Simple and critical zeros of the Riemann zeta-function

Counting critical zeros via pair correlation under a weaker-than-RH assumption. It was known among experts for some time that Montgomery’s Theorem could be made unconditional. Indeed, the starting point of the proof of Montgomery’s explicit formula is an unconditional explicit formula due to Landau from 1909 (see equation (9) on page 353). Setting , where , the same arguments for Montgomery’s explicit formula and theorem carry through. That one could use an unconditional form of Montgomery’s Theorem to extract information about critical zeros was recently discovered by Siegfred Baluyot, Daniel Goldston, Ade Irma Suriajaya, and myself; henceforth BGST-B. In our first project we did not realize this capability of Montgomery’s method, and erroneously stated in the abstract “The method of proof neither requires nor provides any information on whether any of these zeros are or are not on the critical line where .” In a follow-up paper we realized that, in fact, we could extract information on critical zeros using the pair correlation method, albeit upon an assumption weaker than RH. Roughly, we showed that if the nontrivial zeros of are contained in a narrow vertical box centered on the critical line, then the pair correlation method shows that at least 67.25% of the zeros are both simple and critical. I’ll give a sketch of the argument, highlighting some of the key distinctions from Montgomery’s work. Portions of what follows below appear in our preprint.

In addition to making use of the multiplicity of a nontrivial zero , we use a quantity we refer to as the horizontal multiplicity of . (This framework was suggested to us by Soundararajan and simplified our original classification of the ways in which can occur.) For a fixed real number, the horizontal multiplicity is the number of nontrivial zeros of on the horizontal line , counted with multiplicity. Assuming RH, is either zero or the multiplicity of the zero . Note that if , then on the line , there is one simple zero on the critical line.

Let denote the multiset of nontrivial zeros of for which . We can partition into disjoint multisets of zeros according to their horizontal multiplicity: