Radhakrishnan Nair | Searcharxiv

arXiv2021

On good universality and the Riemann hypothesis

We use subsequence and moving average ergodic theorems applied to Boole's transformation and its variants and their invariant measures on the real line to give new characterisations of the Lindelh{ö}f Hypothesis and the Riemann hypothesis. These ideas are then used to study the value distribution of Dirichlet L series, and the zeta functions of Dedekind, Hurwitz and Riemann and their derivatives. This builds on earlier work of R. L. using Birkhoff's ergodic theorem and probability theory.

math.NT↗

arXiv2020

On Polynomials in Primes, Ergodic Averages and Monothetic Groups

Let $G$ denote a compact monothetic group, and let $$ρ(x) = α_k x^k + \ldots + α_1 x + α_0,$$ where $α_0, \ldots , α_k$ are elements of $G$ one of which is a generator of $G$. Let $(p_n)_{n\geq 1}$ denote the sequence of rational prime numbers. Suppose $f \in L^{p}(G)$ for $p> 1$. It is known that if $$A_{N}f(x) := {1 \over N} \sum_{n=1}^{N} f(x + ρ(p_n)) \qquad (N=1,2, \ldots ),$$ then the limit $\lim _{n\to \infty} A_Nf(x)$ exists for almost all $x$ with respect Haar measure. We show that if $G$ is connected then the limit is $\int_{G} f dλ$. In the case where $G$ is the $a$-adic integers, which is a totally disconnected group, the limit is described in terms of Fourier multipliers which are generalizations of Gauss sums.

math.NT↗