dots \log _{k-1}(n)\log _{k}^{1+\varepsilon }(n)}}<+\infty$. We obtain a contradiction and we have the announced conclusion. $\square$

Now let $\varepsilon >0$ and $k\ge 2$ be an integer. If we apply Theorem 4.3 with the sequence $(b_n)$ given by $b_n=\frac{\log _{k+1}(n)}{\log _{k}^{\varepsilon }(n)}$, we obtain the following corollary.

Corollary 4.4

Let $k\ge 2$ be a positive integer and $\varepsilon >0$. Let $\eta >0$ and let $\sum _{n\ge 1} a_n n^{-s}$ be a Dirichlet series in ${\mathcal {D}}_a({\mathbb {C}}_+)$ that satisfies the universal approximation property on $\{it: -\eta \le t\le \eta \}$. Then, we have

$$\begin{aligned} \limsup _{n\rightarrow +\infty }\left( n\vert a_n\vert e^{-\frac{\log (n)}{\log _2(n)\dots \log _{k-1}(n)(\log _{k}(n))^{1+\varepsilon }}}\right) =+\infty . \end{aligned}$$

Now we apply this last result to obtain some information on the admissible size of coefficients of Dirichlet polynomials that approximate continuous functions on closed intervals of the imaginary axis. More precisely, a careful examination of the proof of Theorem 3.1 of [2] shows that the following result holds.

Lemma 4.5

Let $K\in \overline{{\mathbb {C}}}_{-}$ be a compact set with connected complement and $\delta >0.$ Then, for every $N\in {\mathbb {N}},$ the set $\left\{ \sum _{n=N}^{M}a_n n^{-\left( 1-\delta \right) }n^{-s};M\ge N,\vert a_n\vert \le 1\right\}$ is dense in A(K).

On the other hand, Theorem 4.3 allows to obtain the following result. We shall use the following notation: for every integer $k\ge 2$ and every $\tau >0,$ we set

$$\begin{aligned} \delta _{k,\tau }(n)=\frac{1}{\log _2(n)\dots \log _{k-1}(n)\log _{k}^{1+\tau }(n)}\hbox { for all }n\hbox { large enough} (n\ge q_k) \end{aligned}$$

and

$$\begin{aligned} A_{k,\tau }=\left\{ (a_n)\in {\mathbb {C}}^{{\mathbb {N}}}:\vert a_n\vert \le n^{-(1-\tau )}\hbox { for }n<q_k \hbox { and } \vert a_n\vert \le n^{-\left( 1-\delta _{k,\tau }(n)\right) } \hbox { for }n\ge q_k\right\} . \end{aligned}$$

Corollary 4.6

Let $K\in \overline{{\mathbb {C}}}_{-}$ be a compact set with connected complement such that K contains a set of the following type $\left\{ it:a\le t\le b\right\} ,$$a,b\in {\mathbb {R}}.$ Let $k\ge 2$ be a positive integer and $\tau >0$. Then the set $\left\{ \sum _{n=1}^{M}a_n n^{-s};M\ge 1,(a_n)\in A_{k,\tau }\right\}$ is not dense in A(K).

Up to do a translation on the imaginary axis we can assume that there exists $\eta >0$ such that $\{it:-\eta \le t\le \eta \}\subset K$. Hence to prove Corollary 4.6, it suffices to establish the following statement.

Proposition 4.7

Let $k\ge 2$ be a positive integer and $\eta , \tau >0$. Then the set of Dirichlet polynomials defined by $\left\{ \sum _{n=1}^{M}a_n n^{-it};M\ge 1,(a_n)\in A_{k,\tau }\right\}$ is not dense in $C([-\eta ,\eta ])$.

Proof

Set $E_N=\left\{ \sum _{n=N}^{M}a_n n^{-it};M\ge 1,\vert a_n\vert \le n^{-\left( 1-\delta _{k,\tau }(n)\right) }\right\}$ for $N\ge q_k$. We argue by contradiction. Assume that

Hypothesis (H): for every integer $N\ge q_k$, the set $E_N$ is dense in $C([-\eta ,\eta ]).$

