Toward verification of the Riemann hypothesis: Application of the Li criterion

Details Toward verification of the Riemann hypothesis: Application of the Li criterion Toward verification of the Riemann hypothesis: Application of the Li criterion

2005-05-19

arxiv.org/abs/math-ph/0505052v1

Physics

Mathematical Physics

to appear in Math. Physics, Analysis and Geometry; 1 figure

Scientific paper

10.1007/s11040-005-7584-9

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{\lambda_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence \eta_j are valid. The constants \eta_j enter the Laurent expansion of the logarithmic derivative of the zeta function about s=1 and appear to have remarkable characteristics. {\em On our conjecture}, not only does the Riemann hypothesis follow, but an inequality governing the values \lambda_n and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function.

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