Number Theory

A zero-free region is a subset of the complex plane where the Riemann zeta-function does not vanish. Such regions have historically been used to further our understanding of prime-number distributions. In the classical approach, we first assume that a zero exists off the critical line, then arrive at an inequality involving its real and imaginary parts. One notable aspect of the classical argument is that it does not require any knowledge about the relationship between the zeroes. However, it is well known that the location of a hypothetical zero depends strongly on the behaviour of nearby zeroes—for instance, N. Levinson showed in 1969 that if zeroes of the zeta-function are well-spaced near the 1-line, then we can obtain a zero-free region stronger than any that are currently known. In this talk we will discuss some ideas on how one might incorporate information about distributions of hypothetical zeroes to improve existing zero-free regions.

In classical comparative prime number theory, it is customary to assume some kind of linear independence hypothesis about the zeros of the underlying L-functions. These hypotheses are completely out of reach of current methods. However, in the function field case, it is sometimes possible to prove them, or at least to show they hold generically. In this talk I will present recent results in comparative prime number theory over function fields that establish infinite families of “irreducible polynomial races” which we can study unconditionally. Some of those results are joint work with L. Devin, D. Keliher, and W. Li.

In this talk we present new unconditional estimates on $\Delta k(x)$, the remainder term associated with the generalised divisor function, for large $k$. By combining new estimates of exponential sums and Carlson’s exponent, we show that $\Delta k(x) \ll x^{1−1.224(k−2.36)^{−2/3}}$ for $k \geq 58$ and $\Delta k(x) \ll x^{1−1.889k^{−2/3}}$ for all sufficiently large fixed $k$. This is joint work with Andrew Yang.

In joint work with Régis de la Bretéche and Daniel Fiorilli, we consider weighted
moments for the distribution of Frobenius substitutions in conjugacy classes of
Galois groups of normal number field extensions. The question is inspired by work
of Hooley and recent progress by de la Bretéche–Fiorilli in the case of moments for
primes in arithmetic progressions. As in their work, the results I will discuss are
conditional on the Riemann Hypothesis and confirm that the moments considered
should be Gaussian. Time permitting, I will address a different notion of moments
that can be considered in the same context and that leads to non-Gaussian families
for particular Galois group structures.

The Wiener–Ikehara Tauberian theorem is an important theorem giving an asymptotic formula for the sum of coefficients of a Dirichlet series. In this talk, we present a simple and elegant proof of the Wiener–Ikehara Tauberian theorem which relies only on basic Fourier analysis and known estimates for the given Dirichlet series. This method allows us to derive a version of the WienerIkehara theorem with an error term. This is joint work with Prof. M. Ram Murty and Prof. Akshaa Vatwani.