Let also P be a Dirichlet polynomial, $g\in A(K),$$\sigma >0$ and $\varepsilon >0$. We choose $N\ge q_k$ bigger than the degree of P such that $\sum _{n\ge N}n^{-1-\sigma +\delta _{k,\tau }(n)}<\varepsilon .$ In particular, by hypothesis there exists $u=\sum _{n=N}^Ma_n n^{-\left( 1-\delta _{k,\tau }(n)\right) } n^{-it},$ such that

$$\begin{aligned} \vert a_n\vert \le 1\hbox { and }\sup _{t\in [-\eta ,\eta ]}\vert u(it)-g(t)+P(it)\vert <\varepsilon . \end{aligned}$$

Setting $h(t)=u(t)+P(it),$ we deduce

$$\begin{aligned} \sup _{t\in [-\eta ,\eta ]}\vert u(it)\!-\!g(t)\!+\!P(it)\vert<\varepsilon \hbox { and }\Vert h-P\Vert _{\sigma }\!=\!\Vert u(s)\Vert _{\sigma }\le \sum _{n\ge N}n^{-1-\sigma \!+\!\delta _{k,\tau }(n)}<\varepsilon . \end{aligned}$$

Thus we have shown that, under the hypothesis (H), for every Dirichlet polynomial P,  for every continuous function $g\in C([-\eta ,\eta ]),$ for all $\sigma >0$ and $\varepsilon >0,$ there exists a Dirichlet polynomial h such that $\sup _{K}\vert h-g\vert <\varepsilon$ and $\Vert h-P\Vert _{\sigma }<\varepsilon .$ Then following a classical construction [2, 3, 8], we can can build Dirichlet series in ${\mathcal {D}}_a({\mathbb {C}}_+)$ of the form $\sum _{}a_n n^{-s},$ with $\vert a_n\vert \le n^{-(1-\delta _{k,\tau }(n))},$ which satisfies the universal approximation property on $[-\eta ,\eta ]$. Then Theorem 4.3 gives a contradiction. Thus the hypothesis (H) does not hold. Hence there exists a positive integer $N_0=N_0(\tau )\ge q_k$ such that,

$$\begin{aligned} \forall N\ge N_0,\hbox { the set } \left\{ \sum _{n=N}^{M}a_n n^{-\left( 1-\delta _{k,\tau }(n)\right) }n^{-it};M\ge N,\vert a_n\vert \le 1\right\} \hbox { is not dense in }C([-\eta ,\eta ]).\nonumber \\ \end{aligned}$$

(22)

It remains to check that, for any $\varepsilon >0,$ for every integer $N< N_0(\varepsilon )$ the set of Dirichlet polynomials $\left\{ \sum _{n=1}^{M}a_n n^{-it};M\ge 1,(a_n)\in A_{k,\tau }\right\}$ is not dense in $C([-\eta ,\eta ])$. If not, for $\tau >0$ fixed, for any continuous function g and $\varepsilon >0,$ one can find $M\ge 1$ and complex numbers $a_1,\dots ,a_M,$ with $(a_n)\in A_{k,\tau }$, such that

$$\begin{aligned} \sup _{t\in [-\eta ,\eta ]}\left| \sum _{n=1}^{M}a_n n^{-it} - g(t)N_0^{1+\alpha +it}\right| <\varepsilon , \end{aligned}$$

where $N_0=N_0(\tau /2)\ge q_k$ ($N_0(\tau /2)$ is given by (22)) and $\alpha >0$ satisfy the condition

$$\begin{aligned} N_0^{\alpha }\ge n^{\tau }\hbox { for }n=1,\dots , q_k,\hbox { and } \sup _{l\ge q_k}\vert l^{\delta _{k,\tau }(l)-\delta _{k,\tau /2}(lN_0)}\vert \le N_0^{\alpha }. \end{aligned}$$

(23)

We deduce

$$\begin{aligned} \sup _{t\in [-\eta ,\eta ]}\left| \sum _{n=1}^{M}a_n (nN_0)^{-1}N_0^{-\alpha }(nN_0)^{-it} - g(t) \right| <\varepsilon . \end{aligned}$$

Observe that we can rewrite

$$\begin{aligned} \sum _{n=1}^{M}a_n (nN_0)^{-1}N_0^{-\alpha }(nN_0)^{-it}= \sum _{j=N_0}^{MN_0}b_j j^{-it}, \end{aligned}$$

where the coefficients $b_j$ satisfy the following estimates (thanks to (23), $N_0\ge q_k$ and $(a_n)\in A_{k,\tau }$):

