Timothy Trudgian

An improved upper bound for the error in the zero-counting formulae for Dirichlet L-functions and Dedekind zeta-functions

This paper contains new explicit upper bounds for the number of zeroes of Dirichlet L-functions and Dedekind zeta-functions in rectangles.

On the first sign change of θ(x)-x

Let θ(x) = ∑ p≤x log p. We show that θ(x) < x for 2 < x < 1.39 ·10 . We also show that there is an x < exp(727.951332668) for which θ(x) > x.

An improved upper bound for the argument of the Riemann zeta-function on the critical line

This paper concerns the function S(T), the argument of the Rie- mann zeta-function along the critical line. The main result is that |S(T)| � 0.111logT + 0.275 log logT + 2.450, which holds for all Te.

Improvements to Turing’s method

This paper refines the argument of Lehman by reducing the size of the constants in Turings method. This improvement is given in Theorem 1 and scope for further improvements is also given. Analogous improvements to Dirichlet L-functions and Dedekind zeta-functions are also included.

A proof of the conjecture of Cohen and Mullen on sums of primitive roots

We prove that for all q > 61, every non-zero element in the nite eld Fq can be written as a linear combination of two primitive roots of Fq. This resolves a conjecture posed by Cohen and Mullen.

Linear relations of zeroes of the zeta-function

This article considers linear relations between the non-trivial zeroes of the Riemann zeta-function. The main application is an alternative disproof to Mertens’ conjecture by showing that limsupx!1 M(x)x 1=2 1:6383, and liminfx!1 M(x)x 1=2 1:6383:

A still sharper region where ()-() is positive

. It is known that π(x) li(x) infinitely often, although he did not give an estimate on the first counterexample. The smallest number for which π(x) > li(x) is often called Skewes’ number. For a history of refinements to Skewes’ number, see [SD10, §1]. In 2010, Saouter and Demichel [SD10] proved that π(x) > li(x) in a new region around exp(727.951335792). The main approach there was to refine Lehman’s method, first used in [Leh66]. They also remarked that a region around exp(727.951335426) might contain a sign change; their theorems were not strong enough to prove this. In this article, we derive a stronger version of Lehman’s theorem involving a different weight function. This new theorem enables us to certify the preceding candidate region. We can then improve this region by appealing to a combination of theoretical results and numerical computations.

A new upper bound for |ζ(1+it)|

It is known that

Square-full primitive roots

|\zeta(1+ it)|\ll (\log t)^{2/3}

On the Sum of Two Squares and At Most Two Powers of 2

. This paper provides a new explicit estimate, viz.\