关于Bernoulli数及其性质

A prime sum involving Bernoulli numbers

In this note, we propose simple summations for primes, which involve two finite nested sums and Bernoulli numbers. The summations can also be expressed in terms of Bernoulli polynomials.

An introduction to the Bernoulli function

We explore a variant of the zeta function interpolating the Bernoulli numbers based on an integral representation suggested by J. Jensen. The Bernoulli function $\operatorname{B}(s, v) = - s\, ζ(1-s, v)$ can be introduced independently of the zeta function if it is based on a formula first given by Jensen in 1895. We examine the functional equation of $\operatorname{B}(s, v)$ and its representation by the Riemann $ζ$ and $ξ$ function, and recast classical results of Hadamard, Worpitzky, and Hasse in terms of $\operatorname{B}(s, v).$ The extended Bernoulli function defines the Bernoulli number…

素数和与伯努利多项式的同余性质

Congruence properties of prime sums and Bernoulli polynomials

本文研究涉及素数的特定和式的同余性质，利用伯努利数与伯努利多项式表达结果，并进一步导出一个一般的同余关系。

没有公式的伯努利双纽线

The Lemniscate of Bernoulli, Without Formulas

在本文中，我们给纯粹的几何证明著名的性质双纽线的伯努利。

Definitions, notations and proofs for Bernoulli numbers

This is a collection of definitions, notations and proofs for the Bernoulli numbers $B_n$ appearing in formulas for the sum of integer powers, some of which can be found scattered in the large related historical literature in French, English and German. We provide elementary proofs for the original convention with ${\mathcal B}_1=1/2$ and also for the current convention with $B_1=-1/2$, using only the binomial theorem and the concise Blissard symbolic (umbral) notation.