Rasa Steuding

Effective uniform approximation by the Riemann zeta-function

We apply an effective multidimensional Ω result of Voronin in order to obtain effective universality-type theorems for the Riemann zeta-function. We further use this approach to study approximation properties of linear combinations of derivatives of the zeta-function.

Large Gaps Between Zeros of the Zeta-Function on the Critical Line and Moment Conjectures from Random Matrix Theory

AbstractDenote by γn the positive ordinates of the non-trivial zeros of the zeta-function in ascending order. Assuming the Riemann hypothesis and conjectural asymptotic formulae for the (continuous and discrete) 2kth and 4kth moment for the zeta-function originating from random matrix theory, we prove that for any fixed positive integer r more than cN(T) (log T)−4k2 of the ordinates γn ∈ [0, T] satisfy

A modified Möbius μ-function

({\gamma_n+r}-\gamma_n) {{\rm log}\gamma_n \over 2\pi r} \geq \theta \ \ \ \ \ {\rm for \ any} \ \theta \leq {4k \over \pi er}

From Arithmetic to Zeta-Functions

, where c is a computable positive constant depending on k, θ and r.

From Arithmetic to Zeta-Functions : Number Theory in Memory of Wolfgang Schwarz

Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point Pλ of infinite order in the Mordell-Weil group of the elliptic curve Y2 = X3 − n2X over ℚ(λ).