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In this paper, we consider meromorphic extension of the function ζ h r s = ∑ k = 1 ∞ h k r k s , Re s > r (which we call hyperharmonic zeta function) where h n (r) are the hyperharmonic numbers. We establish certain constants, denoted γ h r m , which naturally occur in the Laurent expansion of ζ h r s . Moreover, we show that the constants γ h r m and integrals involving the generalized exponential integral can be written as a finite combination of some special constants. [ABSTRACT FROM AUTHOR]

We derive explicit closed-form formulas for the standard Clausen functions C l 2 m (x) and C l 2 m − 1 (x) in terms of the Hurwitz zeta function, where m is a positive integer. [ABSTRACT FROM AUTHOR]

It is known that multiple zeta values whose weight and depth are of opposite parity can be written in terms of multiple zeta values of lower depth. This theorem is called parity result. Multiple zeta values are special values of the multiple polylogarithms and the parity result is generalized to functional relations satisfied by the multiple polylogarithms. In this paper, we consider q- and elliptic generalizations of the parity result. As a main result of this paper, we establish parity result for functions L k (a , α ; p , q) , which can be considered to be common deformations of q- and elliptic multiple polylogarithms. By taking the trigonometric and classical limits in the main theorem, we obtain q- and elliptic analogues of the parity result. [ABSTRACT FROM AUTHOR]

We construct an interesting family of unbounded sets at infinity having a quasiperiodic structure in their geometry. We study these sets by analyzing their complex dimensions—a generalization of the classical Minkowski dimension, which are defined analytically—as poles of a suitably defined Lapidus zeta function at infinity introduced in a previous work by the author. We define the tube zeta function at infinity and obtain a functional equation that relates it to the Lapidus zeta function at infinity much as in the classical setting of relative fractal drums. Furthermore, under suitable assumptions, we provide some general results about extending the fractal zeta functions at infinity beyond their abscissae of convergence. We also provide a sufficiency condition for Minkowski measurability as well as a bound from above for the upper Minkowski content directly from the corresponding zeta functions at infinity. We show that complex dimensions of quasiperiodic sets at infinity possess a quasiperiodic structure which can be either algebraic or transcendental. Furthermore, we use the quasiperiodic sets at infinity to construct a maximally hyperfractal set at infinity with prescribed Minkowski dimension, i.e., a set for which the abscissa of convergence of the associated fractal zeta function at infinity becomes its natural boundary. [ABSTRACT FROM AUTHOR]

Suppose that π is a unitary cuspidal automorphic representation of G L r (A Q) and L (s , π) is the automorphic L -function related to π , For 1 2 < σ < 1 , let m (σ) ≥ 2 be the supremum of all numbers m such that ∫ 1 T | L (s , π) | m d t ≪ π , ε T 1 + ε. We are interested in the lower bound of m (σ) for 1 - 1 r < σ < 1 . Then as an application, we establish asymptotic formulas for the second, fourth and sixth power moments of L (s , π) . [ABSTRACT FROM AUTHOR]

In 2020 S. M. Gonek, S. W. Graham and Y. Lee formulated the Lindelöf hypothesis for prime numbers and proved that it is equivalent to the Riemann Hypothesis. In this note we show that their result holds in the Selberg class of L-functions. [ABSTRACT FROM AUTHOR]

We generalize a theorem by Titchmarsh about the mean value of Hardy's Z -function at the Gram points to the Hecke L -functions, which in turn implies the weak Gram law for them. Instead of proceeding analogously to Titchmarsh with an approximate functional equation we employ a different method using contour integration. [ABSTRACT FROM AUTHOR]

We construct an absolutely convergent Dirichlet series connected to the classical Hurwitz zeta-function. The shifts of this function approximate analytic functions defined in the right-hand side of the critical strip. [ABSTRACT FROM AUTHOR]

Dirichlet series associated with the Poincaré series attached to SL (2 , Z) are introduced. Integral representations and transformation formulas are given, which describe the Voronoï-type summation formula for the exponential-type generating function of the Riemann zeta-function. As an application, a new proof of the Fourier series expansion of holomorphic Poincaré series is given. [ABSTRACT FROM AUTHOR]

