From Arithmetic to Zeta-Functions

Let S _k (m): = 1k + 2k + ⋯ + (m − 1)k denote a power sum. In 2011 Bernd Kellner formulated the conjecture that for m ≥ 4 the ratio S _k (m + 1)∕S _k (m) of two consecutive power sums is never an integer. We will develop some techniques that allow one to exclude many integers ρ as a ratio and combine them to exclude the integers 3 ≤ ρ ≤ 1501 and, assuming a conjecture on irregular primes to be true, a set of density 1 of ratios ρ. To exclude a ratio ρ one has to show that the Erdős–Moser type equation (ρ − 1)S _k (m) = mk has no non-trivial solutions.

Nagell conjectured in the 1930s that the set of discriminants for which the negative Pell equation has an integral solution has an explicitly given positive proportion within the set of discriminants having no prime factor congruent to 3 modulo 4. In a series of papers, Fouvry and Klüners succeeded in showing that the order of magnitude of such discriminants up to x is indeed x(logx)−1∕2. Here we present a short independent argument that the order of magnitude is at least x(logx)−0. 62.

In support of a still little known, general principle according to which the structure of the set of prime factors of an integer is statistically governed by its actual cardinal, we show that, given any ɛ > 0, the conditional probability that an integer with exactly k prime factors has a divisor in a dyadic interval ]y, 2y] approaches 0 as y → ∞ if 2(1+ɛ)k < logy while it remains larger than a strictly positive constant when 2(1−ɛ)k > logy.

Estimates are obtained for the number of natural numbers below a parameter that do not have a representation as the sum of two squares of primes and a kth power of a prime. These improve earlier bounds in the order of magnitude. The method is then also applied to some related questions.

The Liouville inequality gives a lower bound for the distance between two distinct algebraic numbers in terms of their heights and their degrees. We refine the classical estimate in the special case where one of the algebraic numbers is very close to one of its Galois conjugates.

In this paper, transcendence results and, more generally, results on the algebraic independence of functions and their values are proved via Mahler’s analytic method. Here the key point is that the functions involved satisfy certain types of functional equations as G _d (zd) = G _d (z) − z∕(1 − z) in the case of $$G_{d}(z):=\sum _{h\geq 0}z^{d^{h} }/(1 - z^{d^{h} })$$ for d ∈ { 2, 3, 4, …}. In 1967, these particular functions G _d (z) were arithmetically studied by W. Schwarz using Thue–Siegel–Roth’s approximation method.

We extend results of Jagy and Kaplansky and the present authors and show that for all k ≥ 3 there are infinitely many positive integers n, which cannot be written as x2 + y2 + zk = n for positive integers x, y, z, where for $$k\not\equiv 0\bmod 4$$ a congruence condition is imposed on z. These examples are of interest as there is no congruence obstruction itself for the representation of these n. This way we provide a new family of counterexamples to the Hasse principle or strong approximation.

Mean value theorems for multiplicative arithmetic functions are applied to demonstrate uniformity of sign changes in the Fourier coefficients of automorphic forms.

Let $$\mathcal{E}_{k,l}(\alpha ) =\sum _{q_{m}\equiv l\pmod k}\vert q_{m}\alpha - p_{m}\vert$$ be error sum functions formed by convergents $$p_{m}/q_{m}$$ $$(m \geq 0)$$ of a real number $$\alpha$$ satisfying the arithmetical condition $$q_{m} \equiv l\pmod k$$ with $$0 \leq l <k$$ . The functions $$\mathcal{E}_{k,l}$$ are Riemann-integrable on $$[0,1]$$ , so that the integrals $$\int _{0}^{1}\mathcal{E}_{k,l}(\alpha )\,d\alpha$$ exist as the arithmetical means of the functions $$\mathcal{E}_{k,l}$$ on $$[0,1]$$ . We express these integrals by multiple sums on rational terms and prove upper and lower bounds. In the case when $$l$$ vanishes (i.e. $$k$$ divides $$q_{m}$$ ) and when the smallest prime divisor $$p_{1}$$ of $$k = p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots p_{t}^{a_{t}}$$ satisfies $$p_{1}> k^{\varepsilon }$$ for some positive real number $$\varepsilon$$ , we have found an asymptotic expansion in terms of $$k$$ , namely $$\int _{0}^{1}\mathcal{E}_{k,0}(\alpha )\,d\alpha =\zeta (2)\big(2\zeta (3)k^{2}\big)^{-1} + \mathcal{O}\big(3^{t}k^{-2-\varepsilon }\big)$$ . This result includes all integers $$k$$ which are of the form $$k = p^{a}$$ for primes $$p$$ and integers $$a \geq 1$$ .

