Related papers: Parametric Euler $T$-sums of odd harmonic numbers
Let $T$ be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over $T$, one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in…
This paper investigates the Erd\H{o}s distinct subset sums problem in $\mathbb{Z}^k$. Beyond the classical variance method, using alternative statistical quantities like $\mathbb{E}[\|X\|_1]$ and $\mathbb{E}[\|X\|_3^3]$ can yield better…
We continue our study of the variation of parabolic cohomology (math.AG/0310139) and derive an exact formula for the underlying Poincare duality. As an illustration of our methods, we compute the monodromy of the Picard-Euler system and its…
An easy generalization of Beukers' integrals allows us to conjecture a double integral formula involving the zeta and the gamma functions. A special case of this formula is Sondow's double integral formula for Euler's constant gamma.
We prove a variety of explicit formulas relating special values of generalized hypergeometric functions to lattice sums with four indices of summation. These results are related to Boyd's conjectured identities between Mahler measures and…
In this paper we shall define a special-valued multiple Hurwitz zeta functions, namely the multiple $t$-values $t(\boldsymbol{\alpha})$ and define similarly the multiple star $t$-values as $t^{\star}(\boldsymbol{\alpha})$. Then we consider…
We show that Euler's relation and the Taxi-Cab relation are both solutions of the same equation. General solutions of sums of two consecutive cubes equaling the sum of two other cubes are calculated. There is an infinite number of relations…
This paper, first, we consider the Volterra integral equation for the remainder term in the asymptotic formula for the associated Euler totient function. Secondly, we solve the Volterra integral equation and we split the error term in the…
We present several sequences involving harmonic numbers and the central binomial coefficients. The calculational technique is consists of a special summation method that allows, based on proper two-valued integer functions, to calculate…
In this paper, we consider a "compensated" random sum that arises from numerical approximation of stochastic integrations and differential equations. We show that the compensated sum exhibits some surprising cancellations among its…
We present algorithms to evaluate two types of multiple sums, which appear in higher-order loop computations. We consider expansions of a generalized hypergeometric-type sums, $\sum_{n_1,...,n_N} [Gamma(a1.n+c1) Gamma(a2.n}+c2) ...…
In this paper we investigate congruence relationships of particular finite generalized harmonic numbers sums. We suggest more transparent and simpler method to analyse these sums and present several additional results for certain special…
We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on…
Summation formulas, such as the Euler-Maclaurin expansion or Gregory's quadrature, have found many applications in mathematics, ranging from accelerating series, to evaluating fractional sums and analyzing asymptotics, among others. We show…
We give a new proof for the parabolic Verlinde formula in all ranks based on a comparison of wall-crossings in Geometric Invariant Theory and certain iterated residue functionals. On the way, we develop a tautological variant of Hecke…
An integral formula is given representing the generalized principal Lyapunov estimate for random linear parabolic PDEs. As an application, an upper estimate of the exponent is obtained.
An interplay between the Lambert series and Euler's Pentagonal Number Theorem gives an Euler-type recurrence relation for any given arithmetical function. As consequences of this, we present Euler-type recurrence relations for some…
Euler had considered the problem of finding three integers whose sum, product, and also the sum of the products of the integers, taken two at a time, are all perfect squares. Euler's methods of solving the problem lead to parametric…
In this paper, by generalizing Bombieri and Iwaniec's double large sieve inequality, we obtain an estimation on a type of three-dimensional exponential sums with constant perturbation. As an application of our estimation, we obtain an…
We give a parameterized generalization of the sum formula for quadruple zeta values. The generalization has four parameters, and is invariant under a cyclic group of order four. By substituting special values for the parameters, we also…