In this article, we show that|N(T)?T2?log?(T2?e)|?0.1038log?T+0.2573log?log?T+9.3675 where N(T) denotes the number of non-trivial zeros ?, with 0<Im(?)?T, Riemann zeta function. This improves previous result Trudgian for sufficiently large T. The improvement comes from use various subconvexity bounds and ideas work Bennett et al. on counting Dirichlet L-functions.

Riemann zeta function is an important object of number theory. It was also used for description of disordered systems in statistical mechanics. We show that Riemann zeta function is also useful for the description of integrable model. We study XXX Heisenberg spin 1/2 anti-ferromagnet. We evaluate a probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in ther...