For 0 < α ≤ 1 and 0 < λ ≤ 1, λ rational, we consider the sum of values of the Lerch zeta-function L(λ, α, s) taken at the nontrivial zeros of the Dirichlet L-function L(s, χ), where $$\chi \bmod Q$$ , Q ≥ 1, is a primitive Dirichlet character.

We give an overview of Schur–Weyl dualities involved in the representation theory of orthogonal and symplectic groups and of the properties of a new class of algebras occurring in this context, the Brauer Schur algebras.

The purpose of this article is to describe some questions which have arisen from discussions in the working group of Wolfgang Schwarz in the early 1970s. We concentrate on problems concerning arithmetical functions and deal with investigations of almost-even, limit-periodic, and almost-periodic functions. We give a survey of relevant results by Schwarz and Schwarz–Spilker, respectively, and add corresponding contributions of the author.

In this overview paper, presented at the meeting ELAZ2014, Hildesheim, July 28–August 1, 2014, we present some selected works of the eminent mathematician Wolfgang Schwarz. This choice is personal and reflects the common research interest of the author and Prof. Schwarz.

This article determines the order of magnitude of integers not exceeding x that can be written as sums of two squares of integers that are themselves sums of two squares. The tools include Selberg’s sieve and contour integration in the spirit of the Selberg-Delange method.

In the paper, a joint discrete universality theorem on approximation of a pair of analytic functions by shifts of periodic zeta-functions and periodic Hurwitz zeta-functions is obtained. For the proof the linear independence over $$\mathbb{Q}$$ of a certain set is used.

The essay is the author’s very personal retrospective on the life and work of Wolfgang Schwarz. Besides biographical details of Schwarz’s private life and his professional career, it gives account of his work on multiplicative functions in collaboration with several colleagues.

We give a survey on classical and recent applications of dynamical systems to number theoretic problems. In particular, we focus on normal numbers, also including computational aspects. The main result is a sufficient condition for establishing multidimensional van der Corput sets. This condition is applied to various examples.

The Nyman–Beurling criterion is a well-known approach to the Riemann Hypothesis. Certain integrals over Dirichlet series appearing in this approach can be expressed in terms of cotangent sums. These cotangent sums are also associated with the Estermann zeta function. In this paper improvements as well as further generalizations of asymptotic formulas regarding the relevant cotangent sums are obtained. The main result of this paper is the existence of a unique positive measure μ on $$\mathbb{R}$$ with respect to which normalized versions of these cotangent sums are equidistributed. We also consider the moments of order 2k as a function of k.

Similarly as in number theory one may define the notion of an additive function in the set of all mappings of a finite set into itself. If a mapping is sampled uniformly at random, the function becomes a sum of dependent random variables. Estimation of its variance via the sum of variances of the summands is a non-trivial problem. We give an answer analogously to the Turán-Kubilius inequality, well known in probabilistic number theory.

Adolf Hurwitz is rather famous for his celebrated contributions to Riemann surfaces, modular forms, diophantine equations and approximation as well as to certain aspects of algebra. His early work on an important generalization of Dirichlet’s L-series, nowadays called Hurwitz zeta-function, is the only published work settled in the very active field of research around the Riemann zeta-function and its relatives. His mathematical diaries, however, provide another picture, namely a lifelong interest in the development of zeta-function theory. In this note we shall investigate his early work, its origin, and its reception, as well as Hurwitz’s further studies of the Riemann zeta-function and allied Dirichlet series from his diaries. It turns out that Hurwitz already in 1889 knew about the essential analytic properties of the Epstein zeta-function (including its functional equation) 13 years before Paul Epstein.

Selberg sums are the analogues over finite fields of certain integrals studied by Selberg in 1940s. The original versions of these sums were introduced by R.J. Evans in 1981, and following an elegant idea of G.W. Anderson in 1991 they were evaluated by Anderson, Evans and P.B. van Wamelen. In 2007 the author noted that these sums and certain generalizations of them appear in the study of the distribution of Gauss sums over a rational function field over a finite field. The distribution of Gauss sums is closely related to the distribution of the values of the discriminant of polynomials of a fixed degree. Here we shall take this up further. The main goal here is to establish the basic properties of Selberg sums and to formulate the problems which arise from this point of view.

In the present work we prove a number of results about gaps between consecutive primes. The proofs need the method of Y. Zhang which led to the proof of infinitely many bounded gaps between primes. Several of the results refer to the so-called Polignac numbers which we define as those even integers which can be written in infinitely many ways as the difference of two consecutive primes. Others refer to several 60–70 years old conjecture of Paul Erdős about the distribution of the normalized gaps between consecutive primes and about the distribution of the ratio of consecutive primegaps. The methods involve an extended version of Zhangs method, a property of the GPY weights proved by the author a few years ago and other ideas as well.

