Using the Jackson integral, we obtain the $q$-integral analogue of the Raabe type formulas for Barnes multiple Hurwitz-Lerch zeta functions and Barnes and Vardi's multiple gamma functions. Our results generalize $q$-integral analogue of the Raabe type formulas for the Hurwitz zeta functions and log gamma functions in [N. Kurokawa, K. Mimachi, and M. Wakayama, Jackson's integral of the Hurwitz zeta function, Rend. Circ. Mat. Palermo (2) 56 (2007), no. 1, 43--56]. During the proof we also obtain a new formula on the relationship between the higher and lower orders Hurwitz zeta functions.
We give a result on the asymptotic behavior of the Hurwitz-Lerch multiple zeta functions near non-positive integer points by using the Apostol-Bernoulli polynomials. From this result, we can evaluate limit values at non-positive integer points.
A failed attempt to prove the universality of Lerch zeta function $L(λ,α,s)$ when $λ$ is irrational and $α$ is rational, and for any $λ$ when $α$ is irrational algebraic.
In this paper, we give Hurwitz-Lerch zeta distributions with $0 < σ\ne 1$ by using the Gamma function. Moreover, we define Hurwitz-Lerch type of Euler-Zagier double zeta distributions not only in the region of absolute convergence but also the outside of the region of absolute convergence.
We study analytic properties of multiple zeta-functions of generalized Hurwitz-Lerch type. First, as a special type of them, we consider multiple zeta-functions of generalized Euler-Zagier-Lerch type and investigate their analytic properties which were already announced in our previous paper. Next we give `desingularization' of multiple zeta-functions of generalized Hurwitz-Lerch type, which include those of generalized Euler-Zagier-Lerch type, the Mordell-Tornheim type, and so on. As a result, the desingularized multiple zeta-function turns out to be an entire function and can be expressed as…
As one of the asymptotic formulas for the zeta-function, Hardy and Littlewood gave asymptotic formulas called the approximate functional equation. In 2003, R. Garunkštis, A. Laurinčikas, and J. Steuding (in [1]) proved the Riemann-Siegel type of the approximate functional equation for the Lerch zeta-function $ ζ_L (s, α, λ) = \sum_{n=0}^\infty e^{2πi n λ}(n + α)^{-s} $. In this paper, we prove another type of approximate functional equations for the Hurwitz and Lerch zeta-functions. R. Garunkštis, A. Laurinčikas, and J. Steuding (in \cite{GLS2}) obtained the results on the mean square values o…
Let $0 < a \le 1$, $s,z \in {\mathbb{C}}$ and $0 < |z|\le 1$. Then the Hurwitz-Lerch zeta function is defined by $Φ(s,a,z) := \sum_{n=0}^\infty z^n(n+a)^{-s}$ when $σ:=\Re (s) >1$. In this paper, we show that the Hurwitz zeta function $ζ(σ,a) := Φ(σ,a,1)$ does not vanish for all $0 <σ<1$ if and only if $a \ge 1/2$. Moreover, we prove that $Φ(σ,a,z) \ne 0$ for all $0 <σ<1$ and $0 < a \le 1$ when $z \ne 1$. Real zeros of Hurwitz-Lerch type of Euler-Zagier double zeta functions are studied as well.
For $0 < a \le 1$, $s,z \in {\mathbb{C}}$ and $0 < |z|\le 1$, the Hurwitz-Lerch zeta function is defined by $Φ(s,a,z) := \sum_{n=0}^\infty z^n(n+a)^{-s}$ when $σ:=\Re (s) >1$. In this paper, we show that $Φ(σ,a,z) \ne 0$ when $σ\in (-1,0)$ if and only if [I] $z=1$ and $(3-\sqrt{3}) /6 \le a \le 1/2$ or $(3+\sqrt{3}) /6 \le a \le 1$, [II] $z \in [-1,1)$ and $(1-z)(1-a) \le 1$, [III] $z \not \in {\mathbb{R}}$ and $0<a \le 1$. In addition, we give a new proof of the functional equation of $Φ(s,a,z)$.
Four explicit continued fractions for values of the Lerch transcendent and the Hurwitz zeta function
Institute of Mathematics, Vilnius University（维尔纽斯大学数学研究所）；St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院列别捷夫数学研究所圣彼得堡分部）
In this article, we derive meromorphic continuation of multiple Lerch zeta functions by generalising an elegant identity of Ramanujan. Further, we describe the set of all possible singularities of these functions. Finally, for the multiple Hurwitz zeta functions, we list the exact set of singularities.
