Li's criterion for the Riemann hypothesis - numerical approach

• Autorzy: Maślanka K.
• Języki publikacji: EN

Abstrakty

EN

There has been some interest in a criterion for the Riemann hypothesis proved recently by Xian-Jin Li [9]. The present paper reports on a numerical computation of the first 3300 of Li's coefficients which appear in this criterion. The main empirical observation is that these coefficients can be separated in two parts. One of these grows smoothly while the other is very small and oscillatory. This apparent smallness is quite unexpected. If it persisted till infinity then the Riemann hypothesis would be true.

Słowa kluczowe

EN

Riemann zeta function; Riemann hypothesis; Li's criterion; numerical methods in analytic number theory

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2004
• Tom: Vol. 24, no. 1
• Strony: 103--114
• Opis fizyczny: Bibliogr. 15 poz., rys., tab., wykr.

Twórcy

autor

Maślanka K.

• Astronomical Observatory of Jagiellonian University, ul. Orla 171, 30-244 Cracow, Poland,

Bibliografia

• [1] Biane P., Pitman J., Yor, M.: Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull. Amer. Math. Soc. 38 (2001), 435-465.
• [2] Bombieri E., Lagarias J.C.: Complements to Li's Criterion for the Riemann Hypothesis. J. Number Theory 77 (1999), 274-287.
• [3] Coffey M.: Relations and positivity results for the derivatives of the Riemann £ function. J. Comp. Applied Math. 166 (2004), 525-534.
• [4] Gradshteyn I. S., Ryzhik I. M.: Tables of Integrals, Series, and Products, corr. enl. 4th ed. San Diego, CA: Academic Press 1980.
• [5] Ingham A. E.: The distribution of prime numbers. Cambridge University Press, 1932.
• [6] Keiper J.B.: Power Series Expansion of Riemann's £ Function. Math. Comp. 58 (1992), 765-773.
• [7] Lagarias J.C.: Li Coefficients for Automorphic L-Functions. posted at arXiv:math.NT/0404394, 8 May 2004.
• [8] Lagarias J.C.: private communication, 2003.
• [9] Li X.-J.: The Positivity of a Sequence of Numbers and the Riemann Hypothesis. J. Number Theory 65 (1997), 325-333.
• [10] Li X.-J.: private communication, 2003.
• [11] Maślanka K.: http://arxiv.org/abs/math.NT/0402168. 11 February 2004.
• [12] Riemann G. B. F.: Ueber die Anzahl der Primzahlen unter eine gegebenen Grosse. Monatsberichte Konigl. Preuss. Akad. Wiss. Berlin, 1859, 671.
• [13] Riesz M.: Sur Vhypothe.se de Riemann. Acta Math. 40 (1916), 185-190.
• [14] Voros A.: A Sharpening of Li's Criterion for the Riemann Hypothesis, submitted to C.R. Acad. Sci. Paris, Ser. I; http://arxiv.org/abs/math.NT/0404213, 15 April 2004.
• [15] Voros A.: private communication, 2004.