Let N be a finitely generated nilpotent group. The subgroup zeta function ζ N ⩽ (s) and the normal zeta function ζ N ⊲ (s) of N are Dirichlet series enumerating the finite index subgroups or the finite index normal subgroups of N. We present results about their abscissae of convergence α N ⩽ and α N ⊲ , also known as the degrees of polynomial subgroup growth and polynomial normal subgroup growth of N, respectively. We first prove some upper bounds for the functions N ↦ α N ⩽ and N ↦ α N ⊲ when restricted to the class of torsion-free nilpotent groups of a fixed Hirsch length. We then show that if two finitely generated nilpotent groups have isomorphic C -Mal’cev completions, then their subgroup (resp. normal) zeta functions have the same abscissa of convergence. This follows, via the Mal’cev correspondence, from a similar result that we establish for zeta functions of rings. This result is obtained by proving that the abscissa of convergence of an Euler product of certain Igusa-type local zeta functions introduced by du Sautoy and Grunewald remains invariant under base change. We also apply this methodology to formulate and prove a version of our result about nilpotent groups for virtually nilpotent groups. As a side application of our result about zeta functions of rings, we present a result concerning the distribution of orders in number fields. [ABSTRACT FROM AUTHOR]

We proof a Garrett–Böcherer decomposition of a vector-valued Siegel Eisenstein series E l , 0 2 of genus 2 transforming with the Weil representation of Sp 2 (Z) on the group ring C [ (L ′ / L) 2 ] . We show that the standard zeta function associated to a vector-valued common eigenform f for the Weil representation can be meromorphically continued to the whole s-plane and that it satisfies a functional equation. The proof is based on an integral representation of this zeta function in terms of f and E l , 0 2 . [ABSTRACT FROM AUTHOR]

We provide a simple proof of a Voronoi type identity using the Laplace transform for a class of arithmetical functions whose generating Dirichlet series satisfy a functional equation with gamma factors. As an application, we derivea Voronoi-type identity for Gaussian integers. [ABSTRACT FROM AUTHOR]

In this paper, we prove that fractal zeta functions of orbits of parabolic germs of diffeomorphisms can be meromorphically extended to the whole complex plane. We describe their set of poles (i.e. their complex dimensions) and their principal parts which can be understood as their fractal footprint. We study the fractal footprint of one orbit of a parabolic germ f and extract intrinsic information about the germ f from it, in particular, its formal class. Moreover, we relate complex dimensions to the generalized asymptotic expansion of the tube function of orbits with oscillatory ’coefficients’ as well as to the asymptotic expansion of their dynamically regularized tube function. Interestingly, parabolic orbits provide a first example of sets that have nontrivial Minkowski (or box) dimension and their tube function possesses higher order oscillatory terms, however, they do not posses non-real complex dimensions and are therefore not called fractal in the sense of Lapidus. [ABSTRACT FROM AUTHOR]

Let λ π (r , n) be the Fourier coefficients of a Hecke-Maass cusp form π for S L (3 , Z) and λ f (n) be the Fourier coefficients of a Hecke-eigen form f for S L (2 , Z) . The aim of this article is to get a non-trivial bound on the sum which is non-linear additive twist of the coefficients λ π (m , n) and λ f (n) . More precisely, we have ∑ n = 1 ∞ λ π (r , n) e α n β V n X ≪ π , ϵ α β r 7 6 X 3 4 + 9 β 28 + ϵ. and ∑ n = 1 ∞ λ π (r , n) λ f (n) e α n β V n X ≪ π , f , ϵ (α β) 3 2 r X 3 4 + 29 β 44 + ϵ , where V(x) is a smooth function supported in [1, 2] and satisfying V (j) (x) ≪ j 1 . [ABSTRACT FROM AUTHOR]

