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Results tagged with nt.number-theory
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user 10846
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
9
votes
How to calculate the infinite sum of this double series?
Here is a proof based on Hachino's idea. We have agreed it's enough to
prove that
$$S=\sum_{m=1}^\infty\frac{(-1)^m\tanh(m\pi /2)}{m}=\frac{\log 2-\pi}{4}. $$
Plug in the expansion
$$\tanh(m\pi /2)=\ …
- 1,514
5
votes
Accepted
Relations of eisenstein series with eta quotient
This does not answer your question, but I want to elaborate on my comment to Jeremy's answer that such identities can also be understood from the viewpoint of elliptic functions. I use as a reference …
- 1,514
2
votes
Accepted
Modulo $2$ binomial transform of A243499 applied $k$ times
The definition of $a_1$ given in OEIS is based on a bijection between integer partitions and natural numbers. A partition $\lambda_1\geq\lambda_2\geq\dots\geq\lambda_m>0$ with exactly $m$ parts corres …
- 1,514
15
votes
Computing hypergeometric function at 1
Carlo's answer is correct but doesn't show why the identity holds, so let me explain how to do this easily by hand.
Look more generally at
$$S={}_3F_2\left(\begin{matrix}-m,a,b\\a+1,b+1\end{matrix};1\ …
- 1,514