The irrationality of some number theoretical series

We prove the irrationality of some factorial series. To do so we combine methods from elementary and analytic number theory with methods from the theory of uniform distribution

The exponential sum over squarefree integers

We give an upper bound for the exponential sum over squarefree integers. This establishes a conjecture by Br\"udern and Perelli

Partitions which are p- and q-core

Let p and q be distinct primes, n an integer with n > p2q2. Then there is no partition of n which is at the same time p- and q-core. Hence there is no irreducible representation of Sn which is of p- and q-defect zero at the same time

Sets with more differences than sums

We show that a random set of integers with density 0 has almost always more differences than sums. This proves a conjecture by Martin and O'Bryant

An inequality for means with applications

We show that an almost trivial inequality for the first and second mean of a random variable can be used to give non-trivial improvements on deep results. As applications we improve on results on lower bounds for the Riemann zeta-function on the critical line, the determinant of a skew-symmetric matrix with entries $\pm 1$, and on the maximal order of an irreducible character of the symmetric group

The order of elements in Sylow pp-subgroups of the symmetric group

Define a random variable $\xi_n$ by choosing a conjugacy class $C$ of the Sylow $p$-subgroup of $S_{p^n}$ by random, and let $\xi_n$ be the logarithm of the order of an element in $C$. We show that $\xi_n$ has bounded variance and mean order $\frac{\log n}{\log p}+O(1)$, which differs significantly from the average order of elements chosen with equal probability