arXiv subjects
Publications and source records attributed to Jan-Christoph Schlage-Puchta.
Denote by $σ_k(n)$ the sum of the $k$-th powers of the divisors of $n$, and let $S_k=\sum_{n\geq 1}\frac{σ_k(n)}{n!}$. We prove that Schinzel's conjecture H implies that $S_k$ is irrational, and give an unconditional proof for the case $k=3$.
We show that the Bianchi group $\mathrm{PSL}_2(\mathcal{O}_d)$, where $\mathcal{O}_d$ is the ring of integers in $\Q(\sqrt{d})$, $d<0$, has a free quotient of rank $\geq |d|^{1/4-ε}$, as $|d|\to\infty$.
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.
We show that the large sieve is optimal for almost all exponential sums, thus proving a conjecture by Erdös and Renyi.
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.
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.
Let $G$ be a finite group, $A$ a finite abelian group. Each homomorphism $ϕ:G\to A\wr S_n$ induces a homomorphism $\barϕ:G\to A$ in a natural way. We show that as $ϕ$ is chosen randomly, then the distribution of $\barϕ$ is close to uniform. As application we prove a conjecture of T. Müller on the number of homomorphisms from a finite group into Weyl groups of type $D_n$.
For a finitely generated group $Γ$ denote by $μ(Γ)$ the growth coefficient of $Γ$, that is, the infimum over all real numbers $d$ such that $s_n(Γ)<n!^d$. 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 $μ$.
We describe the structure of a set of integers $A$ of positive density $δ$, such that $A+A$ contains no squarefree integer. It turns out that the behaviour changes abruptly at the values $δ_0=1/4-\frac{2}{π^2}=0.0473...$ and $δ_1=1/18$.
Assuming the Riemann Hypothesis we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Omega-term, we prove that our result is essentially the best possible.
We consider the Dirichlet series associated to the number of representations of an integer as the sum of primes. Assuming the Riemann hypothesis on the distribution of the zeros of the Riemann zeta function we obtain the domain of meromorphic continuation of this series.
We classify singularities of Dirichlet series having Euler products which are rational functions for p and p^{-s} for p a prime number and give examples of natural boundaries from zeta functions of groups and height zeta functions.
We obtain a decidability result for the Davenport constant.
Let n be an integer, and consider finite sequences of elements of the group Z/nZ x Z/nZ. Such a sequence is called zero-sum free, if no subsequence has sum zero. It is known that the maximal length of such a zero-sum free sequence is 2n-2, and Gao and Geroldinger conjectured that every zero-sum free sequence of this length contains an element with multiplicity at least n-2. By recent results of Gao, Geroldinger and Grynkiewicz, it essentially suffices to verify the conjecture for n prime. Now fix a sequence (a_i) of length 2n-2 with maximal multiplicity of elements at most n-3. There are different approeaches to show that (a_i) contains a zero-sum; some work well when (a_i) does contain elements with high multiplicity, others work well when all multiplicities are small. The aim of this article is to initiate a systematic approach to property B via the highest occurring multiplicities. Our main results are the following: denote by m_1 >= m_2 the two maximal multiplicities of (a_i), and suppose that n is sufficiently big and prime. Then (a_i) contains a zero-sum in any of the following cases: when m_2 >= 2/3n, when m_1 > (1-c)n, and when m_2 < cn, for some constant c > 0 not depending on anything.
For a prime number p greater than 6000, the Olson's constant for the group Z_p+Z_p is given by Ol(Z_p+Z_p)=p-1+Ol(Z_p).
Let A be a zero-sum free subset of Z_n with |A|=k. We compute for k\le 7 the least possible size of the set of all subset-sums of A.
We prove some conditions on the existence of natural boundaries of Dirichlet series. We show that generically the presumed boundary is the natural one. We also give an application of natural boundaries in determining asymptotic results.
We determine Davenport's constant for all groups of the form $\Z\_3\oplus \Z\_3\oplus\Z\_{3d}$.