Some problems involving the classical Hardy function $$Z\left(t\right)=\zeta \left(\frac{1}{2}+it\right){\left(\chi \left(\frac{1}{2}+it\right)\right)}^{-1/\phantom{12}2},\zeta \left(s\right)=\chi \left(s\right)\zeta \left(1-s\right)$$ , are discussed. In particular we discuss the odd moments of Z(t) and the distribution of its positive and negative values.

Let σ(n) denote the sum of positive divisors of the integer n, and let ϕ denote Euler's function, that is, ϕ(n) is the number of integers in the interval [1,n] that are relatively prime to n. It has been conjectured by Mąkowski and Schinzel that σ(ϕ(n))/n ≥ 1/2 for all n. We show that σ(ϕ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(ϕ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that ϕ(n-ϕ(n)) < ϕ(n)...

Letting $T_n$ (resp. $U_n$) be the n-th Chebyshev polynomials of the first (resp. second) kind, we prove that the sequences ${\left(X^k{T}_{n-k}\right)}_k$ and ${\left(X^k{U}_{n-k}\right)}_k$ for n - 2⎣n/2⎦ ≤ k ≤ n - ⎣n/2⎦ are two basis of the ℚ-vectorial space ${}_n\left[X\right]$ formed by the polynomials of ℚ[X] having the same parity as n and of degree ≤ n. Also $T_n$ and $U_n$ admit remarkableness integer coordinates on each of the two basis.

We study a special class of $(0,m,2)$-nets in base 2. In particular, we are concerned with the two-dimensional Hammersley net that plays a special role among these since we prove that it is the worst distributed with respect to the star discrepancy. By showing this, we also improve an existing upper bound for the star discrepancy of digital $(0,m,2)$-nets over ${\mathbb{Z}}_2$. Moreover, we show that nets with very low star discrepancy can be obtained by transforming the Hammersley point set in a suitable way.

A recent result of Balandraud shows that for every subset $S$ of an abelian group $G$ there exists a non trivial subgroup $H$ such that $\left|TS\right|\le \left|T\right|+\left|S\right|-2$ holds only if $H\subset Stab\left(TS\right)$. Notice that Kneser’s Theorem only gives $\left\{1\right\}\ne Stab\left(TS\right)$.This strong form of Kneser’s theorem follows from some nice properties of a certain poset investigated by Balandraud. We consider an analogous poset for nonabelian groups and, by using classical tools from Additive Number Theory, extend some of the above results. In particular we obtain short proofs of Balandraud’s...

Let $S$ be a subset of ${\mathbf{F}}_q$, the field of $q$ elements and $h\in {\mathbf{F}}_q\left[x\right]$ a polynomial of degree $d\&gt;1$ with no roots in $S$. Consider the group generated by the image of $\{x-s\mid s\in S\}$ in the group of units of the ring ${\mathbf{F}}_q\left[x\right]/\left(h\right)$. In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]. The bounds have also applications to graph theory and to the bounding of the number of rational points on abelian covers of the projective...