Polygamma theory, the Li/Keiper constants, and validity of the Riemann Hypothesis
Abstract
The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants, that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. We investigate a related set of constants cn, n = 1,2,..., showing in detail that the leading behaviour (1/2) ln n of lambdan/n is absent in cn. Additional results are presented, including a novel explicit representation of cn in terms of the Stieltjes constants gammaj. We conjecture as to the large-n behaviour of cn. Should this conjecture hold, validity of the Riemann hypothesis would follow.
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