In this note, we study the Li coefficients $\lambda _{n,a}$ for the quadrilateral zeta function. Furthermore, we give an arithmetic and asymptotic formula for these coefficients. Especially, we show that for any fixed $n \in {\mathbb {N}}$, there exists $a>0$ such that $\lambda _{2n-1,a}> 0$ and $\lambda _{2n,a} < 0$.

We study the moments $M_k(T;\,\alpha) = \int_T^{2T} |\zeta(s,\alpha)|^{2k}\,dt$ of the Hurwitz zeta function $\zeta(s,\alpha)$ on the critical line, $s = 1/2 + it$ with a rational shift $\alpha \in \mathbb{Q}$ . We conjecture, in analogy with the Riemann zeta function, that $M_k(T;\,\alpha) \sim c_k(\alpha) T (\!\log T)^{k^2}$ . Using heuristics from analytic number theory and random matrix theory, we conjecturally compute $c_k(\alpha)$ . In the process, we investigate moments of products of Dirichlet L-functions on the critical line. We prove some of our conjectures for the cases $k = 1,2$ .

We introduce interpolated multiple Hurwitz polylogs and interpolated multiple Hurwitz zeta values. In addition, we discuss the generating functions for the sum of the polylogs/zeta values of fixed weight, depth, and all heights. The functions are expressed in terms of generalized hypergeometric functions. Compared with the pioneering results of Ohno and Zagier on the generating function, our setup generalizes the results in three directions, namely, at general heights, with a t-interpolation, and as a Hurwitz type. As an application, by fixing the Hurwitz parameter to rational numbers, the generating functions for multiple zeta values with level are given.

We characterize nonempty open subsets of the complex plane where the sum $\zeta (s, \alpha )+ {e}^{\pm i\pi s} \hspace{0.167em} \zeta (s, 1- \alpha )$ of Hurwitz zeta functions has no zeros in $s$ for all $0\leq \alpha \leq 1$. This problem is motivated by the construction of fundamental cardinal splines of complex order $s$.

In this paper we present new explicit simultaneous rational approximations which converge subexponentially to the values of the Bell polynomials at the points where m=1,2,…,a, a∈ℕ, γ is Euler’s constant and ζ is the Riemann zeta function.

We prove a joint universality theorem in the Voronin sense for the periodic Hurwitz zeta-functions.

We provide concise series representations for various $\text{L}$-series integrals. Different techniques are needed below and above the abscissa of absolute convergence of the underlying $\text{L}$-series.

Let

$$\zeta \left( s,x \right)=\sum\limits_{n=0}^{\infty }{\frac{1}{{{\left( n+x \right)}^{s}}}}\left( s>1,x>0 \right)$$

be the Hurwitz zeta function and let

$$Q\left( x \right)=Q\left( x;\alpha ,\beta ;a,b \right)=\frac{{{\left( \zeta \left( \alpha ,x \right) \right)}^{a}}}{{{\left( \zeta \left( \beta ,x \right) \right)}^{{{b}'}}}}$$

where $\alpha ,\beta >1$ and $a,b>0$ are real numbers. We prove: (i) The function $Q$ is decreasing on $\left( 0,\infty \right)$ iff $\alpha a-\beta b\ge \max \left( a-b,0 \right)$. (ii) $Q$ is increasing on $\left( 0,\infty \right)$ iff $\alpha a-\beta b\le \min \left( a-b,0 \right)$. An application of part (i) reveals that for all $x>0$ the function $s\mapsto {{\left[ \left( s-1 \right)\zeta \left( s,x \right) \right]}^{1/\left( s-1 \right)}}$ is decreasing on $\left( 1,\infty \right)$. This settles a conjecture of Bastien and Rogalski.

We slightly modify the definitions of q-Hurwitz ζ-functions and q-L-series constructed by J. Satoh. By using these modified functions, we give some relations for the ordinary Dirichlet L-series. Especially we give an elementary proof of Katsurada’s formula on the values of Dirichlet L-series at positive integers.

The Hurwitz zeta function ζ(s, a) is defined by the series

for 0 < a ≤ 1 and σ = Re(s) > 1, and can be continued analytically to the whole complex plane except for a simple pole at s = 1 with residue 1. The integral functions C(s, a) and S(s, a) are defined in terms of the Hurwitz zeta function as follows:

Using integral representations of C(s, a) and S(s, a), we evaluate explicitly a class of improper integrals. For example if 0 < a < 1 we show that

Let d(n; l, k) be the number of positive divisors of n which lie in the arithmetic progression l mod k. Using the complex integration technique the formula

is proved. This formula holds uniformly in l, k and x satisfying 1 ≦ l ≦ k, (lx)1/2 ≦ k ≦ x1-∊; the exponent α ≦ 1/3.