Let $p(n)$ be the ordinary partition function and let $c(n)=\sum_{k=0}^{n-1}p(k)$, with $c(0)=0$. We prove that, for every $a\geq1$, $b\geq0$, and $m\geq2$, the sequence $c(an+b)$ is not eventually constant modulo $m$, thereby answering a question posed by Chen. The proof combines the algebraicity over $\mathbb C(j)$ of modular functions for finite-index subgroups of $\mathrm{SL}_2(\mathbb Z)$ with Voloch's theorem on the transcendence of elliptic modular functions in positive characteristic.
Let $G$ be a finite group. We prove that the order of the abelianization $G/G'$ divides the order of $H^{2m}(G,\mathbb{Z})$ for every $m\geq1$. As a consequence, for a field $K$, nonvanishing of $Ext^1_{KG}(K,K)$ implies nonvanishing of $Ext^n_{KG}(K,K)$ for every $n\geq1$. This answers a conjecture by Erdmann, Kl\'asz...
Let $A=\mathbb{F}_q[T]$ and $K=\mathbb{F}_q(T)$, and let $\phi$ be a Drinfeld $A$-module over $K$ of rank $r\geq 2$. We study the reduction of $\phi$ at primes $\mathfrak{p} \in A$ lying in a prescribed arithmetic progression, for which $\phi(\mathbb{F}_{\mathfrak{p}})$ has at most $r-1$ cyclic components. We obtain as...
Prassanna Nand Jha, Arnab Saha, A. Vatwani· 0 citations
For every odd prime $p$, we construct a finitely presented group $G_p$ and elements $a,b,c\in\mathbb{F}_p[G_p]$ such that $ab=1$, $ac=0$, and $c\ne0$. Thus $\mathbb{F}_p[G_p]$, and the group algebra over every field of characteristic $p$, fails to be directly finite. The construction combines a single exceptional self-...
Elena Bunina, Winston Lee, A. Miasnikov et al.· 1 citation
Let $p>3$ be a prime, and let $b,c$ be integers with $p\nmid b+2c$. We obtain an explicit congruence modulo $p^2$ for the sum of $\binom{2k}{k}T_{2k}(b,c^2)/(4^k(k+1)(b+2c)^{2k})$ from $k=0$ to $(p-1)/2$, where $T_n(b,c)$ denotes the generalized central trinomial coefficient. This gives a parametric formula for the sum...
If $F$ is a number field and $F = F_0 \subset F_1 \subset F_2\subset \cdots$ is a tower of Galois extensions with $\mathrm{Gal}(F_n/F)$ cyclic of order $p^n$, K. Iwasawa gave a closed formula for the growth of the $p$-part of the class number of $F_n$ whenever $n$ is large enough. We present a new proof of this formula...
Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. We prove that \begin{align*} \sum_{n\leq x^{1/c}}d\left(\left[\frac{x}{n^c}\right]\right)=d_cx^{1/c}+O_{\varepsilon,c}(x^{\theta_c+\varepsilon}), \end{align*} where $d_c$ is a suitable constant, $\theta_c<2/(3c+2)$ for $0<c...
Zhi-Yuan Yang· 0 citations
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