arXiv subjects
Publications and source records attributed to Jörn Steuding.
We provide conditional and unconditional asymptotic formulae for the exponential sums $\sum_γ\,γ^{-iτ}$, where the summation is over the ordinates of the nontrivial zeros $ρ=β+iγ$ of the Riemann zeta-function. In particular, the obtained results are related to the Lindelöf Hypothesis for these ordinates (in the sense of Gonek et al. [10]).
We prove an effective universality theorem of the Riemann zeta-function in short intervals $[T,T+H]$ with $T^{\frac{27}{82}}\le H\le T$ by following an effective multidimensional $Ω$-result of Voronin. Furthermore, we also prove the results in short intervals $[T,T+H]$ with $T^ε\le H\le T$ (for any fixed $ε>0$) under the assumption of the Riemann Hypothesis.
We improve a recent universality theorem for the Riemann zeta-function in short intervals due to Antanas Laurinčikas with respect to the length of these intervals. Moreover, we prove that the shifts can even have exponential growth. This research was initiated by two questions proposed by Laurin\v cikas in a problem session of a recent workshop on universality.
This article deals with applications of Voronin's universality theorem for the Riemann zeta-function $ζ$. Among other results we prove that every plane smooth curve appears up to a small error in the curve generated by the values $ζ(σ+it)$ for real $t$ where $σ\in(1/2,1)$ is fixed. In this sense, the values of the zeta-function on any such vertical line provides an atlas for plane curves. In the same framework, we study the curvature of curves generated from $ζ(σ+it)$ when $σ>1/2$ and we show that there is a connection with the zeros of $ζ'(σ+it)$. Moreover, we clarify under which conditions the real and the imaginary part of the zeta-function are jointly universal.
We prove an equivalent of the Riemann hypothesis in terms of the functional equation (in its asymmetrical form) and the $a$-points of the zeta-function, i.e., the roots of the equation $ζ(s)=a$, where $a$ is an arbitrary fixed complex number.
We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number $a\neq0$ and a function from the Selberg class $\mathcal{L}$, we prove a Riemann-von Mangoldt formula for the number of a-points of the $Δ$-factor of the functional equation of $\mathcal{L}$ and an analog of Landau's formula over these points. From the last formula we derive that the ordinates of these $a$-points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of $\mathcal{L}(s)$ taken at these points.
We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.
For an arbitrary complex number $a\neq 0$ we consider the distribution of values of the Riemann zeta-function $ζ$ at the $a$-points of the function $Δ$ which appears in the functional equation $ζ(s)=Δ(s)ζ(1-s)$. These $a$-points $δ_a$ are clustered around the critical line $1/2+i\mathbb{R}$ which happens to be a Julia line for the essential singularity of $ζ$ at infinity. We observe a remarkable average behaviour for the sequence of values $ζ(δ_a)$.
The class of Dirichlet series associated with a periodic arithmetical function $f$ includes the Riemann zeta-function as well as Dirichlet $L$-functions to residue class characters. We study the value-distribution of these Dirichlet series $L(s;f)$, resp. their analytic continuation in the neighborhood of the critical line (which is the abscissa of symmetry of the related Riemann-type functional equation). In particular, for a fixed complex number $a\neq 0$, we prove for an even or odd periodic $f$ the number of $a$-points of the $Δ$-factor of the functional equation, prove the existence of the mean-value of the values of $L(s;f)$ taken at these points, show that the ordinates of these $a$-points are uniformly distributed modulo one and apply this to show a discrete universality theorem.
We prove explicit formulae for $α$-points of $L$-functions from the Selberg class. Next we extend a theorem of Littlewood on the vertical distribution of zeros of the Riemann zeta-function $ζ(s)$ to the case of $α$-points of the aforementioned $L$-functions. This result implies the uniform distribution of subsequences of $α$-points and from this a discrete universality theorem in the spirit of Voronin is derived.
We study the value-distribution of the Hurwitz zeta-function with algebraic irrational parameter $ζ(s;α)=\sum_{n\geq_0}(n+α)^{-s}$. In particular, we prove effective denseness results of the Hurwitz zeta-function and its derivatives in suitable strips containing the right boundary of the critical strip $1+i\mathbb{R}$. This may be considered as a first "weak" manifestation of universality for those zeta-functions.
In this note we prove that the Selberg zeta-function associated to a compact Riemann surface is pseudo-prime and right-prime in the sense of a decomposition.
We study the values taken by the Riemann zeta-function $ζ$ on discrete sets. We show that infinite vertical arithmetic progressions are uniquely determined by the values of $ζ$ taken on this set. Moreover, we prove a joint discrete universality theorem for $ζ$ with respect to certain permutations of the set of positive integers. Finally, we study a generalization of the classical denseness theorems for $ζ$.
We investigate a descent on simple graphs, starting with the complete graph on $n$ vertices and ending up with the cycle graph by removing one edge after another. We obtain quantitative results showing that graphs with large diameter must have some eigenvalues of large algebraic degree.
In 2000, J. Shallit introduced a special partial ordering of a subset of positive integers and proposed the problem of finding the set of minimal elements with respect to this ordering. Shallit himself solved this problem for the set of primes and also for the set of composite numbers. In this recreational mathematics note, we compute the minimal sets of a few other arithmetically interesting sets and discuss questions on size and shape of minimal sets in general.
We prove an upper bound for the least prime in an irrational Beatty sequence. This result may be compared with Linnik's theorem on the least prime in an arithmetic progression.
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.
We investigate the intersections of the curve $\mathbb{R}\ni t\mapsto ζ({1\over 2}+it)$ with the real axis. We show unconditionally that the zeta-function takes arbitrarily large positive and negative values on the critical line.