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The large values of the Riemann Zeta‐function
Author(s) -
Tsang Kai-Man
Publication year - 1993
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300006987
Subject(s) - mathematics , riemann zeta function , riemann hypothesis , pure mathematics , riemann xi function , arithmetic zeta function , particular values of riemann zeta function , mathematical analysis , prime zeta function
Let |θ| < π/2 and σ ∈ [ 1 2 , 1 ] . By refining Selberg's method, we study the large values of R e { e − i θ log ζ ( σ + i t ) }as t → ∞ For σ close to ½ we obtain Ω + estimates that are as good as those obtained previously on the Riemann Hypothesis. In particular, we show that(sup T < t ⩽ 2 T log | ζ ( 1 2 + i t ) | ) (sup T < t ⩽ 2 T ± S ( t ) ) ≫ log T / log log TandS 1 ( t ) = Ω + (( log t )1 / 2( log log t )− 3 / 2) .Our results supplement those of Montgomery which are good when σ > ½ is fixed.