Let $f_k$ be the $k$-fold convolution $$f_k(n):=(f*f*\cdots*f)(n).$$ I want to evaluate$$\sum_{n\le m}\frac{f_k(n)^2}{n}.$$Is there a classical way of doing this? For simplicity let's consider $f\...

Let $(E_n)_n$ be any finite collection of centred ellipses in $\mathbb{R}^2$. Suppose that $E_n$ are pairwise non-homothetic (i.e. there is no positive constant $c>0$ such that $E_n = c E_m$). Now ...

A result of Selberg (A. Selberg. On the normal density of primes in small intervals, and the difference between consecutive primes. Arch. Math. Naturvid., 47(6):87–105, 1943) says essentially$$\int ...

Let $G=\mathrm{SL}_2(\mathbb C)$, $V$ its standard representation, and $V_m=\operatorname{Sym}^m(V)$ with $m\equiv 2 \pmod 4$. It is classical that$$\Lambda^2 V_m \;\cong\; \bigoplus_{\substack{1\le ...

$\def\R{\mathscr{R}}\def\O{\mathcal{O}}$Let $X$ be a topological space and let $\R$ be a sheaf of unital non-commutative rings over $X$.When $\R$ is commutative, there is much literature on ...

The following fact is well-known, and not hard to prove, but I do not know an explicit reference.Let $R$ be the subring of complex numbers generated by all roots of unity. Then $R$ is free as an ...

For a formal Laurent series $F(q)$, denote its coefficient of $q^j$ by $[q^j](F)$.QUESTION. For integers $r\geq1$, is this true?$$[q^{2r}]\sum_{n\geq1}\frac{q^n}{1-q^{2n}}\sum_{k=1}^n\frac{q^k}{1+q^...

Let $G$ be a smooth connected linear algebraic group over an algebraically closed field. Write $\operatorname{Br}'(BG)$ for the cohomological Brauer group of $BG$, i.e. the group of $\mathbb{G}_m$-...

I have a Markov kernel $(t,A) \mapsto \mathbb{Q}_t(A)$ from a standard Borel space $(T, \mathcal{T})$ into another standard Borel space $(\Omega, \mathcal{F})$. Also, for $t \neq s$, $\mathbb{Q}_t \...

I am looking for bibliography on the following problem.Given $N\in\mathbb{N}$ find $N$ points $p_1,...,p_N\in\mathbb{R}^2$ which(1) maximize $\min_{i,j} |p_i-p_j|$(2) subject to the constraint $\...

Inspired by a recent project Euler problem, I came up with a proof (sketched below) of Cayley's tree formula using the representation theory of $S_n$. I would like to ask for a reference in the ...

BackgroundThe central factorial numbers are described on OEIS sequence A008955. Among the references, "Ramanujan's notebooks, part 1" (edited by Bruce Berndt) is listed. Upon checking this ...

On a smooth manifold $M$ with altas $\mathcal A=\{\phi:U_\phi\subseteq M\to \Bbb R^n\}_\phi$, we can define a tensor $T$ on $M$ as the collection $\{T_\phi\}_{\phi\in\cal A}$ such that whenever $U_\...

For an infinite set S, I believe that all automorphisms of Sym(S) are inner. I would like a reference for this.

A hopefully not too nonsensical question on Daniel Kriz' work (from just revisiting it after a very long delay, my excuses in advance):Daniel Kriz’s work on supersingular $p$-adic $L$-functions (...