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.

Filter by
Sorted by
Tagged with
3 votes
2 answers
302 views

Where to find or how to prove that the ratio of two Bernoulli polynomials is increasing?

It is well known that the classical Bernoulli polynomials $B_j(t)$ are generated by \begin{equation*} \frac{s\operatorname{e}^{ts}}{\operatorname{e}^s-1}=\sum_{j=0}^{\infty}B_j(t)\frac{s^j}{j!}, \quad ...
6 votes
1 answer
461 views

A non-$p$-adic proof of a congruence of Bernoulli numbers

In A Multimodular Algorithm for Computing Bernoulli Numbers, Harvey uses the following congruence for Bernoulli numbers: $$B_k \equiv \frac{k}{1-c^k} \sum_{x=1}^{p-1} x^{k-1} h_c(x)\quad(\text{mod}\ p)...
3 votes
1 answer
416 views

Identities for Bernoulli numbers

I arrived at this formula by inductive reasoning, but I don’t know how to prove it. For any natural numbers $m$ and $k=0,1,2,\ldots, m-1$, $B_i$ - Bernoulli numbers we have: $$\sum_{i=0}^k (-1)^{k-i}\...
4 votes
1 answer
243 views

Kummer's congruence at $p=3$

Let $B_{2k}$ be the Bernoulli numbers of even index and $\varphi(n)$ be Euler's totient function. We recall one instance of Kummer's congruences: for each integer $m\geq1$ and a prime number $p\geq5$, ...
3 votes
0 answers
181 views

Combinatorial interpretation of Sylvester–Lipschitz formula?

If we denote the Bernoulli numbers by $B_n$, then $$ {a^{k+1}(a^{2k} - 1) B_{2k} \over 2k}\in \mathbb{Z}$$ for any $a\in \mathbb{Z}$ and $k\in\mathbb{N}$. This fact is often called the Sylvester–...
2 votes
0 answers
209 views

show that sequence $\{(-1)^n\Upsilon_n\}$ is convergent and strictly decreasing

Edit: Few years ago, I have posted my claim on $\Upsilon$ function regarding prime number but recently I have observed, last observation turns false that's way, (by putting $\Upsilon$ value in ...
0 votes
0 answers
90 views

Is there a prime factor that in the numerator of Bernoulli number $B_{2n}$ strictly greater than $2n$ for any $n>7$?

As well known, Bernoulli number $B_{2n}$ is a rational number which can always be shown as $B_{2n}=V_{2n}/A_{2n}$ , where $\gcd(V_{2n}, A_{2n})=1$. For example, $B_{16}=\frac{3617}{-510}$ and $3617$ ...
0 votes
0 answers
126 views

$p$-divisibility of consecutive Bernoulli numbers

I have asked the same question in MathStackExchange. (Link) Let $B_n$ be $n$-th Bernoulli number. We will write $p \mid B_n$ if prime $p$ divides the numerator of $B_n$. Now $p = 587$, we can show $p \...
2 votes
0 answers
212 views

Zeta function associated with a function $f$

Let the function $f(t) = \cos(at)$, where ($0 < a < 1$). Let us define $$\zeta(z, f) = \frac{1}{\Gamma(z)} \int_0^{+\infty} \frac{t^{z-1}\cos(at)}{e^t-1}\, dt. $$ Is there a general formula that ...
2 votes
0 answers
92 views

A combinatorial triangle for the Bernoulli numbers

Motivation: We informally call an infinite lower triangular matrix $\operatorname{T}(n, k)$ of integers a combinatorial triangle of a sequence of integers or rational numbers if it can be obtained ...
0 votes
0 answers
146 views

Anti-concentration for Bernoulli summation

Suppose $\{ Y_i\}_{i = 1}^n$ is i.i.d. Bernoulli distribution with mean $p$. Denote the sample of $\{ Y_i\}_{i = 1}^n$ as $\overline{Y} = \frac{1}{n} \sum_{i = 1}^n Y_i$. I want to know where there ...
1 vote
0 answers
164 views