• (1) $b_j=0$ if $j\ne nN_0,$ with $n=1,\dots ,M,$

• (2) for $n=1,\dots , q_k-1,$

  $$\begin{aligned} \vert b_{nN_0}\vert =\vert a_n\vert N_0^{-1-\alpha }\le \frac{1}{n^{1-\tau }N_0^{1+\alpha }}\le (nN_0)^{-(1-\delta _{k,\tau /2}(nN_0))} \end{aligned}$$

• (3) for $n= q_k,\dots ,M,$

  $$\begin{aligned} \begin{array}{rcl}\vert b_{nN_0}\vert =\vert a_n\vert N_0^{-1-\alpha }&{}\le &{} n^{-(1-\delta _{k,\tau }(n))}N_0^{-1-\alpha } \\ {} &{}\le &{} \displaystyle (nN_0)^{-(1-\delta _{k,\tau /2}(nN_0))}{n^{\delta _{k,\tau }(n)-\delta _{k,\tau /2}(nN_0)}}{N_0^{-\alpha }}\\ &{}\le &{} (nN_0)^{-(1-\delta _{k,\tau /2}(nN_0))}.\end{array} \end{aligned}$$

Thus for any continuous function g, any $\varepsilon >0,$ we have found a Dirichlet polynomial $\sum _{j=N_0}^{MN_0}b_j j^{-it}$ with $N_0=N_0(\tau /2)$ which is defined in (22) and $\vert b_j\vert \le j^{-(1-\delta _{k,\tau /2}(j))},$ such that

$$\begin{aligned} \sup _{t\in [-\eta ,\eta ]}\left| \sum _{j=N_0}^{MN_0}b_j j^{-it}-g(t)\right| <\varepsilon . \end{aligned}$$

According to (22), we obtain a contradiction. This finishes the proof. $\square$

5 Coefficients of universal Dirichlet series and the Riemann zeta-function

Let us consider the Dirichlet polynomial approximation again. Observe that we can rewrite the statement of [2, Theorem 3.1] as follows.

Lemma 5.1

Let $K\subset \overline{{\mathbb {C}}}_-$ be a compact set with connected complement, g an entire function, $N>0$ a positive integer, $0<\delta <\sigma$ and $\varepsilon >0.$ Then there exist a Dirichlet polynomial $h=\sum _{n=N}^{M}a_nn^{-z}$ such that $\sup _{z\in K}\vert h(z)-g(z)\vert <\varepsilon ,$$\Vert h\Vert _{\sigma }<\varepsilon$ and $\vert a_n\vert \le \frac{1}{n^{1-\delta }},$ for $n=N,\dots , M.$

We end the paper with the Dirichlet version of a result of Gauthier [10]. Although universal Dirichlet series are generic, we do not know exhibit such elements. A similar problem holds in the case of universal power series. In a recent work [10] the Riemann-zeta function was employed to generate the coefficients of universal Taylor series in the sense of Nestoridis. The proof is constructive and is a combination of a lemma of approximation polynomial [10, Lemma 3.2] (which gives a geometric control of growth of coefficients) with Voronin’s Theorem. First let us recall the latter.

Theorem 5.2

(Voronin [18]) For each $z_0$ in the strip $1/2<\sigma <1$ and each $k=0,1,2,\dots ,$ the sequence

$$\begin{aligned} \left\{ \left( \zeta (z_0+im),\zeta '(z_0+im),\dots ,\zeta ^{(k)}(z_0+im)\right) :m=1,2,\dots \right\} \end{aligned}$$

is dense in ${\mathbb {C}}^{k+1}$.

In the case of Dirichlet series, using Lemma 5.1 and following the ideas of [10], we can build an universal Dirichlet series whose coefficients are generated by the Riemann zeta-function. To do this, we need a Dirichlet version of Lemma 3.4 of [10]. Let $(g_n)$ be an enumeration of all Dirichlet polynomials with rational complex coefficients and $(K_n)$ be the sequence of compact sets $K_n=[-n,0]\times [-n,n].$

Lemma 5.3

Let $\sigma$ be a real number with $1/2<\sigma <1$. Let $(\delta _n)$ be a decreasing sequence of positive real numbers with $\delta _1<1/2.$ There exist a sequence $(m_n)$ of integers and an increasing sequence $(k_n)$ of integers (with $k_0=0$) such that

$$\begin{aligned} \left| \zeta (\sigma +im_l)\right| \le \frac{\log (l+2)}{l^{1-\delta _n}}\hbox { for }l=1+k_{n-1},\dots ,k_n, \end{aligned}$$

and the polynomials $Q_n(z)=\sum _{l=1+k_{n-1}}^{k_n}\zeta (\sigma +im_l)l^{-z}$ have the following approximation property

$$\begin{aligned} \sup _{z\in K_{n}}\left| Q_j(z)-g_{n}(z)+Q_1(z)+\dots +Q_{n-1}(z)\right| \le \frac{2}{n}. \end{aligned}$$

