Breadcrumb
Generalised prime systems with periodicity
Wed 23 January 2013, 16:00
Titus Hilberdink
Reading
Organisers: Tim Dokchitser, Daniel Loughran
ABSTRACT
We study Mellin transforms $\hat{N}(s)=\int_{1-}^{\infty} x^{-s} dN(x)$ for which $N(x)-x$ is periodic with period 1 in order to investigate `flows' of such functions to Riemann's $\zeta(s)$ and the possibility of proving the Riemann Hypothesis with such an approach. We show that, excepting the trivial case where $N(x)=x$, the supremum of the real parts of the zeros of any such function is at least $\frac{1}{2}$.
Furthermore, we study generalised prime systems (both discrete and continuous) for which the `integer counting function' $N(x)$ has the above property, and more generally, that $N(x)-cx$ is periodic for some $c>0$. We show that this is extremely rare. In particular, we show that the only such system for which $N$ is continuous is the trivial system with $N(x)-cx$ constant, while if $N$ has finitely many discontinuities per bounded interval, then $N$ must be the counting function of the g-prime system containing the usual primes except for finitely many.