arXiv subjects

Werner Georg Nowak

Publications and source records attributed to Werner Georg Nowak.

5 recordsLinked to original sources

arXiv2013

On the average number of subgroups of the group $\Z_m \times \Z_n$

Let $\Z_m$ be the group of residue classes modulo $m$. Let $s(m,n)$ and $c(m,n)$ denote the total number of subgroups of the group $\Z_m \times \Z_n$ and the number of its cyclic subgroups, respectively, where $m$ and $n$ are arbitrary positive integers. We derive asymptotic formulas for the sums $\sum_{m,n\le x} s(m,n)$, $\sum_{m,n\le x} c(m,n)$ and for the corresponding sums restricted to $\gcd(m,n)>1$, i.e., concerning the groups $\Z_m \times \Z_n$ having rank two.

math.NT↗

arXiv2012

On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions

The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series which involves a product of Riemann zeta-functions of a special form.

math.NT↗

arXiv2004

The lattice point discrepancy of a body of revolution: Improving the lower bound by Soundararajan's method

For a convex body B in three-dimensional Euclidean space, which is invariant under rotations around one coordinate axis and has a smooth boundary of bounded nonzero curvature, the lattice point discrepancy (number of integer points minus volume) of a linearly enlarged copy of B is estimated from below. On the basis of a recent method of K. Soundararajan, an Omega-bound is obtained that improves upon all earlier results of this kind.

math.NT↗

arXiv2003

Primitive lattice points inside an ellipse

Let Q(u,v) be a positive definite binary quadratic form with arbitrary real coefficients. For large real x, one may ask for the number B(x) of primitive lattice points (integer points (m,n) with gcd(m,n) = 1) in the ellipse disc Q(u,v) < x, in particular, for the remainder term R(x) in the asymptotics for B(x). While upper bounds for R(x) depend on zero-free regions of the zeta-function, and thus, in most published results, on the Riemann Hypothesis, the present paper deals with a lower estimate. It is proved that the absolute value of R(x) is, in integral mean, at least a positive constant c times x^{1/4}. Furthermore, it is shown how to find an explicit value for c, for each specific given form Q.

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arXiv2003

The average number of solutions of the Diophantine equation U^2 + V^2 = W^3 and related arithmetic functions

This paper provides asymptotics with a sharp error term for the Dirichlet summatory function of a certain class of arithmetic functions. The result applies, e.g., to the sums over r^2(n) and r(n^3), where r(m) denotes the number of ways to write m as a sum of two squares of integers.

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