All Questions
Tagged with bernoulli-numbers reference-request
8 questions
12
votes
1
answer
598
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Fermat last theorem : proof of a criterion by Cauchy
In 13 Lectures on Fermat's Last Theorem, Ribenboim states the following theorem (on page 7) attributed to Cauchy:
If the first case of Fermat's theorem fails for the exponent $p$, then the sum:
$$ 1^{...
- 121
4
votes
1
answer
260
views
Kummer's congruence at $p=3$
Let $B_{2k}$ be the Bernoulli numbers of even index and $\varphi(n)$ be Euler's totient function. We recall one instance of Kummer's congruences: for each integer $m\geq1$ and a prime number $p\geq5$, ...
- 43.2k
3
votes
0
answers
145
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Interpreting umbral calculus in terms of some kind of extended numbers
I know that currently umbral calculus is developed as some kind of theory of operators and functionals but were there any attempts to put it on a more solid philosophical grounds as study of functions ...
- 10.1k
0
votes
1
answer
127
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Nicely motivated papers or book chapters on the formula for the sum of the $k$-th powers of the first natural numbers [closed]
Do you know of a text where I can find a nicely motivated proof of the formula for $1^{k}+2^{k}+\cdots+n^{k}$?
At the very beginning of page 68 of Professor H. S. Wilf's generatingfunctionology, one ...
- 87
2
votes
0
answers
212
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show that sequence $\{(-1)^n\Upsilon_n\}$ is convergent and strictly decreasing
Edit: Few years ago, I have posted my claim on $\Upsilon$ function regarding prime number but recently I have observed, last observation turns false that's way, (by putting $\Upsilon$ value in ...
- 599
1
vote
2
answers
211
views
Bilinear recurrence relation between even Bernoulli numbers
Throughout this question $n$ is a positive integer greater than 1.
Consider the following well-known identity by Euler,
$$\sum_{k=1}^{n-1} \binom{2n}{2k}B_{2k}B_{2n-2k}=-(2n+1)B_{2n}.$$
Rather ...
- 195
6
votes
2
answers
502
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Any reference for the series expansion of $\Bigr[-\log(1-t)\Bigr]^x$?
Any reference that we can find the following $$\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$$
where $\psi_k(.)$ ...
- 650
16
votes
1
answer
706
views
Connection between Bernoulli numbers and Riemann-Siegel theta function?
I have come across a strange approximation for the Riemann-Siegel theta function involving the Bernoulli numbers - namely that
$$\frac{1}{2} \log \left| B_{2 n}\right|\approx \vartheta (2n)\ ,\quad n ...
- 1,903