Proof

First we apply Lemma 5.1 to find a Dirichlet polynomial $w_1=a_{1,1}+a_{1,2}2^{-z}+\dots +a_{1,k_1}k_1^{-z}$ such that $\vert a_{1,l}\vert \le l^{-1+\delta _1}$, for $l=1,\dots ,k_1$$\sup _{z\in K_{1}}\vert w_1(z)-g_{1}(z)\vert <\frac{1}{2}.$ By Theorem 5.2 there are integers $m_1,\dots ,m_{k_1}$ such that

$$\begin{aligned}\vert \zeta (\sigma +im_l)\vert \le \log (l+2)l^{-1+\delta _1} \hbox { for }l=1,\dots ,k_1, \end{aligned}$$

and

$$\begin{aligned} \sup _{z\in K_{1}}\vert Q_1(z)-g_{1}(z)\vert <1, \end{aligned}$$

with $Q_1(z)=\sum _{l=1}^{k_1}\zeta (\sigma +im_l)l^{-z}.$ By induction, suppose for $j=1,\dots ,n-1$ we have built integers $k_1,\dots , k_{n-1},$ and $m_1,\dots ,m_{k_{n-1}}$ such that

$$\begin{aligned} \left| \zeta (\sigma +im_l)\right| \le \frac{\log (l+2)}{l^{1-\delta _j}}\hbox { for }l=1+k_{j-1},\dots ,k_j, \end{aligned}$$

and the polynomials $Q_j(z)=\sum _{l=1+k_{j-1}}^{k_j}\zeta (\sigma +im_l)l^{-z}$ have the following approximation property

$$\begin{aligned} \sup _{z\in K_{j}}\left| Q_j(z)-g_{j}(z)+Q_1(z)+\dots +Q_{j-1}(z)\right| \le \frac{2}{j}. \end{aligned}$$

We apply Lemma 5.1 again to find a Dirichlet polynomial $w_n(z)=a_{n,k_{n-1}+1}(k_{n-1}+1)^{-z}+\dots +a_{n,k_n}k_n^{-z}$ satisfying $\vert a_{n,l}\vert \le l^{-1+\delta _1}$, for $l=k_{n-1}+1,\dots ,k_n$$\sup _{z\in K_{n}}\vert w_n(z)-g_{n}(z)\vert <\frac{1}{n}.$ By Theorem 5.2 there are integers $m_{k_{n-1}+1},\dots ,m_{k_n}$ such that

$$\begin{aligned}\vert \zeta (\sigma +im_l)\vert \le \log (l+2)l^{-1+\delta _n} \hbox { for }l=k_{n-1}+1,\dots ,k_n, \end{aligned}$$

and

$$\begin{aligned} \sup _{z\in K_{n}}\vert Q_n(z)-g_{n}(z)+Q_1(z)+\dots +Q_{n-1}(z)\vert <\frac{2}{n}, \end{aligned}$$

with $Q_n(z)=\sum _{l=k_{n-1}+1}^{k_n}\zeta (\sigma +im_l)l^{-z}.$ This finishes the proof. $\square$

Now the following statement holds.

Theorem 5.4

Let $\sigma$ be a real number with $1/2<\sigma <1$. There is a sequence $(m_n)$ of integers such that the series $\sum _{n\ge 1}\zeta (\sigma +i m_n)n^{-z}$ is an universal Dirichlet series in ${\mathcal {D}}_a({\mathbb {C}}_+)$.

