All Questions

I would like to ask the following. Let $(a_n)$ be a sequence of natural numbers such that $\sum_{k=1}^{\infty}\frac{1}{a_k}$ converges. Is it true that for infinitely many $m$, there is a $n<m$ ...

I am looking for a closed form for this function $\Lambda:\mathbb{Q}^+\to\mathbb{R}^+$: $$\Lambda(q) = \sum_{m,n\geq 1}\left(\frac{q\wedge\frac{m}{n}}{q\vee\frac{m}{n}}\right)^\alpha\left(\frac{m \...

Let $a(n)$ be A005225 i.e. number of permutations of length $n$ with equal cycles. Here $$ a(n)=n!\sum\limits_{d|n}\frac{1}{d!(\frac{n}{d})^d} $$ Let $$ R(n,q,z)=(q+1)R(n-1,q+1,z)+\sum\limits_{j=0}^{q}...

For integers $n\geq 1$ let $G_n$ be the Gregory coefficients or reciprocal logarithmic numbers, see the Wikipedia [Gregory coefficients.] (https://en.wikipedia.org/wiki/Gregory_coefficients) $${z\...

Let $\sigma(n)=\sum_{1\leq d\mid n}d$ the sum of divisors function then, from the theory of Dirichlet series, it is well-known the value of $$\sum_{n=1}^\infty\frac{\sigma(n)}{n^3},$$ in terms of ...