Open analytical questions for Voros's discretized Keiper–Li sequence: transseries asymptotics, all-$n$ positivity, and remainder bounds

By Li's criterion, the Riemann Hypothesis (RH) is equivalent to positivity of the Keiper–Li coefficients $\lambda_n$, which are analytically elusive and expensive to compute. Voros (2020, 2022) discretized the construction, replacing high derivatives of $\log\xi$ by finite differences on explicit nodes at the even integers, and obtained a variant sequence $\Lambda_n$ in elementary closed form obeying an asymptotic alternative: tempered growth $\log n + \frac12(\gamma-\log\pi-1)$ if RH is true, versus growing oscillations if RH is false. This yields an asymptotic Li-like criterion for RH, but its analytical foundations remain unfinished. The open program that this work explicitly leaves is to rigorously establish the full transseries expansion of $\Lambda_n$ in powers of $1/\log n$ and complex powers of $n$; decide whether $\Lambda_n>0$ for every $n\ge1$ (an affirmative answer would prove RH); derive explicit provable bounds for the remainder $\delta\Lambda_n$; and reduce the roughly $0.76555n$-digit cancellation barrier in evaluating $\Lambda_n$.

Background

The Riemann zeta function $\zeta(x)$, defined for $\mathrm{Re}\,x>1$ by $\sum_{k\ge1}k^{-x}$, extends meromorphically to the whole plane, and its nontrivial zeros are customarily studied through the completed zeta function $2\xi(x)=x(x-1)\pi^{-x/2}\Gamma(x/2)\zeta(x)$, a real entire function whose zeros are exactly those of $\zeta$ in the critical strip $0<\mathrm{Re}\,x<1$. The Riemann Hypothesis (RH) asserts that all these zeros lie on the critical line $\{\mathrm{Re}\,x=1/2\}$. A concrete equivalent is Li's criterion. Keiper and Li introduced the Keiper–Li coefficients: the numbers $\lambda^K_n$ defined by the Taylor expansion $\sum_{n\ge1}\lambda^K_n z^n=\log 2\xi\big(1/(1-z)\big)$, equivalently (Li's normalization, $\lambda^L_n=n\lambda^K_n$) the sums $\lambda^L_n=\sum_\rho[1-(1-1/\rho)^n]$ over all nontrivial zeros $\rho$. Because the conformal map $x=1/(1-z)$ sends the half-plane $\mathrm{Re}\,x>1/2$ onto the unit disk, RH holds if and only if that logarithm is analytic on the whole disk, and this yields Li's criterion: RH is true if and only if $\lambda_n>0$ for all $n$. Only the large-$n$ tail is essential: if RH is verified up to ordinate $T_0$, then $\lambda_n>0$ for $n<T_0^2$ (a result of Oesterlé quoted in Voros's papers), and asymptotically $\lambda^L_n\sim\frac12 n[\log n+\gamma-\log 2\pi-1]$ ($\gamma$ is Euler's constant) under RH, whereas RH-violating zeros $\rho'$ with $\mathrm{Re}\,\rho'>1/2$ contribute exponentially growing oscillations. Bombieri–Lagarias gave complements and a sum-over-zeros framework, Voros (2006) turned the alternative into sharpened criteria, Lagarias (2007) obtained the sharpest known remainder estimates for $\lambda_n$ (in the setting of automorphic $L$-functions, i.e. Dirichlet series generalizing $\zeta$ that carry an analogous functional equation and Euler product), and Arias de Reyna refined the asymptotics. Nevertheless the $\lambda_n$ require high-order derivatives of $\log\xi$ and are strenuous to compute — rigorous evaluations reach only $n\approx10^5$, while detecting any off-line zero would need $n\gtrsim10^{25}$.

