2021/08/08 by Yochay Jerby, Jerby, Yochay
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2108.03716
openalex publication_date 2021/08/08 · openalex created_date 2021/08/16 · openalex updated_date 2026/07/28
For N ∈ ℕ consider the N-th section of the approximate functional equation ζN(s)= ∑n =1 N Bn(s), where Bn(s)= (1)/(2) [ n-s + χ(s) ⋅ ns-1 ]. Our aim in this work is to introduce a new approach for the Riemann hypothesis by studying the way pairs of consecutive zeros of ζN(s) change with respect to N. For the initial stage, it is known that the non-trivial zeros of ζ1(s) all lie on the critical line Re(s)=(1)/(2). In the region 2N ≤ Im(s) ≤ 2 π(N+1) the function ζN(s) serves as an approximation of ζ(s) itself, and it was conjectured by Spira that in this region ζN(s) also admits zeros only on the critical line. We show that the appearance of zeros of a section off the critical line can be realized as the result of two consecutive zeros meeting and pushing each other off the critical line as N changes, a process to which we refer to as a collision of zeros. Based on a study of the properties of ζN(s), we suggest a way of re-arranging the order of summation of the elements Bn(s) in ζN(s) with N= [ (Im(s))/(2) ] that is expected to avoid collisions altogether, we refer to such a re-arrangement as a repelling re-arrangement. In particular, establishing that the suggested repelling re-arrangement indeed avoids collisions for any pair of zeros would imply RH.