Open AccessGeneralization of Recent Method Giving Lower Bound for N o ( T ) of Riemann's Zeta-Function
Author(s) -
Norman Levinson
Publication year - 1974
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.71.10.3984
Subject(s) - combinatorics , riemann hypothesis , generalization , riemann zeta function , function (biology) , polynomial , upper and lower bounds , physics , mathematics , sigma , mathematical analysis , quantum mechanics , evolutionary biology , biology
Leth (s ) = π-s /2τ(s /2). Then,h (s )ζ(s ) ∼h (s )H (s ) +h (1 -s )H (1 -s ) whereH (s ) = Σ(1 - (logn )/logt /2π)n -s ,n ≤t /2π, led toN o (T ) ≥N (T )/3. Here the extension toH (s ) ∼ Σ P (1 - (logn )/logt /2π)n -s is made where P(x ) is a polynomial such that P(0) = 0 andP (x ) + P(1 -x ) = 1. The earlier case is P(x ) =x . The relevant formulas in the general case can be obtained explicitly by the earlier method used for P(x ) =x , and, indeed, in some respects there is greater simplicity for the general case.