Everytopic Seminar

 September 18th, 2007, 1:40 pm – 3:00pm, Goldsmith 226

 

 

Title:  Farey fractions and the Riemann Hypothesis

Speaker: Alan Haynes, Brandeis University

 

 

 

 

Abstract:

Well known theorems originating from the 1920's established that knowing the zero-free region of the Riemann zeta function is equivalent to having an upper bound for the discrepancy of the Farey sequence. It follows that the reduced rationals in the unit interval are as evenly spaced as possible if and only if the Riemann Hypothesis is true. In a paper published in 1971 Martin Huxley proved an analogous statement about the zero free region of any Dirichlet L-function. He showed that the problem of determining the zero free region of such a function is equivalent to that of determining an upper bound for a weighted discrepancy of a subset of the Farey fractions. We will explain all of these facts in more detail and then discuss recent results about properties of special subsets of the Farey fractions.