Related papers: Special values of Kloosterman sums and binomial be…

Assuming the Riemann hypothesis, we obtain a formula for the mean value of the $k$-derivative of $\zeta'/\zeta$, depending on the pair correlation of zeros of the Riemann zeta-function. This formula allows us to obtain new equivalences to…

For $q$ prime, $X \geq 1$ and coprime $u,v \in \mathbb{N}$ we estimate the sums \begin{equation*} \sum_{\substack{p \leq X \substack p \equiv u \hspace{-0.25cm} \mod{v} p \text{ prime}}} \text{Kl}_2(p;q), \end{equation*} where…

A classical result of A. Fleck states that if p is a prime, and n>0 and r are integers, then $$\sum_{k=r(mod p)}\binom {n}{k}(-1)^k=0 (mod p^{[(n-1)/(p-1)]}).$$ Recently R. M. Wilson used Fleck's congruence and Weisman's extension to…

We begin with the observation that the signed generalized Stirling polynomials $P_k(m,x)$, which occur in a generalization of Malmsten's integral, reduce to the falling factorials when $k=m$. The structure of these generalized Stirling…

We revisit a recent bound of I. Shparlinski and T. P. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to…

Multiple q-zeta values are a 1-parameter generalization (in fact, a q-analog) of the multiple harmonic sums commonly referred to as multiple zeta values. These latter are obtained from the multiple q-zeta values in the limit as q tends to…

Let $\zeta(.)$ denote the Riemann zeta function and let $a(.)$ and $A(.)$ respectively denote a multiplicative function and its corresponding summatory function. We consider the correlation $$ \langle a(n)A(n-1) \rangle (T) =…

We establish a series of indefinite integral formulae involving the Hurwitz zeta function and other elementary and special functions related to it, such as the Bernoulli polynomials, ln sin (\pi q), ln Gamma(q) and the polygamma functions.…

We introduce and study "elliptic zeta values", a two-parameter deformation of the values of Riemann's zeta function at positive integers. They are essentially Taylor coefficients of the logarithm of the elliptic gamma function, and share…

The necessary and sufficient conditions for a class of functions $f:\mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$, where $q \geq 2$ is an even positive integer, have been recently identified for $q=4$ and $q=8$. In this article we give an…

We prove that every multiple zeta value is a $\mathbb{Z}$-linear combination of $\zeta(k_1,\dots, k_r)$ where $k_i\geq 2$. Our proof also yields an explicit algorithm for such an expansion. The key ingredient is to introduce modified…

The fundamental objective of this paper is to obtain some interesting properties for $\left(h,q\right)$-Genocchi numbers and polynomials by using the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ and mentioned in the paper…

We give new bounds for $\sum_{{a, m ,n}}\alpha_{m}\beta_n\nu_a {\textrm e}\left(\frac{a\overline m}{n}\right)$ where $\alpha_{m}$, $\beta_n$ and $\nu_a$ are arbitrary coefficients, improving upon a result of Duke, Friedlander and Iwaniec…

The Stieltjes constants $\gamma_k(a)$ appear in the regular part of the Laurent expansion for the Hurwitz zeta function $\zeta(s,a)$. We present summatory results for these constants $\gamma_k(a)$ in terms of fundamental mathematical…

The alternating zeta function zeta*(s) = 1 - 2^{-s} + 3^{-s} - ... is related to the Riemann zeta function by the identity (1-2^{1-s})zeta(s) = zeta*(s). We deduce the vanishing of zeta*(s) at each nonreal zero of the factor 1-2^{1-s}…

In this paper, we construct three ternary linear codes associated with the orthogonal group O^-(2,q) and the special orthogonal groups SO^-(2,q) and SO^-(4,q). Here q is a power of three. Then we obtain recursive formulas for the power…

We consider the class $\Sigma(p)$ of univalent meromorphic functions $f$ on $\ID$ having simple pole at $z=p\in[0,1)$ with residue 1. Let $\Sigma_k(p)$ be the class of functions in $\Sigma(p)$ which have $k$-quasiconformal extension to the…

We revisit several hybrid multiplicative-to-additive type functions from a recent preprint article. These functions, $g(n)$ with Dirichlet generating function (DGF) $\zeta(s)^{-1} (1+P(s))^{-1}$ for $\Re(s) > 1$ where $P(s) = \sum_p p^{-s}$…

In this paper, we give explicit expressions about $q$-harmonic sums on $1-\cdots-1,A,1-\cdots-1$ indices. When $A=1$, many previous authors have studied and showed the identities, expressions, and properties. There are many results for…

Let $q=p^n$ with $n=2m$ and $p$ be an odd prime. Let $0\leq k\leq n-1$ and $k\neq m$. In this paper we determine the value distribution of following exponential(character) sums \[\sum\limits_{x\in \bF_q}\zeta_p^{\Tra_1^m (\alpha…