The Riemann Hypothesis in Li's criterion form: non-negativity of the Li coefficients for every $n\ge 1$

The source (Bombieri and Lagarias, 'Complements to Li's Criterion for the Riemann Hypothesis', J. Number Theory 77 (1999), 274-287) is a standard reference for the equivalent formulation of the Riemann Hypothesis (RH) known as Li's criterion: defining the Li coefficients by $\lambda_n=(1/(n-1)!)\, d^n/ds^n\,[s^{n-1}\log\xi(s)]|_{s=1}$, equivalently as the symmetric-limit zero-sum $\lambda_n=\sum_\rho[1-(1-1/\rho)^n]$ over the nontrivial zeros $\rho$ of $\zeta(s)$, RH holds if and only if $\lambda_n\ge 0$ for every integer $n\ge 1$ - and even if and only if the weaker bound $\lambda_n\ge -c(\varepsilon)e^{\varepsilon n}$ holds for all $n$ for every $\varepsilon>0$. The paper proved this for general symmetric multisets of complex numbers, expressed the $\lambda_n$ arithmetically as limits of truncated von Mangoldt prime-power sums plus explicit constants, and identified each $\lambda_n\ge 0$ with Weil's positivity criterion evaluated on explicit test functions, leaving the positivity conjecture itself unresolved. All four subsidiary tasks explicitly left open by the paper were subsequently carried out or substantially advanced by the follow-up literature (2004-2025): the $L$-function generalization (Selberg class, automorphic $L$-functions over $GL(N)$, function fields, Hecke, Dirichlet, Rankin-Selberg and other $L$-functions), effective computation and asymptotics of the $\lambda_n$, the algebraic derivation of the $\eta_n$-Stieltjes-constant-$\lambda_n$ relations, and various weakenings of the hypotheses of the multiset theorem. The principal open core is untouched: no proof or disproof of the non-negativity of all $\lambda_n$ - equivalently, of the Riemann Hypothesis - exists in the accepted literature; RH has been verified numerically for the first $3\cdot 10^{12}$ zeros (Platt and Trudgian, 2021), and even the asymptotic sign of $\lambda_n$ for large $n$ is known to be equivalent to RH.

Background

Let $\zeta(s)=\sum_{n\ge 1} n^{-s}$ be the Riemann zeta function, continued meromorphically to the complex plane with a single simple pole at $s=1$. Its nontrivial zeros (the zeros other than the 'trivial' ones at the negative even integers) lie in the critical strip $0<\mathrm{Re}(s)<1$, and the Riemann Hypothesis (RH), formulated in 1859 and one of the Clay Millennium Problems, asserts that every nontrivial zero has real part exactly $1/2$. The explicit formulae of prime number theory translate zero-location statements into statements about the distribution of the primes - for example, RH is equivalent to the essentially optimal error term in the prime number theorem - which is why the problem is central to analytic number theory.

The completed zeta function $\xi(s)=\tfrac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$ is entire and satisfies the symmetric functional equation $\xi(s)=\xi(1-s)$. J. B. Keiper (1992), studying power series expansions of $\xi$, introduced the real constants now called the Li or Keiper-Li coefficients,

$$\lambda_n = \left. \frac{1}{(n-1)!} \frac{d^n}{ds^n} \big[ s^{\,n-1} \log \xi(s) \big] \right|_{s=1},$$

and observed that RH would force them to be non-negative. Xian-Jin Li (1997) proved the converse, establishing that RH holds if and only if $\lambda_n\ge 0$ for every integer $n\ge 1$ (and the analogous statement for Dedekind zeta functions, which play the role of $\zeta$ for the integers of a number field). Equivalently, $\lambda_n$ is the value of the conditionally convergent zero-sum $\sum_\rho[1-(1-1/\rho)^n]$, taken over the nontrivial zeros $\rho$ in a symmetric limit over imaginary parts.

The source - E. Bombieri and J. C. Lagarias, 'Complements to Li's criterion for the Riemann Hypothesis' - contributed three observations. First, an abstract multiset criterion (Theorem 1 and Corollary 1): if a multiset $\mathscr{R}$ of complex numbers avoids $0$ and $1$, is invariant under complex conjugation and under $\rho\mapsto 1-\rho$, and satisfies the summability condition $\sum_{\rho\in\mathscr{R}} (1+|\mathrm{Re}(\rho)|)/(1+|\rho|)^2<\infty$, then, with $\lambda_n$ defined by the symmetric limit, all elements of $\mathscr{R}$ lie on the line $\mathrm{Re}=1/2$ if and only if $\lambda_n\ge 0$ for all $n\ge 1$ - indeed already if and only if, for every $\varepsilon>0$, there exists $c(\varepsilon)>0$ with $\lambda_n\ge -c(\varepsilon)e^{\varepsilon n}$ for all $n$. This shows that Li's criterion is a general property of symmetric multisets, not a special feature of $\zeta$. Second, an arithmetic formula (Theorem 2): $\lambda_n$ equals explicit constants (involving $\log 4\pi$, Euler's constant $\gamma$, and values $\zeta(j)$) plus a linear combination of the limits, as $x\to\infty$, of $\sum_{m\le x}\Lambda(m)(\log m)^{j-1}/m-(1/j)(\log x)^j$, where $\Lambda$ is the von Mangoldt function (equal to $\log p$ when $m$ is a power of the prime $p$, and $0$ otherwise). Third, an interpretation of the positivity as Weil positivity: the Guinand-Weil explicit formula (the arithmetic identity relating sums over zeros to sums over prime powers) shows that each condition $\lambda_n\ge 0$ is Weil's positivity criterion - that a certain quadratic form be non-negative - evaluated on the explicit test function $g_n$ whose Mellin transform (the standard integral transform $\int_0^\infty f(x)\,x^{s-1}\,dx$ of multiplicative Fourier analysis) is $1-(1-1/s)^n$ and which satisfies $g_n+\tilde g_n=g_n*\tilde g_n$, where $\tilde f(x)=x^{-1}f(1/x)$ and $*$ is multiplicative convolution.