Ask for a proof of an inequality involving the Bernoulli numbers

Let $B_k$ be the Bernoulli numbers and let \begin{equation} T_k=\frac{2^{2k}}{(2k)!}|B_{2k}|, \quad k\ge1. \end{equation} Prove the inequality \begin{equation*} \frac{\frac{1}{k+2}\sum_{j=0}^{k+1}\...
7 votes
0 answers
245 views

Hankel determinants for some convolutions of Catalan numbers

Let $c(x)=\frac{1-\sqrt{1-4x}}{2x}$ be the generating function of the Catalan numbers and let $$x^k c(x)^{2k}=(c(x)-1)^k =\sum_{n\geq0}c(k,n)x^n.$$ Consider the determinants $$D(k,n,m)= \det\left(c(k,...
1 vote
2 answers
266 views

Ask for a reference or a proof of an identity involving a finite sum and the Bernoulli numbers

Let $B_{n}$ for $n\ge0$ denote the Bernoulli number generated by \begin{equation*} \frac{z}{\textrm{e}^z-1}=\sum_{n=0}^\infty B_n\frac{z^n}{n!}=1-\frac{z}2+\sum_{n=1}^\infty B_{2n}\frac{z^{2n}}{(2n)!},...
2 votes
2 answers
271 views

Ask for a proof of an identity involving the product of two Bernoulli numbers

It is well known that the Bernoulli numbers $B_{k}$ for $k\in\{0,1,2,\dotsc\}$ can be generated by \begin{equation*} \frac{z}{\textrm{e}^z-1}=\sum_{k=0}^\infty B_k\frac{z^k}{k!}=1-\frac{z}2+\sum_{k=1}^...
1 vote
2 answers
165 views

A question about generalized harmonic numbers modulo $p$

Let $p \equiv 1 \pmod{3}$ be a prime and denote $H_{n,m} = \sum_{k = 1}^n 1/k^m$ as the $n,m$-th generalized harmonic number. I'm interested in computing $H_{(p-1)/3,\, 2}$ and $H_{(p-1)/6,\,2}$ ...
6 votes
1 answer
506 views

A novel identity connecting permanents to Bernoulli numbers

For a matrix $[a_{j,k}]_{1\le j,k\le n}$ over a field, its permanent is defined by $$\mathrm{per}[a_{j,k}]_{1\le j,k\le n}:=\sum_{\pi\in S_n}\prod_{j=1}^n a_{j,\pi(j)}.$$ In a recent preprint of mine, ...
6 votes
1 answer
686 views

An operation is defined on polynomials. How do I generalize it to other classes of functions?

I am currently researching divergent integrals. Definition. An extended number is an expression of the form $\int_a^b f(x)\,dx$, where $a,b\in \overline{\mathbb{R}}$ and function $f(x)$ is defined ...
3 votes
0 answers
139 views

Interpreting umbral calculus in terms of some kind of extended numbers

I know that currently umbral calculus is developed as some kind of theory of operators and functionals but were there any attempts to put it on a more solid philosophical grounds as study of functions ...
2 votes
0 answers
81 views

Question about infinitude of $m$-irregular primes

Let $p>2$ be a prime, $\zeta$ a primitive $p$th root of unity, and $m >0$ a square-free integer such that $m \not\equiv 3 \mod 4$ and $\gcd(m,p)=1$. Let $\chi$ be the imaginary quadratic ...
2 votes
1 answer
296 views

Analogue of Fermat's little theorem for Bernoulli numbers

Is the following analogue of Fermat's Little Theorem for Bernoulli numbers true? Let $D_{2n}$ be the denominator of $\frac{B_{2n}}{4n}$ where $B_n$ is the $n$-th Bernoulli number. If $\gcd(a, D_{2n}) ...
1 vote
1 answer
141 views

Simplifying a rational function in terms of Bernoulli numbers and polynomials

Faulhaber's formula expresses a sum over some finite number of naturals to the $m^{th}$ power in terms of the Bernoulli numbers $B_{j}$ (using the $B_{1} = 1/2$ convention) or polynomials $\hat{B}_{j}$...
22 votes
3 answers
1k views

Roots of Bernoulli polynomials - a pattern

Contemplating a question on math.SE, I have stumbled on this: Here, the point labeled $n$ is that root of the $n$th Bernoulli polynomial which has smallest positive imaginary part. Does anyone know ...
0 votes
0 answers
173 views

