A lower bound for the dimension of the $\Q$-vector space spanned by special values of a Dirichlet series with periodic coefficients is given. As a corollary, it is deduced that both special values at even integers and at odd integers contain infinitely many irrational numbers. This result is proved by T.Rivoal if the function considered is the Riemann zeta function, and this paper gives its generalization to more general Dirichlet series.
The study of linear independence of $L(k, χ)$ for a fixed integer $k>1$ and varying $χ$ depends critically on the parity of $k$ vis-à-vis $χ$. This has been investigated by a number of authors for Dirichlet characters $χ$ of a fixed modulus and having the same parity as $k$.The focal point of this article is to extend this investigation to families of Dirichlet characters modulo distinct pairwise co-prime natural numbers. The interplay between the resulting ambient number fields brings in new technical issues and complications hitherto absent in the context of a fixed modulus (consequently a s…
Katz showed that the L-functions of all Dirichlet characters of F_q(t), with conductor a fixed power of a degree one prime, are equidistributed in the limit as q goes to infinity. We generalize this statement to the L-functions of twists of an arbitrary Galois representation by Dirichlet characters, including independence of the L-functions of twists of different representations by the same Dirichlet character. A similar generalization, without the independence statement, for characters with squarefree conductor, was proven by Hall, Keating, and Roditty-Gershon.
We investigate special values of Goss L-functions for Dirichlet characters at s=1 over rings of class number one and prove results on their transcendence and algebraic independence.
The aim of this paper is to provide an analogue of the Ball-Rivoal theorem for $p$-adic $L$-values of Dirichlet characters. More precisely, we prove for a Dirichlet character $χ$ and a number field $K$ the formula $\dim_{K}(K+\sum_{i=2}^{s+1} L_p(i,χω^{1-i}) K )\geq \frac{(1-ε)\log (s)}{2[K:\mathbb{Q}](1+\log 2)}$. As a byproduct, we establish an asymptotic linear independence result for the values of the $p$-adic Hurwitz zeta function.
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.
In this paper, we give Dirichlet series with periodic coefficients that have Riemann's functional equation and real zeros of Dirichlet $L$-functions. The details are as follows. Let $L(s,χ)$ be the Dirichlet $L$-function and $G(χ)$ be the Gauss sum associate with a primitive Dirichlet character $χ$ (${\rm{mod}} \,\, q$). Put $f (s,χ) := q^s L(s,χ) + i^{-κ(χ)} G(χ) L(s,\overlineχ)$, where $\overlineχ$ is the complex conjugate of $χ$ and $κ(χ) :=(1-χ(-1))/2$. Then we prove that $f (s,χ)$ satisfies Riemann's functional equation appearing in Hamburger's theorem if $χ$ is even. In addition, we show…
We address the question of non-vanishing of $L(1,f)$ where $f$ is an algebraic-valued, periodic arithmetical function. We do this by characterizing algebraic-valued, periodic functions $f$ for which $L(1,f)=0$. The case of odd functions was resolved by Baker, Birch and Wirsing in 1973. We apply a result of Bass to obtain a characterization for the even functions. We also describe a theorem of the first two authors which says that it is enough to consider only the even and the odd functions in order to obtain a complete characterization.
For any periodic function $f:{\mathbb N} \to {\mathbb C}$ with period $q$, we study the Dirichlet series $L(s,f):=\sum_{n\geq 1} f(n)/n^s.$ It is well-known that this admits an analytic continuation to the entire complex plane except at $s=1$, where it has a simple pole with residue $$ρ:= q^{-1}\sum_{1\leq a\leq q} f(a).$$ Thus, the function is analytic at $s=1$ when $ρ=0$ and in this case, we study its non-vanishing using the theory of linear forms in logarithms and Dirichlet $L$-series. In this way, we give new proofs of an old criterion of Okada for the non-vanishing of $L(1,f)$ as well as …
Let $a=(a_n)_{n\ge 1}$ be a periodic sequence, $F_a(s)$ the meromorphic continuation of $\sum_{n\ge 1} a_n/n^s$, and $N_a(σ_1, σ_2, T)$ the number of zeros of $F_a(s)$, counted with their multiplicities, in the rectangle $σ_1 < \Re s < σ_2$, $|\Im s | \le T$. We extend previous results of Laurinčikas, Kaczorowski, Kulas, and Steuding, by showing that if $F_a(s)$ is not of the form $P(s) L_χ (s)$, where $P(s)$ is a Dirichlet polynomial and $L_χ(s)$ a Dirichlet L-function, then there exists an $η=η(a)>0$ such that for all $1/2 < σ_1 < σ_2 < 1+η$, we have $c_1 T \le N_a(σ_1, σ_2, T) \le c_2 T$ fo…
We generalize and unify the proofs of several results on algebraic in- dependence of arithmetic functions and Dirichlet series by a theorem of Ax on differential Schanuel conjecture. Along the way, we find counter-examples to some results in the literature.
