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.

We introduce a Selberg type zeta function of two variables which interpolates several higher Selberg zeta functions. The analytic continuation, the functional equation and the determinant expression of this function via the Laplacian on a Riemann surface are obtained.