Sharpenings of Li's Criterion for the Riemann Hypothesis

Abstract

Exact and asymptotic formulae are displayed for the coefficients λ n used in Li's criterion for the Riemann Hypothesis. For n → ∞ we obtain that if (and only if) the Hypothesis is true, λ n ∼ n(A log n + B) (with A > 0 and B explicitly given, also for the case of more general zeta or L-functions); whereas in the opposite case, λ n has a non-tempered oscillatory form.Li's criterion for the Riemann Hypothesis (RH) states that the latter is true if and only if a specific real sequence {λ n } n=1,2,... has all its terms positive [17, 2]. Here we show that it actually suffices to probe the λ n for their large-n behavior, which fully encodes the Riemann Hypothesis by way of a clear-cut and explicit asymptotic alternative. To wit, we first represent λ n exactly by a finite oscillatory sum (9), then by a derived integral formula (12), which can finally be evaluated by the saddle-point method in the n → +∞ limit. As a result, λ n takes one of two sharply distinct and mutually exclusive asymptotic forms: if RH is true, λ n will grow tamely according to (17); if RH is false, λ n will oscillate with an exponentially growing amplitude, in both + and − directions, as described by (18). This dichotomy thus provides a sharp criterion of a new asymptotic type for the Riemann Hypothesis (and for other zeta-type functions as well, replacing (17) by (15)).This work basically reexposes our results of April 2004 announced in [28], but with an uncompressed text; we also update the references and related comments: for instance, we now derive as (24) a large-n expansion surmised by Maślanka [20] in the meantime.

Li coefficients for automorphic L-functions

“…2.3]. While this paper was being completed, an asymptotic formula for the Li coefficients λ n was announced by A. Voros [43], under the Riemann hypothesis. Comparison of his formula with that obtained here in (5.31) led to a simplification of the expression for C 1 (π) in Theorem 5.1. done while the author was at AT&T Labs-Research, whom he thanks for support.…”

On Li’s coefficients for the Rankin–Selberg L-functions

“…Voros has proved that it actually suffices to probe the Li coefficients (1) for their large n behavior. Namely, the Riemann hypothesis true is equivalent to the tempered growth of λ n (as 1 2 n log n), determined by its archimedean part, while the Riemann hypothesis false is equivalent to the oscillations of λ n with exponentially growing amplitude, determined by its finite part (for details, see [30,Sect. 3.3.]).…”

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

“…[26] for λ E (n) may be written as (2) In Ref. [39], the rate of growth of the sum S 1 (n) is conjectured. An integral expression for the Li constants is written and a saddle point method is applied.…”