The paper also left four explicitly acknowledged open ends: (i) the summability hypotheses of the multiset theorem could be relaxed, with details left to the reader; (ii) the generalization of the explicit-formula treatment and the arithmetic formulas to other $L$-functions was asserted to involve no additional difficulties but was not carried out; (iii) effective numerical evaluation of the limiting procedures in Theorem 2, with rigorous error control, was not developed; (iv) the direct algebraic derivation of the relations between the numbers $\eta_n$ defined by $-\zeta'/\zeta(s+1)=1/s+\sum_{n\ge 0}\eta_n s^n$ (each $\eta_n$ is a polynomial in the Stieltjes constants $\gamma_n$, the Laurent coefficients of $\zeta$ at $s=1$) and the $\lambda_n$ was left as an exercise.

What is unconditionally known is limited but informative. RH has been verified numerically for the first $3\cdot 10^{12}$ zeros (Platt and Trudgian, 2021); $\lambda_n$ has been computed and found positive for thousands of indices $n$ (Maslanka, 2004), and explicit discretized variants are computable up to $n=5\cdot 10^5$ (Voros, 2020). However, the asymptotic law $\lambda_n\sim n(A\log n+B)$ with $A>0$ is equivalent to RH (Voros, 2006), so even the eventual sign of the $\lambda_n$ is as hard as RH itself. The classical Davenport-Heilbronn example - a linear combination of Dirichlet $L$-functions which satisfies a functional equation but has zeros off the critical line and fails the corresponding positivity - shows that functional-equation symmetry alone cannot imply the criterion, so any proof must use the Euler product and the arithmetic of $\zeta$ in an essential way.

Problem Statement

Prove or disprove the Riemann Hypothesis: determine whether every nontrivial zero $\rho$ of the Riemann zeta function $\zeta(s)$ (i.e., every zero with $0<\mathrm{Re}(\rho)<1$) satisfies $\mathrm{Re}(\rho)=1/2$.

In the equivalent form that the source leaves open, determine whether the Li coefficients

$$\lambda_n = \left. \frac{1}{(n-1)!} \frac{d^n}{ds^n} \big[ s^{\,n-1} \log \xi(s) \big] \right|_{s=1} = \sum_{\rho} \Big[ 1 - \Big( 1 - \frac{1}{\rho} \Big)^{n} \Big],$$

where $\xi(s)=\tfrac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$ is the completed zeta function, satisfy $\lambda_n\ge 0$ for every integer $n\ge 1$. The zero-sum runs over the nontrivial zeros of $\zeta$, truncated to $|\mathrm{Im}(\rho)|\le T$ and taken in the limit $T\to\infty$ (the symmetric limit). Equivalently, determine whether for every $\varepsilon>0$ there exists $c(\varepsilon)>0$ such that $\lambda_n\ge -c(\varepsilon)\,e^{\varepsilon n}$ for all $n$. By Li's theorem and the Bombieri-Lagarias multiset criterion, these statements hold if and only if the Riemann Hypothesis holds, so any complete resolution of this question - in either direction - resolves the Riemann Hypothesis itself.

