Related papers: Stringent bounds for the non-zero Bernoulli number…

In this paper, we present series representations of the remainders in the expansions for $2/(e^t+1)$, $\mbox{sech} t$ and $\coth t$. For example, we prove that for $t > 0$ and $N\in\mathbb{N}:=\{1, 2, \ldots\}$, \[\mbox{sech}\,…

We establish in this paper sharp lower bounds for the $2k$-th moment of the derivative of the Riemann zeta function on the critical line for all real $k \geq 0$.

Let $\Omega\subset \mathbb{R}^N$ ($N\geq 3$) be an open domain which is not necessarily bounded. The sharp constant and extremal functions to the following kind of double-variable inequalities $$ S_{\alpha,\beta,\lambda,\mu}(\Omega)…

We prove, assuming the Riemann Hypothesis, that \int_{T}^{2T} |\zeta(1/2+it)|^{2k} dt \ll_{k} T log^{k^{2}} T for any fixed k \geq 0 and all large T. This is sharp up to the value of the implicit constant. Our proof builds on well known…

The aim of this article is to generalize in several variables some formulae for Eisenstein series in one variable. For example the formula $2\zeta(2k) = (2\pi)^{2k} \frac{B_{2k}}{(2k)!} = Res_{z=0}(\frac{1}{z^{2k}(1-e^z)})$ for the values…

We introduce a Bernoulli operator,let $\mathbf{B}$ denote the operator symbol,for n=0,1,2,3,... let ${\mathbf{B}^n}: = {B_n}$ (where ${B_n}$ are Bernoulli numbers,${B_0} = 1,B{}_1 = 1/2,{B_2} = 1/6,{B_3} = 0$...).We obtain some formulas for…

We obtain lower bounds of the correct order of magnitude for the 2k-th moment of the Riemann zeta function for all k > 1. Previously such lower bounds were known only for rational values of k, with the bounds depending on the height of the…

We prove lower bounds for the discrete negative $2k$th moment of the derivative of the Riemann zeta function for all fractional $k\geqslant 0$. The bounds are in line with a conjecture of Gonek and Hejhal. Along the way, we prove a general…

In this paper, new sharp bounds for circular functions are proved. We provide some improvements of previous results by using infinite products, power series expansions and a generalisation of the so-called Bernoulli inequality. New proofs,…

Let $\Omega$ be a cone in $\mathbb{R}^{n}$ with $n\ge 2$. For every fixed $\alpha\in\mathbb{R}$ we find the best constant in the Rellich inequality $\int_{\Omega}|x|^{\alpha}|\Delta u|^{2}dx\ge C\int_{\Omega}|x|^{\alpha-4}|u|^{2}dx$ for…

In this paper, we find the greatest values $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$, $\alpha_{4}$, $\alpha_{5}$, $\alpha_{6}$, $\alpha_{7}$, $\alpha_{8}$ and the least values $\beta_{1}$, $\beta_{2}$, $\beta_{3}$, $\beta_{4}$, $\beta_{5}$,…

We obtain an improvement of the Beckner's inequality $\| f\|^{2}_{2} -\|f\|^{2}_{p} \leq (2-p) \| \nabla f\|_{2}^{2}$ valid for $p \in [1,2]$ and the Gaussian measure. Our improvement is essential for the intermediate case $p \in (1,2)$,…

In this note we establish a uniform bound for the distribution of a sum $S_n=X_1+\cdots+X_n$ of independent non-homogeneous Bernoulli trials. Specifically, we prove that $\sigma_n \mathbb{P}(S_n\!=\!j)\leq\eta$ where $\sigma_n$ denotes the…

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$…

T. Erd\'{e}lyi, A.P. Magnus and P. Nevai conjectured that for $\alpha, \beta \ge - {1/2} ,$ the orthonormal Jacobi polynomials ${\bf P}_k^{(\alpha, \beta)} (x)$ satisfy the inequality \begin{equation*} \max_{x \in…

We study long-range Bernoulli percolation on $\mathbb{Z}^d$ in which each two vertices $x$ and $y$ are connected by an edge with probability $1-\exp(-\beta \|x-y\|^{-d-\alpha})$. It is a theorem of Noam Berger (CMP, 2002) that if…

We prove that \[ \sum_{k,{\ell}=1}^N\frac{(n_k,n_{\ell})^{2\alpha}}{(n_k n_{\ell})^{\alpha}} \ll N^{2-2\alpha} (\log N)^{b(\alpha)} \] holds for arbitrary integers $1\le n_1<\cdots < n_N$ and $0<\alpha<1/2$ and show by an example that this…

The aim of this paper is to find the numerical solutions of the second order linear and nonlinear differential equations with Dirichlet, Neumann and Robin boundary conditions. We use the Bernoulli polynomials as linear combination to the…

We consider computing the Riemann zeta function $\zeta(s)$ and Dirichlet $L$-functions $L(s,\chi)$ to $p$-bit accuracy for large $p$. Using the approximate functional equation together with asymptotically fast computation of the incomplete…

A new definition for the Dirichlet beta function for positive integer arguments is discovered and presented for the first time. This redefinition of the Dirichlet beta function, based on the polygamma function for some special values,…