Asymptotically similar functions with opposite parity, were they considered, are they useful? Case of polynomials

So, can we transform an even function into an odd function and vice versa? Let's consider this method: Transformation even->odd: Suppose $f_{even}(x)$ is a function which satisfies the following ...
1 vote
1 answer
159 views

Prove $ \sum_{i=0}^{2a+1} {2a+1 \choose i} B_{2a+1-i} [ (n+1)^i+(-n)^i ] =0 $ for Bernoulli numbers $B_{n}$

Prove for the Bernoulli numbers $B_n$, that for all $a \in \mathbb{N}$, that $ \sum_{i=0}^{2a+1} {2a+1 \choose i} B_{2a+1-i} [ (n+1)^i+(-n)^i ] =0 $. As much as this is a neat identity, it's a crucial ...
3 votes
1 answer
484 views

Conjecture on bernoulli numbers and binomial coefficients

Crossposted from https://math.stackexchange.com/questions/4116414/conjecture-on-bernoulli-numbers-and-binomial-coefficients In playing around with some formulas, I have come up with the following ...
0 votes
1 answer
123 views

Nicely motivated papers or book chapters on the formula for the sum of the $k$-th powers of the first natural numbers [closed]

Do you know of a text where I can find a nicely motivated proof of the formula for $1^{k}+2^{k}+\cdots+n^{k}$? At the very beginning of page 68 of Professor H. S. Wilf's generatingfunctionology, one ...
18 votes
0 answers
623 views

Bernoulli & Betti numbers (of manifolds) and the prime 34511

The purpose of this question is to resolve a mystery surrounding the prime 34511 that has got me bogged down for a while now. If you only care about the number theory and not the motivation coming ...
10 votes
2 answers
359 views

Do Bernoulli polynomials know about face vectors?

This question is grounded firmly in numerology. It originates in an observation about some Bernoulli polynomials and the regular icosahedron. Let $F_{k+1}(n)=\sum_{i=1}^n i^k$ be the sum of the ...
3 votes
0 answers
154 views

Combinatorial interpretation for Möbius-poly-Bernoulli numbers

The Möbius-Bernoulli numbers ,are related to Dedekind Sums $$\sum_{d|n}\frac{t\mu(d)}{e^{td}-1}=\sum_{k=0}^\infty M_k(n)\frac{t^k}{k!}$$ where $|t|<\frac{2\pi}{n}$, and $M_k(1)=B_k$. We define the ...
100 votes
10 answers
15k views

Why do Bernoulli numbers arise everywhere?

I have seen Bernoulli numbers many times, and sometimes very surprisingly. They appear in my textbook on complex analysis, in algebraic topology, and of course, number theory. Things like the criteria ...
1 vote
1 answer
184 views

Claim on divisibility of a power sum

Let $x,y,z$ are integer and $x,y>0$ Define $S(x,y)=1^y+2^y+3^y+...+x^y$ Can it be shown that If given $z\ne0$ then $S(x,y)\equiv z\pmod{x}$ have finitely many solution of $x$ with respect to $y$. ...
2 votes
0 answers
118 views

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

For any irreducible polynomial $f$ of degree $d$ in finite field $\mathbb F$ we have that $$x^{|\mathbb F|^d}-x\equiv0\bmod f$$ and thus the polynomial $x^{|\mathbb F|^d}-x$ is the analog of factorial ...
1 vote
2 answers
208 views

Bilinear recurrence relation between even Bernoulli numbers

Throughout this question $n$ is a positive integer greater than 1. Consider the following well-known identity by Euler, $$\sum_{k=1}^{n-1} \binom{2n}{2k}B_{2k}B_{2n-2k}=-(2n+1)B_{2n}.$$ Rather ...
16 votes
1 answer
576 views

What is this sequence?

This is again a question that I asked at Stack Exchange, but got no answer so far, so I am trying here. Let: $$ a_n=\sum_{k\ge0}(k+1) {n+2\brack k+2}(n+2)^kB_k$$ $B_k$ is the Bernoulli number. ${n\...
45 votes
16 answers
8k views

What does the generating function $x/(1 - e^{-x})$ count?

