Ordinary Dirichlet Series Research Articles

VARIANTS OF A MULTIPLIER THEOREM OF KISLYAKOV

AbstractWe prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislyakov and the Kahane–Salem–Zygmund inequality. As a by-product, we show various multiplier theorems for spaces of trigonometric polynomials on the n-dimensional torus $\mathbb {T}^n$ or Boolean cubes $\{-1,1\}^N$ . Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application, we show that various recent $\ell _1$ -multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane–Salem–Zygmund inequality.

On the Non-Trivial Zeros of Dirichlet Functions

The purpose of this research is to extend to the functions obtained by meromorphic continuation of general Dirichlet series some properties of the functions in the Selberg class, which are all generated by ordinary Dirichlet series. We wanted to put to work the powerful tool of the geometry of conformal mappings of these functions, which we built in previous research, in order to study the location of their non-trivial zeros. A new approach of the concept of multiplier in Riemann type of functional equation was necessary and we have shown that with this approach the non-trivial zeros of the Dirichlet function satisfying a Reimann type of functional equation are either on the critical line, or they are two by two symmetric with respect to the critical line. The Euler product general Dirichlet series are defined, a wide class of such series is presented, and finally by using geometric and analytic arguments it is proved that for Euler product functions the symmetric zeros with respect to the critical line must coincide.

Vector-valued general Dirichlet series

With early contributions due to, among others, Besicovitch, Bohr, Bohnenblust, Hardy, Hille, Riesz, Neder and Landau, the last 20 years show a substantial revival of systematic research on ordinary Dirichlet series $\sum a_n n^{-s}$, and more recently eve

A two-variable Dirichlet series and its applications

We define a two-variable Dirichlet series associated with two arithmetic functions, which is related to the Riemann zeta function, the Dirichlet L-function, the Dirichlet series associated to the harmonic numbers, and truncated multiple zeta functions. Using the periodic Euler-Maclaurin summation formula, we obtain a representation in terms of an ordinary Dirichlet series, which leads to the explicit evaluation of its values at nonpositive integers. We also find a reciprocity formula, which provides some symmetric formulas involving Bernoulli and associated numbers.

Joint universality of periodic zeta-functions with multiplicative coefficients

The periodic zeta-function is defined by the ordinary Dirichlet series with periodic coefficients. In the paper, joint universality theorems on the approximation of a collection of analytic functions by nonlinear shifts of periodic zeta-functions with multiplicative coefficients are obtained. These theorems do not use any independence hypotheses on the coefficients of zeta-functions.

Variants of a theorem of Helson on general Dirichlet series

A result of Helson on λ-Dirichlet series ∑ane−λns states that, whenever (an) is 2-summable and λ=(λn) satisfies a certain condition introduced by Bohr, then for almost all homomorphism ω:(R,+)→T the Dirichlet series ∑anω(λn)e−λns converges on the open right half plane [Re>0]. For ordinary Dirichlet series ∑ann−s Hedenmalm and Saksman related this result with the famous Carleson-Hunt theorem on pointwise convergence of Fourier series, and Bayart extended it within his theory of Hardy spaces Hp of such series. The aim here is to prove variants of Helson's theorem within our recent theory of Hardy spaces Hp(λ),1≤p≤∞, of λ-Dirichlet series. To be more precise, in the reflexive case 1<p<∞ we extend Helson's result to Dirichlet series in Hp(λ) without any further condition on the frequency λ, and in the non-reflexive case p=1 to the wider class of frequencies satisfying the so-called Landau condition (more general than Bohr's condition). In both cases we add relevant maximal inequalities. Finally, we try to indicate how our study of Helson's theorem can be employed to understand that for a fixed but arbitrary frequency λ certain key stones of the structure theory of λ-Dirichlet series in fact turn out to be equivalent.

Riesz means in Hardy spaces on Dirichlet groups

Given a frequency $$\lambda =(\lambda _n)$$ , we study when almost all vertical limits of a $$\mathcal {H}_1$$ -Dirichlet series $$\sum a_n e^{-\lambda _ns}$$ are Riesz summable almost everywhere on the imaginary axis. Equivalently, this means to investigate almost everywhere convergence of Fourier series of $$H_1$$ -functions on so-called $$\lambda $$ -Dirichlet groups, and as our main technical tool we need to invent a weak-type $$(1, \infty )$$ Hardy-Littlewood maximal operator for such groups. Applications are given to $$H_1$$ -functions on the infinite dimensional torus $$\mathbb {T}^\infty $$ , ordinary Dirichlet series $$\sum a_n n^{-s}$$ , as well as bounded and holomorphic functions on the open right half plane, which are uniformly almost periodic on every vertical line.