References

1. Enrico Bombieri and Jeffrey C. Lagarias, Complements to Li's Criterion for the Riemann Hypothesis, Journal of Number Theory 77(2) (1999) 274-287, DOI: 10.1006/jnth.1999.2392, https://doi.org/10.1006/jnth.1999.2392
2. Xian-Jin Li, The positivity of a sequence of numbers and the Riemann hypothesis, Journal of Number Theory 65(2) (1997) 325-333, DOI: 10.1006/jnth.1997.2137, https://doi.org/10.1006/jnth.1997.2137
3. J. B. Keiper, Power series expansions of Riemann's xi function, Mathematics of Computation 58(198) (1992) 765-773, DOI: 10.1090/S0025-5718-1992-1122072-5, https://doi.org/10.1090/S0025-5718-1992-1122072-5
4. Jeffrey C. Lagarias, Li coefficients for automorphic L-functions, Annales de l'Institut Fourier 57(5) (2007) 1689-1740, DOI: 10.5802/aif.2311, arXiv: math/0404394, https://arxiv.org/abs/math/0404394
5. Francis C. S. Brown, Li's criterion and zero-free regions of L-functions, Journal of Number Theory 111 (2005) 1-32, DOI: 10.1016/j.jnt.2004.07.016, https://doi.org/10.1016/j.jnt.2004.07.016
6. Pedro Freitas, A Li-type criterion for zero-free half-planes of Riemann's zeta function, Journal of the London Mathematical Society 73 (2006) 399-414, DOI: 10.1112/S0024610706022599, https://doi.org/10.1112/S0024610706022599
7. André Voros, Sharpenings of Li's Criterion for the Riemann Hypothesis, Mathematical Physics, Analysis and Geometry 9 (2006) 53-63, DOI: 10.1007/s11040-005-9002-8, arXiv: math/0506326, https://arxiv.org/abs/math/0506326
8. Krzysztof Maslanka, An Explicit Formula Relating Stieltjes Constants and Li's Numbers, arXiv: math/0406312 (2004), https://arxiv.org/abs/math/0406312
9. Krzysztof Maslanka, Effective method of computing Li's coefficients and their properties, arXiv: math/0402168 (2004), https://arxiv.org/abs/math/0402168
10. Mark W. Coffey, Toward Verification of the Riemann Hypothesis: Application of the Li Criterion, Mathematical Physics, Analysis and Geometry 8 (2005) 211-255, DOI: 10.1007/s11040-005-7584-9, arXiv: math-ph/0505052, https://arxiv.org/abs/math-ph/0505052
11. Mark W. Coffey, Summation properties of the eta_j and Li constants, Journal of Computational and Applied Mathematics 233 (2009) 667-673, DOI: 10.1016/j.cam.2009.02.034, https://doi.org/10.1016/j.cam.2009.02.034
12. Mark W. Coffey, The Stieltjes constants, their relation to the eta_j coefficients, and representation of the Hurwitz zeta function, Analysis 30(4) (2010), DOI: 10.1524/anly.2010.1048, arXiv: 0706.0343, https://arxiv.org/abs/0706.0343
13. Donal F. Connon, A recurrence relation for the Li/Keiper constants in terms of the Stieltjes constants, arXiv: 0902.1691 (2009), https://arxiv.org/abs/0902.1691
14. Lejla Smajlović, On Li's criterion for the Riemann hypothesis for the Selberg class, Journal of Number Theory 130 (2010) 828-851, DOI: 10.1016/j.jnt.2009.10.012, https://doi.org/10.1016/j.jnt.2009.10.012
15. Sami Omar and Kamel Mazhouda, The Li criterion and the Riemann hypothesis for the Selberg class II, Journal of Number Theory 130 (2010) 1098-1108, DOI: 10.1016/j.jnt.2009.08.012, https://doi.org/10.1016/j.jnt.2009.08.012
16. Sami Omar and Saber Bouanani, Li's criterion and the Riemann hypothesis for function fields, Finite Fields and Their Applications 16 (2010) 477-485, DOI: 10.1016/j.ffa.2010.09.004, https://doi.org/10.1016/j.ffa.2010.09.004
17. S. K. Sekatskii, Generalized Bombieri-Lagarias' Theorem and Generalized Li's Criterion with its Arithmetic Interpretation, Ukrainian Mathematical Journal 66 (2014) 415-431, DOI: 10.1007/s11253-014-0940-9, arXiv: 1304.7895, https://arxiv.org/abs/1304.7895
18. André Voros, Discretized Keiper/Li Approach to the Riemann Hypothesis, arXiv: 1703.02844 (2017), journal version: Experimental Mathematics 29(4) (2020) 452-469, DOI: 10.1080/10586458.2018.1482480, https://arxiv.org/abs/1703.02844
19. André Voros, From asymptotic to closed forms for the Keiper/Li approach to the Riemann Hypothesis, arXiv: 2204.01036 (2022), https://arxiv.org/abs/2204.01036
20. Masatoshi Suzuki, Li coefficients as norms of functions in a model space, Journal of Number Theory 252 (2023) 177-194, DOI: 10.1016/j.jnt.2023.05.007, arXiv: 2301.05779, https://arxiv.org/abs/2301.05779
21. Dave Platt and Tim Trudgian, The Riemann hypothesis is true up to 3·10^12, Bulletin of the London Mathematical Society 53 (2021) 792-797, DOI: 10.1112/blms.12460, https://doi.org/10.1112/blms.12460
22. Renato Spigler, A Brief Survey on the Riemann Hypothesis and Some Attempts to Prove It, Symmetry 17(2) (2025) 225, DOI: 10.3390/sym17020225, https://doi.org/10.3390/sym17020225
23. H. Davenport and H. Heilbronn, On the zeros of certain Dirichlet series, Journal of the London Mathematical Society s1-11(3) (1936) 181-185, DOI: 10.1112/jlms/s1-11.3.181, https://doi.org/10.1112/jlms/s1-11.3.181