Applications of mean value theorems to the theory of the Riemann zeta function

Abstract

Abstract In the first half of these lectures we discuss mean value theorems for functions representable by Dirichlet series and sketch several applications to the distribution of zeros of the Riemann zeta function. These include the clustering of zeros about the critical line, Levinson's result that a third of the zeros are on the critical line, and a conditional result on the number of simple zeros. The second half focuses on mean values of Dirichlet polynomials, particularly “long” ones. We then show how these can be used to investigate the pair correlation of the zeros of the zeta function and to conjecture the sixth and eighth power moments of the zeta function on the critical line. What is a Mean Value Theorem? By a mean value theorem we mean an estimate for the average of a function. When F ( s ) has a convergent Dirichlet series expansion in some half–plane Re s > σ 0 of the complex plane, we typically take the average over a vertical segment: The path of integration here need not lie in this half–plane. For example, we would like to know the size of the integrals for σ ≥ ½ and k a positive integer. Here F ( s ) = ζ( s ) k and its Dirichlet series converges only for σ > 1. There are many variations. For example, one can consider a discrete mean value where the points σ r + it r lie in ℂ.