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 $\mu$

Feasibility of Integer Knapsacks

Given an integer mxn matrix A satisfying certain regularity assumptions, we consider the set F(A) of all integer vectors b such that the associated knapsack polytope P(A,b)={x: Ax=b, x>=0} contains an integer point. When m=1 the set F(A) is known to contain all consecutive integers greater than the Frobenius number associated with A. In this paper we introduce the diagonal Frobenius number g(A) which reflects in an analogous way feasibility properties of the problem and the structure of F(A) in the general case. We give an optimal upper bound for g(A) and also estimate the asymptotic growth of the diagonal Frobenius number on average