Open AccessDeduction of Semi-Optimal Mollifier for Obtaining Lower Bound for N 0 ( T ) for Riemann's Zeta-Function
Author(s) -
Norman Levinson
Publication year - 1975
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.72.1.294
Subject(s) - riemann hypothesis , mathematics , function (biology) , term (time) , key (lock) , upper and lower bounds , riemann zeta function , smoothing , combinatorics , euler's formula , usable , mathematical analysis , computer science , physics , statistics , quantum mechanics , computer security , evolutionary biology , world wide web , biology
A mollifier played a key role in showingN 0 (T ) > 1/3N (T ) for largeT in ref. 1 [Levinson, N. (1974)Advan. Math. 13, 383-436]. A basic problem in ref. 1 was that of obtaining an upper bound for a sum of two terms, one larger than the other. Here a deductive procedure is given for finding a mollifier that actually minimizes the larger term. An Euler-Lagrange equation is obtained. (Optimization of the sum of both the major and minor terms appears to be formidable.) The actual improvement effected by the optimized mollifier over the ad hoc mollifier of ref. 1 is unfortunately only 1.4%. To obtain a usable mollifier it is necessary to blur the optimization procedure by smoothing at several stages of the deduction. The procedure is of more interest than the particular application because of the small improvement in this case.