Discretized Keiper/Li Approach to the Riemann Hypothesis

ABSTRACT

The Keiper–Li sequence {λ_n} is most sensitive to the Riemann Hypothesis asymptotically (n → ∞), but highly elusive both analytically and numerically. We deform it to fully explicit sequences, simpler to analyze and to compute (up to n = 5 · 10⁵ by G. Misguich). This also works on the Davenport–Heilbronn counterexamples, thus we can demonstrate explicit tests that selectively react to zeros off the critical line.

Keywords:

• Keiper-Li sequence
• Li's criterion
• Riemann Hypothesis
• Riemann zeros
• steepest-descent method
• Dirichlet L-functions
• Davenport-Heilbronn functions

2010 AMS Subject Classification:

• 11M26
• 30B40
• 30E15
• 41A60

Notes

1 Erratum: in [Voros 06,Voros 10, chap. 11], we missed the overall (−) sign (with no effect on our conclusions), which we rectified in [Voros 14].

2 One still has the freedom to spread the x_m further out by skipping some locations (2j), (or inversely, to keep some residual degeneracy in the numerics if ever that helps it).

3 Erratum: our asymptotic statements for λ_n in the RH false case have wrong sign.

4 Eqs. (35), (40), (51) have typos, fixed in the present work.