For a rational valued periodic function, we associate a Dirichlet series and provide a new necessary and sufficient condition for the vanishing of this Dirichlet series specialized at positive integers. This question was initiated by Chowla, and carried out by Okada for a particular infinite sum. Our approach relies on the decomposition of the Dirichlet characters in terms of primitive characters. Using this, we find some new family of natural numbers for which a conjecture of Erdös holds. We also characterize rational valued periodic functions for which the associated Dirichlet series vanishe…
The Linear Independence hypothesis (LI), which states roughly that the imaginary parts of the critical zeros of Dirichlet L-functions are linearly independent over the rationals, is known to have interesting consequences in the study of prime number races, as was pointed out by Rubinstein and Sarnak. In this paper, we prove that a function field analogue of LI holds generically within certain families of elliptic curve L-functions and their symmetric powers. More precisely, for certain algebro-geometric families of elliptic curves defined over the function field of a fixed curve over a finite …
Series acceleration formulas are obtained for Dirichlet series with periodic coefficients. Special cases include Ramanujan's formula for the values of the Riemann zeta function at the odd positive integers exceeding two, and related formulas for values of Dirichlet L-series and the Lerch zeta function. Such formulas have been made famous by Lerch, Ramanujan, Grosswald, Berndt, and others. The approach herein is based on the partial fraction expansion of the hyperbolic cotangent, and as such is somewhat more elementary than previous approaches, yet is quite general.
The class of Dirichlet series associated with a periodic arithmetical function $f$ includes the Riemann zeta-function as well as Dirichlet $L$-functions to residue class characters. We study the value-distribution of these Dirichlet series $L(s;f)$, resp. their analytic continuation in the neighborhood of the critical line (which is the abscissa of symmetry of the related Riemann-type functional equation). In particular, for a fixed complex number $a\neq 0$, we prove for an even or odd periodic $f$ the number of $a$-points of the $Δ$-factor of the functional equation, prove the existence of th…
The purpose of this paper is to generalize our earlier work on the logarithm of the Riemann zeta-function to linear combinations of logarithms of primitive Dirichlet $L$-functions with constant real coefficients. Under the assumption of suitable hypotheses, we prove that as $T\to \infty $, a sequence of the form $(a_1\log|L(ρ,χ_n)|+\dots+a_1\log|L(ρ,χ_n)|)$ has an approximate Gaussian distribution with mean $0$ and variance $ \tfrac{1}{2}\big({a_1}^2+\dots+{a_n}^2 \big)\log\log T$. Here $a_1, \dots, a_n \in \mathbb{R}$, each of the $χ_i$ is a primitive Dirichlet character modulo $M_i$ with $M_…
In a recent paper with Sprang and Zudilin, the following result was proved: if $a$ is large enough in terms of $\varepsilon>0$, then at least $2^{(1-\varepsilon)\frac{\log a}{\log \log a}}$ values of the Riemann zeta function at odd integers between $3$ and $a$ are irrational. This improves on the Ball-Rivoal theorem, that provides only $\frac{1-\varepsilon}{1+\log 2} \log a$ such irrational values -- but with a stronger property: they are linearly independent over the rationals.In the present paper we generalize this recent result to both $L$-functions of Dirichlet characters and Hurwitz zeta…
We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $SL_3(\Zz)$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky--Nelson, corresponding to multiplicative Dirichlet characters, and applies in particular to trace functions of small conductor modulo primes.
Two remarks related with the mixed joint universality for a polynomial Euler product and a periodic Hurwitz zeta-function with a transcendental parameter are given. One is the mixed joint functional independence, and the other is a generalized universality, which includes several periodic Hurwitz zeta-functions.