Voros's discretization (2020, surveyed in the 2022 source) replaces differentiation by finite differences. Splitting the order-$(n+1)$ pole at $z=0$ into simple poles displaced by Möbius transformations of the unit disk ($z\mapsto(z-\tilde z)/(1-\tilde z^{*}z)$, which keep RH-sensitivity by preserving the disk) and placing the nodes at $z_m=1-(2m)^{-1}$, i.e. at the even integers $x=2m$ where $2\xi(2m)=|B_{2m}|(2\pi)^m/(2m-3)!!$ is explicit in the Bernoulli numbers $B_k$ (the rational numbers defined by $t/(e^t-1)=\sum_k B_kt^k/k!$), yields $\Lambda_n=\sum_{m=1}^{n}(-1)^mA_{nm}\Phi_m,\qquad\Phi_m=\log 2\xi(2m),$ with explicitly computable real coefficients $A_{nm}=2^{m-n}(2(n+m)-1)!!/((2m-1)(n-m)!(2m)!)$ (formula (25) of the source); for instance $\Lambda_1=\frac32\log(\pi/3)\approx0.06918$. The established main result is an asymptotic alternative: if RH is false, $\Lambda_n\sim\sum_{\mathrm{Re}\,\rho'>1/2}F_n(\rho')$, where $F_n(\rho')$ is the contribution of the individual zero $\rho'$, itself given asymptotically (for $n\gg|T|$, non-uniformly in $T$) by $F_n(\rho')\sim e^{i\varphi(\rho')}(-1)^n(2n)^{t+iT}/(|T|^{2+t}\log n)$ for $\rho'=1/2+t+iT$ with $t>0$ — an oscillation whose amplitude grows like $n^t/\log n$; if RH is true, $\Lambda_n=\log n+\tfrac12(\gamma-\log\pi-1)+o(1),$ and the remainder is denoted $\delta\Lambda_n$. This entails an asymptotic Li-like criterion: RH is true if and only if $\Lambda_n>0$ for all $n>n_0$ for some threshold $n_0$. The framework was validated on the Davenport–Heilbronn functions — Dirichlet series satisfying a functional equation like $\zeta$'s but possessing zeros off the critical line, hence explicit counterexamples to RH — where the analog sequence $\Lambda_{-,n}$ indeed displays the predicted growing oscillation from its first off-line zero. Computationally, the coefficients $A_{nm}$ reach size $(3+2\sqrt2)^n$ near $m\approx n/\sqrt2$, so about $0.76555n$ decimal digits cancel in the sum; $\Lambda_n$ has been certified up to $n=500000$ (all values positive, via the MPFR arbitrary-precision floating-point library). The right-hand side of the RH-false branch is only the leading term of a double expansion in integer powers of $1/\log n$ and complex powers of $n$ — a transseries (a combined asymptotic expansion mixing power, logarithmic, and exponential-in-$\log n$ contributions) — whose rigorous properties, like the rest of the analytical theory of $\Lambda_n$, remain to be developed.

Problem Statement

Place Voros's discretized Keiper–Li sequence $\{\Lambda_n\}$ on a rigorous analytical and effective computational footing as a test of the Riemann Hypothesis. The source (Voros 2022, §3.3–3.4 and §4, based on Voros 2020) explicitly leaves the following linked questions open. They are logically independent: resolving any one of them does not resolve the others.

(1) Transseries asymptotics. The RH-false branch $\Lambda_n\sim\sum_{\mathrm{Re}\,\rho'>1/2}F_n(\rho')$ is stated to be only the start of a double expansion in integer powers of $1/\log n$ and complex powers of $n$ — a transseries in $1/\log n$ — 'whose analytical properties wholly remain to be investigated'. Establish this full expansion rigorously: prove the transseries structure with a quantified remainder, specify its summation sense, and handle the fact that the known leading form of $F_n(\rho')$ is not uniform in $T=\mathrm{Im}\,\rho'$, so termwise summation over all RH-violating zeros is currently unjustified.

(2) All-$n$ positivity. It is established that RH is true if and only if $\Lambda_n>0$ for all $n>n_0$ for some threshold $n_0$; it is open whether $n_0=0$ works, i.e. whether $\Lambda_n>0$ for every $n\ge1$ (only verified numerically, up to $n=500000$), and, assuming RH, how small the minimal $n_0$ can be proved to be. Decide this question; note that a proof that $\Lambda_n>0$ for all $n\ge1$ would imply RH itself.

(3) Remainder bounds. Derive explicit provable bounds for the remainder $\delta\Lambda_n=\Lambda_n-\log n-\frac12(\gamma-\log\pi-1)$ on the RH-true branch, in the spirit of the estimates Lagarias obtained for $\lambda_n$, strong enough that the numerical detection of a weak growing oscillation from an RH-violating zero (as rehearsed on the Davenport–Heilbronn analogs) becomes a certified procedure rather than a heuristic signal-processing observation.

(4) Conditioning and computation. Reduce the cancellation barrier: the evaluation of $\Lambda_n$ loses about $0.76555n$ decimal digits to cancellation because the coefficients $A_{nm}$ grow like $(3+2\sqrt2)^n$. Reduce this precision requirement while preserving the asymptotic RH-sensitivity.

References

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