Related papers: Dirichlet eta and beta functions at negative integ…
We propose a generic algorithm for computing the inverses of a multiplicative function under the assumption that the set of inverses is finite. More generally, our algorithm can compute certain functions of the inverses, such as their power…
An explicit identity of sums of powers of complex functions presented via this a closed-form formula of Riemann zeta function produced at any given non-zero complex numbers. The closed-form formula showed us Riemann zeta function has no…
We present a unified approach which gives completely elementary proofs of three weighted sum formulae for double zeta values. This approach also leads to new evaluations of sums relating to the harmonic numbers, the alternating double zeta…
We develop a systematic and fully explicit approach to the evaluation of binomial sums involving reciprocals of binomial coefficients based on Beta integral techniques. Starting from a simple integral representation, we provide a derivation…
We study a special Dirichlet series studied before by Apostol and Matsuoka and specify its values at negative integers. These values are related to a certain convolution of Bernoulli numbers
Explicit bounds on the tails of the zeta function $\zeta$ are needed for applications, notably for integrals involving $\zeta$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $\zeta$. Two…
The finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers $\beta$ having the negative finiteness property, that is the set of finite $(-\beta)$-expansions is equal to…
We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also…
Contour integral representations for Riemann's Zeta function and Dirichelet's Eta (alternating Zeta) function are presented and investigated. These representations flow naturally from methods developed in the 1800's, but somehow they do not…
In 1999, Arakawa and Kaneko introduced a zeta function whose special values at negative integers yield the poly-Bernoulli numbers and investigated its relation to multiple zeta values. Since the poly-Bernoulli numbers appear in this…
In this paper, we introduce and study the Dirichlet series enumerating (proper) equivalence classes of full rank subforms/sublattices of a given quadratic form/lattice, focusing on the positive definite binary case. We obtain formulas…
We study rather general multiple zeta-functions whose denominators are given by polynomials. The main aim is to prove explicit formulas for the values of those multiple zeta-functions at non-positive integer points. We first treat the case…
We develop a unified analytical and dynamical framework for the qualitative study of the one-parameter family of generalized Dirichlet eta functions $\eta_{a}(t)=\sum_{m\ge0}(-1)^{m}(am+1)^{-t}$, $a>0$, $t>0$, which includes the classical…
We consider the sum of the reciprocals of the middle prime factor of an integer, defined according to multiplicity or not. We obtain an asymptotic expansion in the first case and an asymptotic formula involving an implicit parameter in the…
We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special…
Let $\gamma$ denote imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Certain sums over the $\gamma$'s are evaluated, by using the function $G(s) = \sum_{\gamma>0}\gamma^{-s}$ and other techniques. Some integrals…
In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.
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 Dirichlet eta function can be divided into $n$-th partial sum $\eta_{n}(s)$ and remainder term $R_{n}(s)$. We focus on the remainder term which can be approximated by the expression for $n$. And then, to increase reliability, we make…
A sharp explicit estimate is proved for the difference $e^\beta-\alpha$ when $\alpha$ and $\beta$ are nonzero algebraic numbers.