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The Distribution of the Summatory Function of the Möbius Function
Author(s) -
Ng Nathan
Publication year - 2004
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/s0024611504014741
Subject(s) - mathematics , conjecture , möbius function , riemann hypothesis , riemann zeta function , distribution (mathematics) , function (biology) , limiting , order (exchange) , pure mathematics , distribution function , combinatorics , mathematical analysis , physics , quantum mechanics , evolutionary biology , biology , mechanical engineering , finance , engineering , economics
The summatory function of the Möbius function is denoted M ( x ). In this article we deduce conditional results concerning M ( x ) assuming the Riemann hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the Riemann zeta function. Assuming these conjectures, we show that M ( x ), when appropriately normalized, possesses a limiting distribution, and also that a strong form of the weak Mertens conjecture is true. Finally, we speculate on the lower order of M ( x ) by studying the constructed distribution function. 2000 Mathematics Subject Classification 11M26, 11N56.
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