University of Lethbridge

In this talk, we will discuss an estimate for a discrete mean value of the Riemann zeta function and its derivatives multiplied by Dirichlet polynomials. Assuming the Riemann Hypothesis, we obtain a lower bound for the 2kth moment of all the...

We discuss the role of long Dirichlet polynomials in number theory. We first survey some applications of mean values of long Dirichlet polynomials over primes in the theory of the Riemann zeta function which includes central limit theorems and pair...

It is conjectured that there is no projective plane of order 12. Balanced splittable Hadamard matrices were introduced in 2018. In 2023, it was shown that a projective plane of order 12 is equivalent to a balanced multi-splittable Hadamard matrix of...

In this talk, we prove that |ζ(σ+it)|≤ 70.7 |t|4.438(1-σ)^{3/2} log2/3|t| for 1/2≤ σ ≤ 1 and |t| ≥ 3, combining new explicit bounds for the Vinogradov integral with exponential sum estimates. As a consequence, we improve the explicit zero-free region...

By using an identity relating a sum to an integral, we obtain a family of identities for the averages $M(X)=\sum_{n\leq X} \mu(n)$ and $m(X)=\sum_{n\leq X} \mu(n)/n$. Further, by choosing some specific families, we study two summatory functions...

In the late 90's, Keating and Snaith used random matrix theory to predict the exact leading terms of conjectural asymptotic formulas for all integral moments of the Riemann zeta-function. Prior to their work, no number-theoretic argument or heuristic...

In the 1960s Shanks conjectured that the $\zeta(\rho)$, where $\rho$ is a non-trivial zero of zeta, is both real and positive in the mean. Conjecturing and proving this result has a rich history, but efforts to generalise it to higher moments have so...

The size function h0 for a number field is analogous to the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. Van der Geer and Schoof conjectured that h0 attains its maximum at the trivial class of Arakelov divisors if that...

This talk will begin with a study on explicit bounds for ψ(x) starting with the work of Rosser in 1941. It will also cover various improvements over the years including the works of Rosser and Schoenfeld, Dusart, Faber-Kadiri, Platt-Trudgian, Büthe...

Historically, the notion of Pólya fields dates back to some works of George Pólya and Alexander Ostrowski, in 1919, on entire functions with integer values at integers; a number field $K$ with ring of integers $\mathcal{O}_K$ is called a Pólya field...