Large normal subgroup growth and large characteristic subgroup growth

The maximal normal subgroup growth type of a finitely generated group is $n^{\log n}$. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let $\Gamma$ be a group and $\Delta$ a subgroup of finite index. Suppose $\Delta$ has normal subgroup growth of type $n^{\log n}$, does $\Gamma$ has normal subgroup growth of type $n^{\log n}$? We give a positive answer in some cases, generalizing a result of M\"uller and the second author and a result of Gerdau. For instance, suppose $G$ is a profinite group and $H$ an open subgroup of $G$. We show that if $H$ is a generalized Golod-Shafarevich group, then $G$ has normal subgroup growth of type of $n^{\log n}$. We also use our methods to show that one can find a group with characteristic subgroup growth of type $n^{\log n}$

The irrationality of a number theoretical series

Denote by $\sigma_k(n)$ the sum of the $k$-th powers of the divisors of $n$, and let $S_k=\sum_{n\geq 1}\frac{\sigma_k(n)}{n!}$. We prove that Schinzel's conjecture H implies that $S_k$ is irrational, and give an unconditional proof for the case $k=3$

The subgroup growth spectrum of virtually free groups

For a finitely generated group $\Gamma$ denote by $\mu(\Gamma)$ the growth coefficient of $\Gamma$, that is, the infimum over all real numbers $d$ such that $s_n(\Gamma)<n!^d$. We show that the growth coefficient of a virtually free group is always rational, and that every rational number occurs as growth coefficient of some virtually free group. Moreover, we describe an algorithm to compute $\mu$