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Complements to Li's Criterion for the Riemann Hypothesis

✍ Scribed by Enrico Bombieri; Jeffrey C. Lagarias

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
131 KB
Volume
77
Category
Article
ISSN
0022-314X

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✦ Synopsis

In a recent paper Xian-Jin Li showed that the Riemann Hypothesis holds if and only if * n = \ [1&(1&1Γ‚) n ] has * n >0 for n=1, 2, 3, ... where \ runs over the complex zeros of the Riemann zeta function. We show that Li's criterion follows as a consequence of a general set of inequalities for an arbitrary multiset of complex numbers \ and therefore is not specific to zeta functions. We also give an arithmetic formula for the numbers * n in Li's paper, via the Guinand Weil explicit formula, and relate the conjectural positivity of * n to Weil's criterion for the Riemann Hypothesis. 1999 Academic Press * n = 1 (n&1)! d n ds n [s n&1 log !(s)] } s=1 ,

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