We study the analogue of the Montgomery's pair correlation function F (α) for a family of Γ 1 (q) L-functions. Assuming the Generalized Riemann Hypothesis, we evaluate this analogous pair correlation function in the range | α | < 2 , where the off-diagonal terms also contribute to the main term. As applications, we obtain that more than 93.5% of the low-lying zeros of Γ 1 (q) L-functions are simple, and we achieve upper and lower bounds for the number of pairs of zeros which are closer than β times the average spacing apart. [ABSTRACT FROM AUTHOR]

In this article, we prove that a general version of Alladi's formula with Dirichlet convolution holds for arithmetical semigroups satisfying Axiom A or Axiom A # . As applications, we apply our main results to certain semigroups coming from algebraic number theory, arithmetical geometry, and graph theory. [ABSTRACT FROM AUTHOR]

In this paper, we prove the universality theorem for the iterated integrals of the logarithm of the Riemann zeta-function on a line parallel to the real axis. [ABSTRACT FROM AUTHOR]

We obtain an asymptotic formula to estimate the counting function of fixed points for the Riemann zeta function and its derivatives. A result on fixed points of more general Dirichlet series will also be given. [ABSTRACT FROM AUTHOR]

In this article, we will prove the following sub-convex bound for G L (3) × G L (2) Rankin Selberg L-functions. [ABSTRACT FROM AUTHOR]

Suppose ∑ n = 1 ∞ a (n) n - s be a Dirichlet series in the Selberg class of degree d and let E(x) be the arithmetical error term of ∑ n ⩽ x a (n) . By the truncated Tong-type formula of E(x), we can get two kinds of the mean square estimates of E(x) in short intervals of Jutila's type. Using the estimates, we are able to improve some previous results established by Lü and Wang [9]. [ABSTRACT FROM AUTHOR]

The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem. [ABSTRACT FROM AUTHOR]

By the Mishou theorem (Mishou in Lith Math J 47:32–47, 2007), the set of shifts (ζ (s + i τ) , ζ (s + i τ , α)) , τ ∈ R , of the Riemann zeta-function and Hurwitz zeta-function with transcendental parameter simultaneously approximating a pair of analytic functions has a positive lower density. In the paper, two absolutely convergent Dirichlet series close to the Riemann and Hurwitz zeta-functions are constructed, and it is proved that the set of their shifts approximating a pair of analytic functions has a positive density. For the proof, probabilistic arguments are applied. [ABSTRACT FROM AUTHOR]

We give explicit evaluations of the linear and non-linear Euler sums of hyperharmonic numbers h n r with reciprocal binomial coefficients. These evaluations enable us to extend closed form formula of Euler sums of hyperharmonic numbers to an arbitrary integer r. Moreover, we reach at explicit formulas for the shifted Euler-type sums of harmonic and hyperharmonic numbers. All the evaluations are provided in terms of the Riemann zeta values, harmonic numbers and linear Euler sums. [ABSTRACT FROM AUTHOR]

Let Z(s) be the Selberg zeta-function for the modular group. We consider the existence of the second moments of Z(s) and of its reciprocal on σ = 1 . The existence of such moments is related to the properties of certain Beurling natural numbers. Here the behavior of the counting function and the distribution of minimal gaps between these Beurling natural numbers are important. We also obtain unconditional upper bounds for these moments. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study Ramanujan sums c J (I) , where I and J are integral ideals in an arbitrary quadratic number field. In particular, the asymptotic behaviour of sums of c J (I) over both I and J is investigated. [ABSTRACT FROM AUTHOR]

In 1985, Howard introduced degenerate Cauchy polynomials together with degenerate Bernoulli polynomials. His degenerate Bernoulli polynomials have been studied by many authors, but his degenerate Cauchy polynomials have been forgotten. In this paper, we introduce some kinds of hypergeometric degenerate Cauchy numbers and polynomials from the different viewpoints. By studying the properties of the first one, we give their expressions and determine the coefficients. Concerning the second one, called H-degenerate Cauchy polynomials, we show several identities and study zeta functions interpolating these polynomials.