Effective method of computing Li's coefficients and their properties
Krzysztof Maslanka
TL;DR
This work develops an effective numerical method to compute Li's coefficients $\lambda_n$ associated with Li's criterion for the Riemann hypothesis. It introduces a precise trend-oscillation decomposition, with $\overset{-}{\lambda}_n$ given by $\frac{1}{\Gamma(n)} \left. \frac{d^{n}}{ds^{n}} [ s^{n-1} \ln(\pi^{-s/2}\Gamma(1+s/2)) ] \right|_{s=1}$ and $\overset{\sim}{\lambda}_n$ by $\frac{1}{\Gamma(n)} \left. \frac{d^{n}}{ds^{n}} [ s^{n-1} \ln((s-1)\zeta(s)) ] \right|_{s=1}$, along with their expression in terms of Stieltjes constants $\gamma_n$ and the related $\eta_n$ via a recurrence for coefficients $c_n^{(k)}$; the oscillatory part is $\overset{\sim}{\lambda}_n = -\sum_{j=1}^{n} \binom{n}{j} \eta_{j-1}$. Using these, the author computed thousands of $\eta_n$ and $\lambda_n$ up to $n \approx 3300$, confirming a strictly growing trend with very small oscillations. The numerical results support a reformulated RH criterion: if $\overset{\sim}{\lambda}_n \le \overset{-}{\lambda}_n$ for all $n$, RH holds, though the paper cautions that numerical evidence alone cannot prove RH. Overall, the work combines analytic derivations with high-precision computations to provide a practical framework for investigating Li's coefficients and their implications for RH.
Abstract
In this paper we present an effective method for computing certain real coefficients $λ_{n}$ which appear in a criterion for the Riemann hypothesis proved by Xian-Jin Li. With the use of this method a sequence of over three-thousand $λ_{n}$'s has been calculated. This sequence reveals a peculiar and unexpected behavior: it can be split into a strictly growing \textit{trend} and some tiny \textit{oscillations} superimposed on this trend.