Toward Verification of the Riemann Hypothesis: Application of the Li Criterion

摘要

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence {λk}, 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 ηj are valid. The constants ηj enter the Laurent expansion of the logarithmic derivative of the zeta function about s=1 and appear to have remarkable characteristics. On our conjecture, not only does the Riemann hypothesis follow, but an inequality governing the values λn and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function.