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 in $σ\in (0,1)$.
Moreover, we prove that $Q(s,a)$ has infinitely many complex zeros in the region of absolute convergence and the critical strip when $a \in {\mathbb{Q}} \cap (0,1/2) \setminus \{1/6, 1/4, 1/3\}$. The Lerch formula, Hadamard product formula, Riemann-von Mangoldt formula for $Q(s,a)$ are also shown.
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.
The probabilistic study of the value-distributions of zeta-functions is one of the modern topics in analytic number theory. In this paper, we study a certain probability measure related to the value-distribution of the Lerch zeta-function. We prove that it has a density function which we can explicitly construct. Moreover, we prove an asymptotic formula for the number of zeros of the Lerch zeta-function on the right side of the critical line, whose main term is associated with the density function.
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 span…
This is the first of four papers that study algebraic and analytic structures associated to the Lerch zeta function. This paper studies "zeta integrals" associated to the Lerch zeta function using test functions , and obtains functional equations for them. Special cases include a pair of symmetrized four-term functional equations for combinations of Lerch zeta functions, found by A. Weil, for real parameters (a,c) with 0< a, c< 1. It extends these functions to real a, and c, and studies limiting cases of these functions where at least one of a and c takes the values 0 or 1. A main feature is t…
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.
A. Speiser proved that the Riemann hypothesis is equivalent to the absence of non-real zeros of the derivative of the Riemann zeta-function left of the critical line. His result has been extended by N. Levinson and H.L. Montgomery to the statement that the Riemann zeta-function and its derivative have approximately the same number of non-real zeros left of the critical line. We obtain the Levinson-Montgomery type result for the Lerch zeta-function with equal parameters. For the Lerch zeta-function, the analogue of the Riemann hypothesis is usually not true and its zeros usually are distributed…
For most values of parameters $λ$ and $α$, the zeros of the Lerch zeta-function $L(λ, α, s)$ are distributed very chaotically. In this paper we consider the special case of equal parameters $L(λ, λ, s)$ and show by calculations that the nontrivial zeros either lie extremely close to the critical line $σ= 1/2$ or are distributed almost symmetrically with respect to the critical line. We also investigate this phenomenon theoretically.
Starting from a recent result expressing the Lerch zeta function as a fractional derivative, we consider further fractional derivatives of the Lerch zeta function with respect to different variables. We establish a partial differential equation, involving an infinite series of fractional derivatives, which is satisfied by the Lerch zeta function.
We provide efficient methods to evaluate the Riemann zeta, the Lerch zeta and the Dirichlet $L$-functions. The method uses the Riemann-Siegel (RS) type formulas and a modified double exponential (MDE) quadrature method near the saddle point of appropriate integrands. We provide simplified derivations of the RS formulas, containing finite series sums and residual integrals, for the Lerch and the Dirichlet $L$-functions. The MDE method allows us to remove the contribution of the singularities near the contour of integration for the residual integrals. The method allows us to evaluate the residua…
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）
We derive and prove a new formulation of the Lerch zeta function as a fractional derivative of an elementary function. We demonstrate how this formulation interacts very naturally with basic known properties of Lerch zeta, and use the functional equation to obtain a second formulation in terms of fractional derivatives.
This paper gives a representation-theoretic interpretation of the Lerch zeta function and related Lerch $L$-functions twisted by Dirichlet characters. These functions are associated to a four-dimensional solvable real Lie group $H^{J}$, called here the sub-Jacobi group, which is a semi-direct product of $GL(1, {\mathbb R})$ with the Heisenberg group $H({\mathbb R})$. The Heisenberg group action on L^2-functions on the Heisenberg nilmanifold $H({\mathbb Z}) \backslash H({\mathbb R})$ decomposes as $\bigoplus_{N \in {\mathbb Z}} H_N$, where each space $H_N~ (N \neq 0)$ consists of $|N|$ copies o…
For $0 < a \le 1/2$, we define the quadrilateral zeta function $Q(s,a)$ using the Hurwitz and periodic zeta functions and show that $Q(s,a)$ satisfies Riemann's functional equation studied by Hamburger, Heck and Knopp. Moreover, we prove that for any $0 < a \le 1/2$, there exist positive constants $A(a)$ and $T_0(a)$ such that the number of zeros of the quadrilateral zeta function $Q(s,a)$ on the line segment from $1/2$ to $1/2 +iT$ is greater than $A(a) T$ whenever $T \ge T_0(a)$.
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…
The classical Lerch's formula states the following normalized product: $$\prod_{n=0}^\infty(x+n)=\frac{\sqrt{2π}}{Γ(x)},\quad \textrm{Re}(x)>0, $$ where $Γ(x)$ is the Euler gamma function. In this note, by using Barnes' multiple zeta function and its alternating form, we obtain two kinds of generalizations of Lerch's formula, which imply the product $$ \prod_{n=1}^\infty n=\sqrt{2π} $$ (in the sense of zeta regularization) and the product $$\frac{2\cdot2}{1\cdot 3}\frac{4\cdot4}{3\cdot 5}\frac{6\cdot6}{5\cdot 7}\cdots=\fracπ{2}$$ (Wallis' formula in 1656), respectively.
For a complex variable $s$ and real parameters $a$ and $λ$ with $a>0$, let $ϕ(s,a,λ)$ denote the Lerch zeta-function with a complex variable, $ϕ^{\ast}(s,a,λ)$ a slight modification of $ϕ(s,a,λ)$ defined by extracting the (possible) singularity of $ϕ(s,a,λ)$ at $s=1$, and $(ϕ^{\ast})^{(m)}(s,a,λ)$ for any $m\in\mathbb{Z}$ the $m$th derivative with respect to $s$ if $m\geq0$, while if $m\leq0$ the $|m|$-th primitive defined with its initial point at $s+\infty$. The present paper aims to study asymptotic aspects of $(ϕ^{\ast})^{(m)}(s,a,λ)$, transformed through the Laplace-Mellin and Riemann-Lio…
In this paper, we show that all real zeros of the bilateral Hurwitz zeta function $Z(s,a):=ζ(s,a) + ζ(s,1-a)$ with $1/4 \le a \le 1/2$ are on only the non-positive even integers exactly same as in the case of $(2^s-1) ζ(s)$. We also prove that all real zeros of the bilateral periodic zeta function $P(s,a):={\rm{Li}}_s (e^{2πia}) + {\rm{Li}}_s (e^{2πi(1-a)})$ with $1/4 \le a \le 1/2$ are on only the negative even integers just like $ζ(s)$. Moreover, we show that all real zeros of the quadrilateral zeta function $Q(s,a):=Z(s,a) + P(s,a)$ with $1/4 \le a \le 1/2$ are on only the negative even int…
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 provide a multidimensional weighted Euler--MacLaurin summation formula on polytopes and a multidimensional generalization of a result due to L. J. Mordell on the series expansion in Bernoulli polynomials. These results are consequences of a more general series expansion; namely, if $χ_{τ\mathcal{P}}$ denotes the characteristic function of a dilated integer convex polytope $\mathcal{P}$ and $q$ is a function with suitable regularity, we prove that the periodization of $qχ_{τ\mathcal{P}}$ admits an expansion in terms of multivariate Bernoulli polynomials. These multivariate polynomials are re…
