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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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 ...
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A conjecture on p-divisibility of Bernoulli numbers
Is anyone aware of the history of the following conjecture on the $p$-divisibility of (the numerators) of Bernoulli numbers?
CONJECTURE: For $p$ an odd prime, and $k$ even with $2 \leq k \leq p-3$, $...
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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 $$D(k,n,m)= \det\left(c(k,...
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divisibility by Bernoulli numbers of discriminant of Hecke algebra over the space of modular forms of level 1
For the space of modular forms and the space of cusp forms (here I only care about the level $1$ case), we have the action by Hecke algebras. Therefore, we can calculate the discriminant of this ...
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Combinatorial interpretation of Sylvester–Lipschitz formula?
If we denote the Bernoulli numbers by $B_n$, then
$$ {a^{k+1}(a^{2k} - 1) B_{2k} \over 2k}\in \mathbb{Z}$$
for any $a\in \mathbb{Z}$ and $k\in\mathbb{N}$. This fact is often called the Sylvester–...
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Interpreting umbral calculus in terms of some kind of extended numbers
I know that currently umbral calculus is developed as some kind of theory of operators and functionals but were there any attempts to put it on a more solid philosophical grounds as study of functions ...
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Combinatorial interpretation for Möbius-poly-Bernoulli numbers
The Möbius-Bernoulli numbers ,are related to Dedekind Sums
$$\sum_{d|n}\frac{t\mu(d)}{e^{td}-1}=\sum_{k=0}^\infty M_k(n)\frac{t^k}{k!}$$ where $|t|<\frac{2\pi}{n}$, and $M_k(1)=B_k$.
We define the ...
user160903
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Values of Bernoulli polynomials at roots of unity
I am wondering if there are any nice results on the values of Bernoulli polynomials at roots of unity, besides those at 1 or -1.
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Zeta function associated with a function $f$
Let the function $f(t) = \cos(at)$, where ($0 < a < 1$). Let us define
$$\zeta(z, f) = \frac{1}{\Gamma(z)} \int_0^{+\infty} \frac{t^{z-1}\cos(at)}{e^t-1}\, dt.
$$
Is there a general formula that ...
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A combinatorial triangle for the Bernoulli numbers
Motivation: We informally call an infinite lower triangular matrix
$\operatorname{T}(n, k)$ of integers a combinatorial triangle of a sequence of integers or
rational numbers if it can be obtained ...
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Question about infinitude of $m$-irregular primes
Let $p>2$ be a prime, $\zeta$ a primitive $p$th root of unity, and $m >0$ a square-free integer such that $m \not\equiv 3 \mod 4$ and $\gcd(m,p)=1$. Let $\chi$ be the imaginary quadratic ...
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Analogues over finite fields of certain integers defined multiplicatively in $\mathbb Z$
For any irreducible polynomial $f$ of degree $d$ in finite field $\mathbb F$ we have that $$x^{|\mathbb F|^d}-x\equiv0\bmod f$$ and thus the polynomial $x^{|\mathbb F|^d}-x$ is the analog of factorial ...
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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 ...
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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}\...
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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 $3617$ ...
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$p$-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 \...
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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 ...
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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 ...
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