A subset $S$ of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of $S$ is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, $p$-groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general...

Improving on some results of J.-L. Nicolas [15], the elements of the set $𝒜=𝒜(1+z+z^3+z^4+z^5)$, for which the partition function $p(𝒜,n)$ (i.e. the number of partitions of $n$ with parts in $𝒜$) is even for all $n\ge 6$ are determined. An asymptotic estimate to the counting function of this set is also given.

In this paper we consider the asymptotic formula for the number of the solutions of the equation $p_1+p_2+p_3=N$ where $N$ is an odd integer and the unknowns $p_i$ are prime numbers of the form $p_i=\left[n^{1/{\gamma }_i}\right]$. We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case ${\gamma }_1={\gamma }_2={\gamma }_3=\gamma$ our theorem implies that every sufficiently large odd integer $N$ may be written as the sum of three Piatetski-Shapiro primes of type $\gamma$ for $50/53$ < $\gamma$ < $1$.

A partition of a positive integer n is a nonincreasing sequence of positive integers with sum $n.$ Here we define a special class of partitions. 1. Let $t\ge 1$ be a positive integer. Any partition of n whose Ferrers graph have no hook numbers divisible by t is known as a t- core partitionof $n.$ The hooks are important in the representation theory of finite symmetric groups and the theory of cranks associated with Ramanujan’s congruences for the ordinary partition function [3,$$4,$$6]. If $t\ge 1$ and $n\ge 0$, then we define...

Introduction. The problem of determining the formula for $P_S\left(n\right)$, the number of partitions of an integer into elements of a finite set S, that is, the number of solutions in non-negative integers, $h_{s₁},...,h_{s_k}$, of the equation hs₁ s₁ + ... + hsk sk = n, was solved in the nineteenth century (see Sylvester [4] and Glaisher [3] for detailed accounts). The solution is the coefficient of$xⁿin$[(1-xs₁)... (1-xsk)]-1, expressions for which they derived. Wright [5] indicated a simpler method by which to find part of the solution...

We examine the $q$-Pell sequences and their applications to weighted partition theorems and values of $L$-functions. We also put them into perspective with sums of tails. It is shown that there is a deeper structure between two-variable generalizations of Rogers-Ramanujan identities and sums of tails, by offering examples of an operator equation considered in a paper published by the present author. The paper starts with the classical example offered by Ramanujan and studied by previous authors noted...

We prove a 2-terms Weyl formula for the counting function $N\left(\mu \right)$ of the spectrum of the Laplace operator in the Euclidean disk with a sharp remainder estimate $O\left({\mu }^{2/3}\right)$.