# Questions tagged [bernoulli-numbers]

The Bernoulli numbers are the rational numbers $B_n$ defined as the coefficients in the expansion $\frac{x}{e^x-1} = \sum_{n \geq 0} B_n \frac{x^n}{n!}$. They vanish when $n$ is odd and greater than $2$. They appear in the values at integers of the Riemann $\zeta$ function. These classical numbers play an important role in number theory and in several other places in mathematics.

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### References for the heuristics on the uniform distribution of the Bernoulli Numbers modulo a prime

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### Claim on divisibility of a power sum

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### Analogues over finite fields of certain integers defined multiplicatively in $\mathbb Z$

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### Exact and simple expressions for the (inverse) of the $n$-th Bernoulli number

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### show that $n\Upsilon_{n-1} \equiv -1 \pmod{n}$ iff $n$ is prime

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### Bilinear recurrence relation between even Bernoulli numbers

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### A question on Bernoulli polynomials

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### Coefficients of shifted Bernoulli polynomials

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### 2-adic valuation of $L(0,\chi)$ for a Dirichlet character

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### What is this sequence?

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### Bernoulli sum meets golden number

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### Roots of Bernoulli polynomials - a pattern

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### Transformation converting power series to Bernoulli polynomial series

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### Any reference for the series expansion of $\Bigr[-\log(1-t)\Bigr]^x$?

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### Values of Bernoulli polynomials at roots of unity

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### Connection between Bernoulli numbers and Riemann-Siegel theta function?

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### divisibility by Bernoulli numbers of discriminant of Hecke algebra over the space of modular forms of level 1

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### Estimate of the sum Taylor's coefficients

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### Probability of the maximum of a throw of an infinite number of $n$-sided dice being $k$ [closed]

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### Why do $12$ and $120$ occur very often in the denominators of $\zeta(-n)$ for odd $n$?

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### How naturally can functions defined by parametric integrals be interpolated from $\mathbb N$ to $\mathbb R^+$?

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### For a given even integer $k >14$ is there always a prime $p$ such that $k \leq p-3$ and $p|B_k$?

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### What is $p$-adic Fourier series?

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### zeta(3) in Euler's Section 153

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### computing Bernoulli numbers

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### Sign of coefficients

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### p-adic poly-Bernoulli numbers

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### Coefficients in Hirzebruch polynomial and divisibility of Bernoulli numbers: reference request

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### Riemann's $\zeta$ function and the uniform distribution on $[-1,0]$

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### Why do Bernoulli numbers arise everywhere?

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### A conjecture on p-divisibility of Bernoulli numbers

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