Proof

Let $(\delta _n)$ be a decreasing sequence of positive real numbers with $\delta _1<1/2.$ Let us consider the Dirichlet series $f(z)=\sum _{l=1}^{+\infty }\zeta (\sigma +i m_l)l^{-z}$ given by Lemma 5.3. Observe that

$$\begin{aligned} f(z)=\sum _{n=1}^{+\infty }\sum _{l=k_{n-1}+1}^{k_n}\zeta (\sigma +i m_l)l^{-z}. \end{aligned}$$

Moreover for all $\tau >0,$ there exists $n_{\tau },$ so that for every integer $n\ge n_{\tau },$$\delta _n<\tau .$ Set $\varepsilon _{\tau }=\tau -\delta _{n_{\tau }}>0$. Thus we have

$$\begin{aligned} \begin{array}{rcl}\displaystyle \Vert f\Vert _{\tau }&{}=&{}\displaystyle \Vert \sum _{n=1}^{n_{\tau }}\sum _{l=k_{n-1}+1}^{k_n}\zeta (\sigma +i m_l)l^{-z} \Vert _{\tau } +\sum _{l=k_{n_{\tau }}}^{+\infty }\vert \zeta (\sigma +i m_l)\vert l^{-\tau }\\ {} &{}\le &{} \displaystyle \sum _{l=1}^{k_{n_{\tau }}}\vert \zeta (\sigma +i m_l)\vert l^{-\tau }+ \sum _{l=k_{n_{\tau }}}^{+\infty }\log (l+2)l^{-\varepsilon _{\tau }-1}<+\infty .\end{array} \end{aligned}$$

Hence f belongs to ${\mathcal {D}}_a({\mathbb {C}}_+).$ Now let $K\subset \overline{{\mathbb {C}}}_{-}$ be a compact set with connected complement, P be a Dirichlet polynomial and $\varepsilon >0.$ By definition, there exists a positive integer N such that for all $n\ge N,$$K\subset K_n$ and one can find $n_1\ge \max (N,4/\varepsilon )$ such that $\sup _{K}\vert P-g_{n_1}\vert <\varepsilon /2.$ By construction, the partial sum $\sum _{l=1}^{k_{n_1}}\zeta (\sigma +i m_l)l^{-z}$ satisfies

$$\begin{aligned} \sup _{z\in K_{n_1}}\left| \sum _{l=1}^{k_{n_1}}\zeta (\sigma +i m_l)l^{-z}-g_{n_1}(z)\right|< \frac{2}{n_1}<\frac{\varepsilon }{2}. \end{aligned}$$

Therefore we have by the triangle inequality $\sup _{z\in K}\left| \sum _{l=1}^{k_{n_1}}\zeta (\sigma +i m_l)l^{-z}-P(z)\right| <\varepsilon ,$ and f is an universal Dirichlet series. $\square$

6 Open problems

We conclude the paper with some open problems.

• (1) Let $K\in \overline{{\mathbb {C}}}_{-}$ be a compact set with connected complement and $\delta >0.$ We know that, for every $N\ge 1,$ the set $\left\{ \sum _{n=N}^{M}a_n n^{-\left( 1-\delta \right) }n^{-s};M\ge N,\vert a_n\vert \le 1\right\}$ is dense in A(K). On the other hand, Corollary 4.6 ensures that this result does not hold anymore if one replaces $\delta$ by a sequence $(\delta _n)$ defined by $\delta _n=\left( \log _2(n)\dots \log _{k-1}(n)\log _k^{1+\tau }(n)\right) ^{-1}$ (for all $k\ge 2$ and $\tau >0$). Nevertheless the following problem remains open: is the set of Dirichlet polynomials $\left\{ \sum _{n=N}^{M}a_n n^{-\left( 1-\frac{1}{\log _2(n+2)}\right) }n^{-s};M\ge N,\vert a_n\vert \le 1\right\}$ dense in A(K), for all $N\ge 1$?

• (2) Let $\sigma <0,$ does there exist a Dirichlet series $\sum _{k\ge 1}{a_k}k^{-s}$ belonging to ${\mathcal {D}}_a({\mathbb {C}}_+)$ which is universal in the strip $\{s\in {\mathbb {C}}:\sigma \le \mathfrak {R}(s)\le 0\},$ satisfying $\sum _{k=1}^n a_k k^{-s}\rightarrow \infty ,$ as n tends to infinity, almost everywhere in the half-plane $\{z\in {\mathbb {C}}:\mathfrak {R}(s)<\sigma \}$?

• (3) An universal Dirichlet series f cannot be logarithmically summable at any point of its line of convergence. However we don’t know whether the sequence of its logarithmic means $\left( \left( \frac{1}{\log (n)}\right) \sum _{k=1}^n k^{-1} D_k(f)(s)\right)$ still satisfies the universal approximation property.

• (4) The sequence of Cesàro means of partial sums of an universal Dirichlet series remains universal, but it would be interesting to know whether the converse implication holds.