In this note we prove that for all $a \in \mathbb{N}$, $x \in \mathbb{R}_+ \cup \{0\}$, and $s \in \mathbb{C}$ with $\Re(s) > a + 2$, the (alternating) weighted series of the Hurwitz zeta function, $$ \sum_{k \geq 1} (\pm 1)^k (k + x)^aζ(s,k + x), $$ resolves into a finite combination of Hurwitz (Lerch) zeta functions. This applies in Marichal and Zenaïdi's theory on analogues of the Bohr-Mollerup theorem for higher-order convex functions.
We consider the series $\sum_{n=1}^{\infty} z^{n} (a_{n} + x)^{-s}$ where $a_{n}$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under appropriate conditions, we prove that it can be continued to a meromorphic function on the complex $s$-plane. Thus we may associate a Lerch-type zeta function $φ(z,s,x)$ to a general recurrence. This subsumes all previous results which dealt only with the ordinary zeta and Hurwitz cases and degrees $2$ and $3$. Our method generalizes a formula of Ramanujan for the classical Hurwitz-Riemann zeta functions. We determine the poles and…
This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of two-variable Hecke operators $\{ T_m: \, m \ge 1\}$ given by $T_m(f)(a, c) = \frac{1}{m} \sum_{k=0}^{m-1} f(\frac{a+k}{m}, mc)$ acting on certain spaces of real-analytic functions, including Lerch zeta functions for various parameter values. It determines the action of various related operators on these function spaces. It characterizes Lerch zeta functions (for fixed $s$ in the following way. It shows that there is for each $s \in {\bf C}$ a two-dimensional vector space spanned…
Let $0 < a \le 1/2$ and define the quadrilateral zeta function by $2Q(s,a) := ζ(s,a) + ζ(s,1-a) + {\rm{Li}}_s (e^{2πia}) + {\rm{Li}}_s(e^{2πi(1-a)})$, where $ζ(s,a)$ is the Hurwitz zeta function and ${\rm{Li}}_s (e^{2πia})$ is the periodic zeta function. In the present paper, we show that there exists a unique real number $a_0 \in (0,1/2)$ such that $Q(σ, a_0)$ has a unique double real zero at $σ= 1/2$ when $σ\in (0,1)$, for any $a \in (a_0,1/2]$, the function $Q(σ, a)$ has no zero in the open interval $σ\in (0,1)$ and for any $a \in (0,a_0)$, the function $Q(σ, a)$ has at least two real zeros…
The Hurwitz-type Euler zeta function is defined as a deformation of the Hurwitz zeta function: \begin{equation*} ζ_E(s,x)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+x)^s}. \end{equation*} In this paper, by using the method of Fourier expansions, we shall evaluate several integrals with integrands involving Hurwitz-type Euler zeta functions $ζ_E(s,x)$. Furthermore, the relations between the values of a class of the Hurwitz-type (or Lerch-type) Euler zeta functions at rational arguments have also been given.
We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within the article satisfy expansions by generalized harmonic number sequences, or the partial sums of the Hurwitz zeta function, which are analogous to known properties for the Stirling numbers of the first kind and for the known transformation coefficients employed to enumerate variants of the polylogarithm function series. Applications of the new results we prove …
The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet $L$ series, to Eisenstein series, and to general Mellin transforms. The Hurwitz number…
The need to evaluate Logarithmic integrals is ubiquitous in essentially all quantitative areas including mathematical sciences, physical sciences. Some recent developments in Physics namely Feynman diagrams deals with the evaluation of complicated integrals involving logarithmic functions. This work deals with a systematic review of logarithmic integrals starting from Malmsten integrals to classical collection of Integrals, Series and Products by I. S. Gradshteyn and I. M. Ryzhik [1] to recent times. The evaluation of these types of integrals involves higher transcendental functions (i.e., Hur…
The standard twist $F(s,α)$ of $L$-functions $F(s)$ in the Selberg class has several interesting properties and plays a central role in the Selberg class theory. It is therefore natural to study its finer analytic properties, for example the functional equation. Here we deal with a special case, where $F(s)$ satisfies a functional equation with the same $Γ$-factor of the $L$-functions associated with the cusp forms of half-integral weight; for simplicity we present our results directly for such $L$-functions. We show that the standard twist $F(s,α)$ satisfies a functional equation reflecting $…
