For 0 < a ? 1 / 2 , we define the quadrilateral zeta function Q ( s ) using Hurwitz and periodic functions show that satisfies Riemann's functional equation studied by Hamburger, Heck Knopp. Moreover, prove for any there exist positive constants A T such number of zeros on line segment from 1/2 to + i is greater than whenever ? .

We study the extended Euler sums and the alternating extended Euler sums and establish their explicit expressions in terms of Riemann zeta functions and Hurwitz zeta functions. Comparing with the existing results, ours are simpler and thus yield significantly better accuracy when Matlab is used for numerical calculation.

As the first step of research on functional equations for multiple zeta-functions, we present a candidate of the functional equation for a class of two variable double zeta-functions of the Hurwitz–Lerch type, which includes the classical Euler sum as a special case.

A generalized modular relation of the form $$F(z, w, \alpha )=F(z, iw,\beta )$$ , where $$\alpha \beta =1$$ and $$i=\sqrt{-1}$$ is obtained in course evaluating an integral involving Riemann $$\Xi $$ -function. This involves a surprising new generalization Hurwitz zeta function $$\zeta (s, a)$$ which we denote by _w(s, . We show that satisfies beautiful theory generalizing In particular, it sho...

We use a variant of a method of Goncharov, Kontsevich, and Zhao [Go2, Z] to meromorphically continue the multiple Hurwitz zeta function ζd(s; θ) = ∑ 0<n1<···<nd (n1 + θ1) −s1 · · · (nd + θd)d , θk ∈ [0, 1), to C, to locate the hyperplanes containing its possible poles, and to compute the residues at the poles. We explain how to use the residues to locate trivial zeros of ζd(s; θ).

We present a variety of series representations of the Stieltjes and related constants, the Stieltjes constants being the coefficients of the Laurent expansion of the Hurwitz zeta function ζ(s, a) about s = 1. Additionally we obtain series and integral representations of a sum Sγ(n) formed as an alternating binomial series from the Stieltjes constants. The slowly varying sum Sγ(n) + n is an impo...