Eigenvalue Density, Li's Positivity, and the Critical Strip


  • Yang-Hui He London Institute, Royal Institution; City, University of London; & University of Oxford
  • Vishnu Jejjala
  • Djordje Minic


Zero-counting formula, Riemann zeta function, Li coefficients


We rewrite the zero-counting formula within the critical strip of the Riemann zeta function as a cumulative density distribution;this subsequently allows us to formally derive an integral expression for the Li coefficients associated with the Riemann Xi-function which, in particular, indicate that their positivity criterion is obeyed, whereby entailing the criticality of the non-trivial zeros. We conjecture the validity of this and related expressions without the need for the Riemann Hypothesis and also offer a physical interpretation of the result and discuss the Hilbert-Polya approach.


B. Riemann, “Über die Anzahl der Primzahlen unter einer gegebenen Größe,” Monatsberichte der Königlichen Preußischen Akademie der Wissenschaften zu Berlin. Aus dem Jahre 1859, vol. 50, no. 3, pp. 671– 680, 1860.

E. Bombieri, “Problems of the millennium: The Riemann hypothesis.” Available at http://www.claymath.org/millennium/, 2000.

J. Conrey, “The Riemann hypothesis,” Notices of the American Mathematical Society, vol. 50, no. 3, pp. 341– 353, 2003.

D. Hilbert Unpublished work, c. 1914.

G. P´olya Unpublished work, c. 1914.

A. Selberg, “Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series,” The Journal of the Indian Mathematical Society, vol. 20, no. 1–3, p. 47, 1956.

D. Hejhal, The Selberg Trace Formula for PSL(2,R), Vol. I. Lecture Notes in Mathematics, Heidelberg: Springer– Verlag Berlin, 1st ed., 1976.

A. Weil, “Sur les ‘formules explicites’ de la théorie des nombres premiers,” Meddelanden Från Lunds Universitet Mat. Sem., no. 252, 1952.

A. Weil, Oeuvres Scientifiques - Collected Papers, Vol. II. Heidelberg: Springer Berlin, 1st ed., 2009.

A. P. Guinand, “A summation formula in the theory of prime numbers,” Proceedings of the London Mathematical Society, vol. s2-50, no. 1, pp. 107–119, 1948.

A. P. Guinand, “Summation formulae and self-reciprocal functions (III),” The Quarterly Journal of Mathematics, vol. os-13, no. 1, pp. 30–39, 1942.

H. Montgomery, “The pair correlation of zeros of the zeta function,” in Proceedings of Symposia in Pure Mathematics: Analytic Number Theory (H. Diamond, ed.), vol. 24, (St. Louis, Missouri), pp. 181–193, American Mathematical Society, 1973.

A. M. Odlyzko, “On the distribution of spacings between zeros of the zeta function,” Mathematical Computation, vol. 48, pp. 273–308, 1987.

X. Gourdon and P. Sebah, “Computation of zeros of the zeta function.” See also http://numbers.computation.free.fr/.

M. Watkins. See http://www.maths.ex.ac.uk/zeta.

M. V. Berry and J. P. Keating, “H=xp and the Riemann zeros,” in Supersymmetry and Trace Formulae: Chaos and Disorder (I. V. Lerner, J. P. Keating, and D. E. Khmelnitskii, eds.), pp. 355–367, Boston, MA: Springer US, 1999.

M. V. Berry and J. P. Keating, “The Riemann zeros and eigenvalue asymptotics,” SIAM Review, vol. 41, no. 2, pp. 236–266, 1999.

A. Connes, “Trace formula in noncommutative geometry and the zeros of the Riemann zeta function,” Selecta Mathematica, vol. 5, no. 1, p. 29, 1999.

A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives. New York: American Mathematical Society, 1st ed., 2008.

G. Sierra, “H = x p with interaction and the Riemann zeros,” Nuclear Physics B, vol. 776, no. 3, pp. 327–364, 2007.

G. Sierra, “The Riemann zeros and the cyclic renormalization group,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2005, no. 12, p. P12006, 2005.

G. Sierra, “A quantum mechanical model of the Riemann zeros,” New Journal of Physics, vol. 10, no. 3, p. 033016, 2008.

G. Sierra and P. K. Townsend, “Landau levels and Riemann zeros,” Physical Review Letters, vol. 101, no. 11, 2008.

X.-J. Li, “The positivity of a sequence of numbers and the Riemann hypothesis,” Journal of Number Theory, vol. 65, no. 2, pp. 325–333, 1997.

E. Bombieri and J. C. Lagarias, “Complements to Li’s criterion for the Riemann hypothesis,” Journal of Number Theory, vol. 77, no. 274, pp. 274–287, 1999.

H. Edwards, Riemann’s Zeta Function. New York: Academic Press, 1st ed., 1974.

E. Titchmarsh, The Theory ofthe Riemann Zeta Function. Oxford: Clarendon Press, 1st ed., 1986.

A. Ivic, The Riemann Zeta Function. New York: John Wiley, 1st ed., 1986.

J. Conrey, A. Ghosh, and S. Gonek, “Simple Zeros of the Riemann Zeta-Function,” Proceedings of the London Mathematical Society, vol. 76, no. 3, pp. 497–522, 1998.

X. Gourdon, “The 1013 first zeros of the Riemann zeta function, and zeros computation at very large height.” Available at http://numbers.computation.free.fr, 2004.

A. Weil, “Numbers of solutions of equations in finite fields,” Bulletin of the American Mathematical Society, vol. 55, no. 5, pp. 497–508, 1949.

P. Deligne, “La conjecture de Weil : I,” Publications Mathématiques de l’IHÉS, vol. 43, pp. 273–307, 1974.

P. Deligne, “La conjecture de Weil : II,” Publications Mathématiques de l’IHÉS, vol. 52, pp. 137–252, 1980.

E. Freitag and R. Kiehl, Étale Cohomology and the Weil Conjecture. Berlin: Springer–Verlag, 1st ed., 1988.

A. Polyakov, Gauge Fields and Strings. Chur: Harwood, 1st ed., 1987.




How to Cite

He, Y.-H.; Jejjala, V.; Minic, D. Eigenvalue Density, Li’s Positivity, and the Critical Strip. inSTEMM 2022, 1, 1-14.