2015/10/31 by Michael Milgram
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Critical line #Differential equation #Function (biology) #Functional equation #Line (geometry) #Mathematical analysis #Mathematics #Pure mathematics #Quantum chaos and dynamical systems #Riemann hypothesis #Riemann zeta function #Simple (philosophy) #Zero (linguistics) #advanced mathematical theories #math.CA #math.NT #msc:11M06 #msc:11M26 #msc:11M99 #msc:26A09 #msc:30B40 #msc:30C15 #msc:30E20 #msc:33B99 #msc:33C47 #msc:33F99
paper · pdf · doi:10.1080/23311835.2016.1179246
published as Cogent Mathematics, Vol. 3, number 1, June 14, 2016, Article: 1179246 · This is the final published version, Cogent Mathematics, 2016
openalex publication_date 2016/04/19 · arxiv created 2016/09/10 · arxiv updated 2018/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
An equivalent, but variant form of Riemann’s functional equation is explored, and several discoveries are made. Properties of Riemann’s zeta function ζ(s), from which a necessary and sufficient condition for the existence of zeros in the critical strip, are deduced. This in turn, by an indirect route, eventually produces a simple, solvable, differential equation for arg(ζ(s)) on the critical line s=1/2+iρ, the consequences of which are explored, and the “LogZeta" function is introduced. A singular linear transform between the real and imaginary components of ζ and ζ′ on the critical line is derived, and an implicit relationship for locating a zero (ρ=ρ0) on the critical line is found between the arguments of ζ(1/2+iρ) and ζ′(1/2+iρ). Notably, the Volchkov criterion, a Riemann Hypothesis (RH) equivalent, is analytically evaluated and verified to be half equivalent to RH, but RH is not proven. Numerical results are presented, some of which lead to the identification of <i>anomalous zeros</i>, whose existence in turn suggests that well-established, traditional derivations such as the Volchkov criterion and counting theorems require re-examination. It is proven that the derivative ζ′(1/2+iρ) will never vanish on the perforated critical line (ρ≠ρ0). Traditional asymptotic and counting results are obtained in an untraditional manner, yielding insight into the nature of ζ(1/2+iρ) as well as very accurate asymptotic estimates for distribution bounds and the density of zeros on the critical line.