Convergence formulae for double Dirichlet series

Regular convergence of double Dirichlet series was originally studied by Kojima (Rep Tohoku Univ 9:351–400, 1920), where the author proved a formula for what he called the related abscissae of regular convergence, allowing him to describe the sets of regular convergence of such double series in an extensive study. In this work we give a shorter and more comprehensive proof for his main result, a formula for the related abscissae. We also give two new formulae which are easier to work with and which extend the traditional formulae of convergence for the abscissa of an ordinary Dirichlet series to the double case. These new formulae can be adapted for the general double ones with some particular frequencies which grow at a pace very similar to the pace at which the sequence $$\{\log n\}_n$$ grows. Finally, we apply these new formulae to give examples of double Dirichlet series whose sets of regular convergence are not trivially obtained from the one variable case.

$$\pmb {{\mathcal {H}}_{p}}$$-Theory of General Dirichlet Series

Inspired by results of Bayart on ordinary Dirichlet series $$\sum a_n n^{-s}$$, the main purpose of this article is to start an $${\mathcal {H}}_p$$-theory of general Dirichlet series $$\sum a_n e^{-\lambda _{n}s}$$. Whereas the $${\mathcal {H}}_p$$-theory of ordinary Dirichlet series, in view of an ingenious identification of Bohr, may be seen as a sub-theory of Fourier analysis on the infinite dimensional torus $${\mathbb {T}}^\infty $$, the $${\mathcal {H}}_p$$-theory of general Dirichlet series is build as a sub-theory of Fourier analysis on certain compact abelian groups, including the Bohr compactification $${\overline{{\mathbb {R}}}}$$ of the reals. Our approach allows to extend various important facts on Hardy spaces of ordinary Dirichlet series to the much wider setting of $${\mathcal {H}}_p$$-spaces of general Dirichlet series.

Hardy spaces of general Dirichlet series — a survey

The main purpose of this article is to survey on some key elements of a recent $\mathcal{H}_p$-theory of general Dirichlet series $\sum a_n e^{-\lambda_{n}s}$, which was mainly inspired by the work of Bayart and Helson on ordinary Dirichlet series $\sum a_n n^{-s}$. In view of an ingenious identification of Bohr, the $\mathcal{H}_p$-theory of ordinary Dirichlet series can be seen as a sub-theory of Fourier analysis on the infinite dimensional torus $\mathbb{T}^\infty$. Extending these ideas, the $\mathcal{H}_p$-theory of $\lambda$-Dirichlet series is build as a sub-theory of Fourier analysis on what we call $\lambda$-Dirichlet groups. A number of problems is added.

On the meromorphic continuation of Beatty Zeta-functions and Sturmian Dirichlet series

For a positive irrational number α, we study the ordinary Dirichlet series ζα(s)=∑n≥1⌊αn⌋−s and Sα(s)=∑n≥1(⌈αn⌉−⌈α(n−1)⌉)n−s. We prove relations between them and Jα(s)=∑n≥1({αn}−12)n−s. Motivated by the previous work of Hardy and Littlewood, Hecke and others regarding Jα, we show that ζα and Sα can be continued analytically beyond the imaginary axis except for a simple pole at s=1. Based on the latter results, we also prove that the series ζα(s;β)=∑n≥0(⌊αn⌋+β)−s can be continued analytically beyond the imaginary axis except for a simple pole at s=1.

Geometric Aspects of Quasi-Periodic Property of Dirichlet Functions

The concept of quasi-periodic property of a function has been introduced by Harald Bohr in 1921 and it roughly means that the function comes (quasi)-periodically as close as we want on every vertical line to the value taken by it at any point belonging to that line and a bounded domain Ω. He proved that the functions defined by ordinary Dirichlet series are quasi-periodic in their half plane of uniform convergence. We realized that the existence of the domain Ω is not necessary and that the quasi-periodicity is related to the denseness property of those functions which we have studied in a previous paper. Hence, the purpose of our research was to prove these two facts. We succeeded to fulfill this task and more. Namely, we dealt with the quasi-periodicity of general Dirichlet series by using geometric tools perfected by us in a series of previous projects. The concept has been applied to the whole complex plane (not only to the half plane of uniform convergence) for series which can be continued to meromorphic functions in that plane. The question arise: in what conditions such a continuation is possible? There are known examples of Dirichlet series which cannot be continued across the convergence line, yet there are no simple conditions under which such a continuation is possible. We succeeded to find a very natural one.

Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables

Let $\mathcal{H}_\infty$ be the set of all ordinary Dirichlet series $D=\sum_n a_n n^{-s}$ representing bounded holomorphic functions on the right half plane. A multiplicative sequence $(b_n)$ of complex numbers is said to be an $\ell_1$-multiplier for $\mathcal{H}_\infty$ whenever $\sum_n |a_n b_n| < \infty$ for every $D \in \mathcal{H}_\infty$. We study the problem of describing such sequences $(b_n)$ in terms of the asymptotic decay of the subsequence $(b_{p_j})$, where $p_j$ denotes the $j$th prime number. Given a multiplicative sequence $b=(b_n)$ we prove (among other results): $b$ is an $\ell_1$-multiplier for $\mathcal{H}_\infty$ provided $|b_{p_j}| < 1$ for all $j$ and $\overline{\lim}_n \frac{1}{\log n} \sum_{j=1}^n b_{p_j}^{*2} < 1$, and conversely, if $b$ is an $\ell_1$-multiplier for $\mathcal{H}_\infty$, then $|b_{p_j}| < 1$ for all $j$ and $\overline{\lim}_n \frac{1}{\log n} \sum_{j=1}^n b_{p_j}^{*2} \leq 1$ (here $b^*$ stands for the decreasing rearrangement of $b$). Following an ingenious idea of Harald Bohr it turns out that this problem is intimately related with the question of characterizing those sequences $z$ in the infinite dimensional polydisk $\mathbb{D}^\infty$ (the open unit ball of $\ell_\infty$) for which every bounded and holomorphic function $f$ on $\mathbb{D}^\infty$ has an absolutely convergent monomial series expansion $\sum_{\alpha} \frac{\partial_\alpha f(0)}{\alpha!} z^\alpha$. Moreover, we study analogous problems in Hardy spaces of Dirichlet series and Hardy spaces of functions on the infinite dimensional polytorus $\mathbb{T}^\infty$.

Weak Product Spaces of Dirichlet Series

Let $\mathscr{H}^2$ denote the space of ordinary Dirichlet series with square summable coefficients, and let $\mathscr{H}^2_0$ denote its subspace consisting of series vanishing at $+\infty$. We investigate the weak product spaces $\mathscr{H}^2\odot\mathscr{H}^2$ and $\mathscr{H}^2_0\odot\mathscr{H}^2_0$, finding that several pertinent problems are more tractable for the latter space. This surprising phenomenon is related to the fact that $\mathscr{H}^2_0\odot\mathscr{H}^2_0$ does not contain the infinite-dimensional subspace of $\mathscr{H}^2$ of series which lift to linear functions on the infinite polydisc. The problems considered stem from questions about the dual spaces of these weak product spaces, and are therefore naturally phrased in terms of multiplicative Hankel forms. We show that there are bounded, even Schatten class, multiplicative Hankel forms on $\mathscr{H}^2_0 \times \mathscr{H}^2_0$ whose analytic symbols are not in $\mathscr{H}^2$. Based on this result we examine Nehari's theorem for such Hankel forms. We define also the skew product spaces associated with $\mathscr{H}^2\odot\mathscr{H}^2$ and $\mathscr{H}^2_0\odot\mathscr{H}^2_0$, with respect to both half-plane and polydisc differentiation, the latter arising from Bohr's point of view. In the process we supply square function characterizations of the Hardy spaces $\mathscr{H}^p$, for $0 < p < \infty$, from the viewpoints of both types of differentiation. Finally we compare the skew product spaces to the weak product spaces, leading naturally to an interesting Schur multiplier problem.

On the functional properties of Bessel zeta-functions

Zeta-functions associated with modified Bessel functions are introduced as ordinary Dirichlet series whose coefficients are $J$-Bessel and $K$-Bessel functions. Integral representations, transformation formulas, a power series expansion involving the Riem

Bohr’s absolute convergence problem for ℋp-Dirichlet series in Banach spaces

The Bohr-Bohnenblust-Hille Theorem states that the width of the strip in the complex plane on which an ordinary Dirichlet series $\sum_n a_n n^{-s}$ converges uniformly but not absolutely is less than or equal to 1/2, and this estimate is optimal. Equivalently, the supremum of the absolute convergence abscissas of all Dirichlet series in the Hardy space $\mathcal{H}_\infty$ equals 1/2. By a surprising fact of Bayart the same result holds true if $\mathcal{H}_\infty$ is replaced by any Hardy space $\mathcal{H}_p$, $1 \le p < \infty$, of Dirichlet series. For Dirichlet series with coefficients in a Banach space $X$ the maximal width of Bohr's strips depend on the geometry of $X$; Defant, Garc\'ia, Maestre and P\'erez-Garc\'ia proved that such maximal width equal $1- 1/\ct(X)$, where $\ct(X)$ denotes the maximal cotype of $X$. Equivalently, the supremum over the absolute convergence abscissas of all Dirichlet series in the vector-valued Hardy space $\mathcal{H}_\infty(X)$ equals $1- 1/\ct(X)$. In this article we show that this result remains true if $\mathcal{H}_\infty(X)$ is replaced by the larger class $\mathcal{H}_p(X)$, $1 \le p < \infty$.

Construction of an Ordinary Dirichlet Series with Convergence Beyond the Bohr Strip

An ordinary Dirichlet series has three abscissae of interest, describing the maximal regions where the Dirichlet series converges, converges uniformly, and converges absolutely. The paper of Hille and Bohnenblust in 1931, regarding the region on which a Dirichlet series can converge uniformly but not absolutely, has prompted much investigation into this region, the However, a related natural question has apparently gone unanswered: For a Dirichlet series with non-trivial Bohr strip, how far beyond the Bohr strip might the series converge? We investigate this question by explicit construction, creating Dirichlet series which converge beyond their Bohr strip.

Orthogonal Dirichlet polynomials with arctangent density

Let {λj}j=1∞ be a strictly increasing sequence of positive numbers with λ1=1. We find a simple explicit formula for the orthogonal Dirichlet polynomials {ϕn} formed from linear combinations of {λj−it}j=1n, associated with the arctangent density. Thus ∫−∞∞ϕn(t)ϕm(t)¯dtπ(1+t2)=δmn. We obtain formulae for their Christoffel functions, and deduce their asymptotics, as well as universality limits, and spacing of zeros for their reproducing kernels. We also investigate the relationship between ordinary Dirichlet series, and orthogonal expansions involving the {ϕn}, and establish Markov–Bernstein inequalities.

Extending Landau's Theorem on Dirichlet Series with Non-Negative Coefficients

A classical theorem of Landau states that, if an ordinary Dirichlet series has non-negative coefficients, then it has a singularity on the real line at its abscissa of convergence. In this article, we relax the condition on the coefficients while still arriving at the same conclusion. Specifically, we write $a_n$ as $|a_n| e^{i \theta _n}$ and we consider the sequences $\{ |a_n| \}$ and $\{ \cos{\theta _n} \}$. Let $M \in \mathbb{N}$ be given. The condition on $\{ |a_n| \}$ is that, dividing the sequence sequentially into vectors of length $M$, each vector lies in a certain convex cone $B \subset [0,\infty)^M$. The condition on $\{ \cos{\theta _n} \}$ is (roughly) that, again dividing the sequence sequentially into vectors of length $M$, each vector lies in the negative of the polar cone of $B$. We demonstrate the additional freedom allowed in choosing the $\theta _n$, compared to Landau's Theorem. We also obtain sharpness results.

On the boundary behaviour of the Hardy spaces of Dirichlet series and a frame bound estimate

A range of Hardy-like spaces of ordinary Dirichlet series, called the Dirichlet–Hardy spaces ℋp, p ≧ 1, have been the focus of increasing interest among researchers following a paper of Hedenmalm, Lindqvist and Seip [16]. The Dirichlet series in these spaces converge on a certain half-plane, where one may also define the classical Hardy spaces Hp. In this paper, we compare the boundary behaviour of elements in ℋp and Hp. Moreover, Carleson measures for the spaces ℋp are studied. Our main result shows that for certain cases the following statement holds true. Given an interval on the boundary of the half-plane of definition and a function in the classical Hardy space, it is possible to find a function in the corresponding Dirichlet–Hardy space such that their difference has an analytic continuation across this interval.