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Vinogradov's Integral and Bounds for the Riemann Zeta Function
Author(s) -
Ford Kevin
Publication year - 2002
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611502013655
Subject(s) - mathematics , combinatorics , riemann hypothesis , exponential function , riemann zeta function , upper and lower bounds , bounding overwatch , lambda , sigma , function (biology) , prime (order theory) , mathematical analysis , physics , quantum mechanics , artificial intelligence , evolutionary biology , computer science , biology
The main result is an upper bound for the Riemann zeta function in the critical strip: ζ ( σ + i t ) ⩽ A | t | B ( 1 − σ ) 3 / 2log 2 / 3 | t | with A = 76.2 and B = 4.45, valid for ½ ⩽ σ ⩽ 1 and |t| ⩾ 3. The previous best constant B was 18.5. Tools include a variant of the Korobov–Vinogradov method of bounding exponential sums, an explicit version of T. D. Wooley's bounds for Vinogradov's integral, and explicit bounds for mean values of exponential sums over numbers without small prime factors, also using methods of Wooley. An auxiliary result is the exponential sum bound S ( N , t ) ⩽ 9.463 N 1 − 1 / ( 133.66 λ 2 ) , where N is a positive integer, t is a real number, λ = log ( t )/(log N ) and S ( N , t ) = max 0 < u ⩽ 1max N < R ⩽ 2 N| ∑ N < n ⩽ R( n + u ) − i t | .2000 Mathematical Subject Classification : primary 11M06, 11N05, 11L15; secondary 11D72, 11M35.