A note on the zeros of zeta and L-functions

2015/03/03 by Emanuel Carneiro, Vorrapan Chandee, Micah B. Milinovich · 23 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic number field #Analytic Number Theory Research #Bounding overwatch #Class (philosophy) #Class number #Combinatorics #Finite Group Theory Research #Function (biology) #Generalization #Geometry #Mathematical analysis #Mathematics #Order (exchange) #Pure mathematics #Quadratic equation #Riemann Xi function #Riemann hypothesis #Riemann zeta function #Upper and lower bounds #Z function #Zero (linguistics) #math.NT #msc:11M06 #msc:11M26 #msc:11M36 #msc:11M41 #msc:41A30

paper · pdf · doi:10.1007/s00209-015-1485-9

published in Mathematische Zeitschrift 281(1-2), 315-332 (Springer Science+Business Media)

arxiv created 2015/03/03 · openalex publication_date 2015/05/28 · arxiv updated 2021/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Let πS(t) denote the argument of the Riemann zeta-function at the point s=\tfrac12+it. Assuming the Riemann hypothesis, we give a new and simple proof of the sharpest known bound for S(t). We discuss a generalization of this bound for a large class of L-functions including those which arise from cuspidal automorphic representations of GL(m) over a number field. We also prove a number of related results including bounding the order of vanishing of an L-function at the central point and bounding the height of the lowest zero of an L-function.

Cited by

• The positivity technique and low-lying zeros of Dirichlet L-functions
• Upper bounds on gaps between zeros of L-functions
• Littlewood's estimates for L-functions in the hyperelliptic ensemble
• Conditional estimates on the argument of Dirichlet L-functions with applications to low-lying zeros
• Extremal functions in de Branges and Euclidean spaces