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math.NT2026

Central non-vanishing of Dirichlet -functions

Benjamin Durkan, Andrew Pearce-Crump

We prove that, for every sufficiently large modulus , at least of the primitive Dirichlet characters modulo have . We also esta…

math.NT2026

Sharp lower bounds for shifted moments of Dedekind zeta functions

Benjamin Durkan, Nilmoni Karak, Kamalakshya Mahatab

Let be fixed number fields, and let be the compositum of their Galois closures. Assuming GRH for , we prove a sharp lower bound for products of shifted De…

math.NT2026

A sharp almost sure upper bound for partial sums of random multiplicative functions

Benjamin Durkan, Andrew Pearce-Crump

We prove that, for either a Steinhaus random multiplicative function or a Rademacher random multiplicative function , and every , almost surely $|\sum_{n\le x}f(n…

math.NT2026

Sharp Upper Bounds for Moments of Dedekind Zeta Functions

Benjamin Durkan, Nilmoni Karak, Kamalakshya Mahatab

Assuming the Generalised Riemann Hypothesis, we establish conjecturally sharp upper bounds for shifted moments of products of Dedekind zeta functions of arbitrary number fields. Th…

math.NT2026

Amplified moments of the Riemann zeta function

Benjamin Durkan, Timothy Page

We establish asymptotic formulae for two-piece amplified second and fourth moments of the Riemann zeta function. As applications, we obtain unconditional effective lower bounds for…

math.NT2026

Generalisations of the Landau--Gonek Theorem and applications to mean values of zeta

Benjamin Durkan, Christopher Hughes, Andrew Pearce-Crump

The Landau--Gonek Theorem evaluates summed over the non-trivial zeros of the Riemann zeta function. Their result shows great sensitivity to the arithmetic nature of . We p…

math.NT2026

The discrete second moment of mixed derivatives of the Riemann zeta function

Benjamin Durkan, Christopher Hughes, Andrew Pearce-Crump

We establish the full asymptotic for the discrete second moment of the Riemann zeta function of mixed derivatives evaluated at the zeta zeros, providing both unconditional and cond…

math.NT2023

On the Hardy-Ramanujan Theorem

Benjamin Durkan

In this note we prove an effective version of the Hardy--Ramanujan Theorem. For every and every non-negative function on the non-negative integers, we show $$\frac{1}{…