Let $x$ be a formal (or small, since the function is analytic) variable, and consider the power series $$ A(x) = \frac{x}{1 - e^{-x}} = \sum_{m=0}^\infty \left( -\sum_{n=1}^\infty \frac{(-x)^n}{(n+1)!}...
3 votes
0 answers
227 views

Values of Bernoulli polynomials at roots of unity

I am wondering if there are any nice results on the values of Bernoulli polynomials at roots of unity, besides those at 1 or -1.
3 votes
1 answer
179 views

A question on Bernoulli polynomials

Denote by $B_r$ the $r$-th Bernoulli polynomial. Are there any positive integers $r, x$ such that. $B_r(x)$ divides $B_r(x+1)$ or vice versa ?
7 votes
3 answers
646 views

Transformation converting power series to Bernoulli polynomial series

I wonder, can anyone describe an expression or formula of a transform that converts $$\sum_{k=0}^\infty \frac{a_k x^k}{k!}$$ into $$\sum_{k=0}^\infty \frac{a_k B_k(x)}{k!}$$ where $B_k(x)$ are ...
3 votes
2 answers
518 views

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

I seek a reference for the fact that "coefficients of the Hirzebruch $L$-polynomial have odd denominators". The coefficients are $$\frac{2^{2k}(2^{2k-1}-1)B_k}{(2k)!}$$ where $B_k$ is the Bernoulli ...
8 votes
2 answers
377 views

Coefficients of shifted Bernoulli polynomials

I stumbled across the following curious empirical properties of the Bernoulli polynomials $B_n(x)$. Can anyone provide a reference or proof? Let $k\in\mathbb{Z}$, $k\geq 2$. Then (empirically): The ...
7 votes
1 answer
183 views

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

Let $\chi : (\mathbb Z/f\mathbb Z)^\times \to K = \mathbb Q(\mu_{\phi(f)})$ be a primitive Dirichlet character. Assume moreover that it is not quadratic, that is, $\chi^2$ is not the trivial character....
14 votes
1 answer
305 views

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

Let $k$ be a sufficiently large positive even integer. (I think $k > 14$ should do.) Can one always find a prime $p$ such that $p$ divides the numerator of the $k$-th Bernoulli number $B_k$ and $k \...
18 votes
5 answers
3k views

Bernoulli sum meets golden number

Let $B_n$ denote the Bernoulli numbers and let $\phi=\frac{1+\sqrt{5}}2$ be the golden ratio. I encountered the following infinite sum and would like to ask: Question. Is this true? If so, any ...
3 votes
1 answer
2k views

What is $p$-adic Fourier series?

Q1: Can we define Fourier series for a function $\mathbb{Z}_p\to \mathbb{Q}_p$? Q2: There are (in a real case) Bernoulli polynomials which have the most simple Fourier expansion: $$B_n(\{x\})=-\frac{...
3 votes
2 answers
642 views

p-adic poly-Bernoulli numbers

We can define p-adic Bernoulli polynomials by using q-integral on $\mathbb{Z}_p$ and Taekyun Kim's method. But how can we define p-adic poly-Bernoulli numbers and polynomials by using integral on $\...
6 votes
2 answers
498 views

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

Any reference that we can find the following $$\Bigr[-\log(1-t)\Bigr]^x = t^x + x t^x \sum_{k=0}^\infty \psi_k(x+k)\,t^{k+1}; \quad \mbox{for all} \, x\in \mathbb R, \, |t|<1$$ where $\psi_k(.)$ ...
5 votes
1 answer
264 views

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

It is well known that several definite integrals $I_n$ containing a parameter $n\in\mathbb N$ can be expressed recursively (e.g. doing integration by parts) in terms of $I_{n-1} $ or $I_{n-2} $, and ...
16 votes
1 answer
675 views

Connection between Bernoulli numbers and Riemann-Siegel theta function?

I have come across a strange approximation for the Riemann-Siegel theta function involving the Bernoulli numbers - namely that $$\frac{1}{2} \log \left| B_{2 n}\right|\approx \vartheta (2n)\ ,\quad n ...
6 votes
0 answers
123 views

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

For the space of modular forms and the space of cusp forms (here I only care about the level $1$ case), we have the action by Hecke algebras. Therefore, we can calculate the discriminant of this ...