Alomair et al. (J Math Cryptol 4(2):121–148, 2010, Lemma 3.1) noticed the following result which seems not to appear previously explicitly in the literature: Given a nonzero $$a \in \mathbb{Z}_{n}$$ , the ring of residues modulo n, such that gcd(a, n) = d | b, not only there exists an element $$x \in \mathbb{Z}_{n}$$ such that $$x \cdot a \equiv b\pmod n$$ , but that there even exists an invertible element $$x \in \mathbb{Z}_{n}^{{\ast}}$$ such that $$x \cdot a \equiv b\pmod n$$ . Their sufficient and necessary condition for this says that gcd(b∕d, n∕d) = 1 with d as above.A typical structure result on finite commutative semigroup says that the multiplicative semigroup of $$\mathbb{Z}_{n}$$ decomposes into the so-called maximal subsemigroups belonging to the idempotents of $$\mathbb{Z}_{n}$$ . Each such semigroup contains a maximal subgroup having for its identity the corresponding idempotent. In general this subgroup is a proper subset of the maximal subsemigroup containing it. However, the group of elements of $$\mathbb{Z}_{n}$$ coprime to n is an example of the case when this maximal subsemigroup and the maximal subgroup coincide (both evidently belonging to the idempotent 1).In what follows we prove that if a congruence $$x \cdot a \equiv b\pmod n$$ is solvable there always exists a solution in the maximal semigroup belonging to the idempotent given by the divisor δ = gcd(b∕d, n∕d) and if δ is a unitary divisor of n then there even exists a solution in the maximal subgroup belonging to the idempotent given by δ.

This article reflects some of the authors’ perspectives on and personal experiences with the interplay between number theory and graph theory. To start with, we touch on a few historical milestones of the rewarding relations between the two disciplines. Later on, we shall address more recent developments without any ambition to strive for completeness.

The leading coefficient of the nth Stern polynomial defined by Klavžar et al. (Adv Appl Math 39:86–95, 2007) is expressed in terms of the binary expansion of n.

We show that universal elliptic Carmichael numbers do not exist, answering a question of Silverman. Moreover, we show that the probability that an integer n, which is not a prime power, is an elliptic Carmichael number for a random curve E with good reduction modulo n, is bounded above by $$\mathcal{O}(\log ^{-1}n)$$ . If we choose both n and E at random, the probability that n is E-Carmichael is bounded above by $$\mathcal{O}(n^{-1/8+\epsilon })$$ .

This paper provides a survey of results on the greatest prime factor, the number of distinct prime factors, the greatest squarefree factor and the greatest m-th powerfree part of a block of consecutive integers, both without any assumption and under assumption of the abc-conjecture. Finally we prove that the explicit abc-conjecture implies the Erdős–Woods conjecture for each k ≥ 3.

Let f(n) be the Fibonacci-sequence defined by f(n + 2) = f(n) + f(n + 1), f(0) = 0, f(1) = 1, and let t _s (n) : = gcd( f(n) + s, f(n + 1) + s), s integer. In 2011 K.-W. Chen has proved that the function t _s (n) is bounded if s exceeds 1. THEOREM: Let n and s be integers. (1) t_s(n) divides s2+ (−1)n; (2) if m := s4− 1 is not 0, then t_s(n) is simply periodic; a period p is defined by $$f(\,p) \equiv 0\bmod m$$ , $$f(\,p + 1) \equiv 0\bmod m$$ . There are explicit formulas of t _s (n) and generalisations to a wider class of recursive second-order sequences.

In 1962, Erdős proved that every real number can be represented as a sum and as a product of two Liouville numbers. This has been generalized by Rieger and Schwarz. In this note we shall give an analysis of these results and their proofs. Moreover, we consider a certain subclass of Liouville numbers and prove similar results for this subclass. Since Wolfgang Schwarz had been very much interested in the history of mathematics, and the author shares this interest, he could not resist to include a few historical remarks (and footnotes) on transcendental numbers and Baire’s category theorem which might be interesting for the reader.

For an algebraic number field $$K\neq \mathbb{Q}$$ we prove that the unit disc is a natural boundary of the power series $$\sum _{\mathfrak{a}\neq \mathfrak{o}}z^{N(\mathfrak{a})}$$ , $$\mathfrak{a}$$ running through the integral ideals of K and N denoting the norm function. As an application, we deduce the same result for power series $$\sum _{\mathfrak{a}\neq \mathfrak{o}}g(\mathfrak{a})\,z^{N(\mathfrak{a})}$$ with specific multiplicative coefficients $$g(\mathfrak{a})$$ thereby extending known results to algebraic number fields.

A simple proof of Hardy’s theorem on the existence of power series which converge uniformly on the unit circle but not absolutely.

A quasiplatonic curve can be defined over the rationals if it has a regular dessin with abelian automorphism group.