Related papers: Contour integration underlies fundamental Bernoull…

Let $(X,T,\mu,d)$ be a metric measure-preserving system for which $3$-fold correlations decay exponentially for Lipschitz continuous observables. Suppose that $(M_k)$ is a sequence satisfying some weak decay conditions and suppose there…

One familiar with the Euler zeta function, which established the remarkable relationship between the prime and composite numbers, might naturally ponder the results of the application of this special function in cases where there is no…

The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and Lison\v{e}k states that by inserting all cyclic permutations of some initial blocks of 2's into the multiple zeta value $ \zeta(1,3,\ldots,1,3) $ and summing, one obtains…

In this paper, by using the method of Contour Integral Representations and the Theorem of Residues and integral representations of series, we discuss the analytic representa- tions of parametric Euler sums that involve harmonic numbers…

In this paper, by introducing a new operation in the vector space of analytic functions, the author presents a method for derivating the well-known formulas: $\zeta(1-k)=-\frac{B_k}{k}$ and $\zeta(1-n,a)=-\frac{B_n(a)}{n}$ , where $\zeta$,…

We define an integral of real-valued functions with respect to a measure that takes its values in the extended positive cone of a partially ordered vector space $E$. The monotone convergence theorem, Fatou's lemma, and the dominated…

Summing the values of the real portion of the logarithmic integral of n^rho, where rho is one of a consecutive series of zeros of the Riemann zeta function, reveals an unexpected periodicity to the sum. This is the cycle problem.

The modified B_{n}^{*} = \sum_{r=0}^{n} \binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 introduced by D. Zagier in 1998 are extended to the polynomial case by replacing $B_{r}$ by the Bernoulli polynomials $B_{r}(x)$. Properties of these new…

This article gives a direct formula for the computation of B(n) using the asymptotic formula $$B (n) \approx 2 {\frac {n!}{{\pi}^{n}{2}^{n}}}$$ where n is even and $n >> 1$. This is simply based on the fact that $\zeta (n)$ is very near 1…

We believe that Euler constant is not just the "renormalized" value of the Riemann zeta function in 1. In a sense that we shall clarify it is in fact the normal and natural value of zeta of 1. In this paper we first propose a limit…

As a sequel to [1] and [2], I present some recent progress on Bessel integrals $\int_0^{\infty}{\rmd u}\; uK_0(u)^{n}$, $\int_0^{\infty}{\rmd u}\; u^{3}K_0(u)^{n}$, ... where the power of the integration variable is odd and where $n$, the…

This paper presents a new approach towards the Riemann Hypothesis. On iterative expansion of integration term in functional equation of the Riemann zeta function we get sum of two series functions. At the `non-trivial' zeros of zeta…

The author derives new family of series representations for the values of the Riemann Zeta function $\zeta(s)$ at positive odd integers. For $n\in\mathbb{N}$, each of these series representing $\zeta(2n+1)$ converges remarkably rapidly with…

Contour integration is a crucial technique in many numeric methods of interest in physics ranging from differentiation to evaluating functions of matrices. It is often important to determine whether a given contour contains any poles or…

It is known that the numbers which occur in Apery's proof of the irrationality of zeta(2) have many interesting congruence properties while the associated generating function satisfies a second order differential equation. We prove…

In this note, we show that the values of integrals of the log-tangent function with respect to any square-integrable function on $\left[0 , \frac{\pi}{2} \right]$ may be determined by a finite or infinite sum involving the Riemann…

Because of its relation to the distribution of prime numbers, the Riemann zeta function {\zeta} (s) is one of the most important functions in mathematics. The zeta function is defined by the following formula for any complex number s with…

To evaluate Riemann's zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special…

During a first St. Petersburg period Leonhard Euler, in his early twenties, became interested in the Basel problem: summing the series of inverse squares (posed by Pietro Mengoli in mid 17th century). In the words of Andre Weil (1989) "as…

Beuker's [2] considers the following integral $$ \int_{0}^{1}\int_{0}^{1} \frac{-\log xy}{1-xy} P_n(x)P_n(y)\ dx dy$$If $d_n=\text{LCM}(1,2,...,n)$, then $$ 0<\frac{|A_n+B_n\zeta(3)|}{d_n^3}<2(\sqrt{2}-1)^{4n} \zeta(3) $$ for some…