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Applications of the L ‐functions ratios conjectures
Author(s) -
Conrey J. B.,
Snaith N. C.
Publication year - 2007
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/plms/pdl021
Subject(s) - mathematics , variety (cybernetics) , order (exchange) , computation , function (biology) , riemann zeta function , riemann hypothesis , zero (linguistics) , pure mathematics , discrete mathematics , statistics , algorithm , linguistics , philosophy , finance , evolutionary biology , economics , biology
In upcoming papers by Conrey, Farmer and Zirnbauer there appear conjectural formulas for averages, over a family, of ratios of products of shifted L ‐functions. In this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest in number theory, such as lower order terms in the zero statistics of L ‐functions, mollified moments of L ‐functions and discrete averages over zeros of the Riemann zeta function. In particular, using the ratios conjectures we easily derive the answers to a number of notoriously difficult computations.