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Sharpenings of Li’s criterion for the Riemann hypothesis. (English) Zbl 1181.11055
Summary: Exact and asymptotic formulae are displayed for the coefficients \(\lambda_n\) used in Li’s criterion for the Riemann Hypothesis [see X.-J. Li, J. Number Theory, 65, No. 2, 325–333 (1997; Zbl 0884.11036), E. Bombieri and J.C. Lagarias, J. Number Theory 77, 274–287 (1999; Zbl 0972.11079)]. For \(n \rightarrow \infty\) we obtain that if (and only if) the Hypothesis is true, \(\lambda_n\sim n (A \log n + B)\) (with \(A>0\) and \(B\) explicitly given, also for the case of more general zeta or \(L\)-functions); whereas in the opposite case, \(\lambda_n\) has a non-tempered oscillatory form.

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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[1] Biane, P., Pitman, J. and Yor, M.: Probability laws related to the Jacobi theta and Riemann zeta functions, Bull. Amer. Math. Soc. 38 (2001), 435–465 [Sec. 2.3]. · Zbl 1040.11061 · doi:10.1090/S0273-0979-01-00912-0
[2] Bombieri, E. and Lagarias, J. C.: Complements to Li’s criterion for the Riemann hypothesis, J. Number Theory 77 (1999), 274–287. · Zbl 0972.11079 · doi:10.1006/jnth.1999.2392
[3] Coffey, M. W.: Relations and positivity results for the derivatives of the Riemann \(\xi\) function, J. Comput. Appl. Math. 166 (2004), 525–534. · Zbl 1107.11033 · doi:10.1016/j.cam.2003.09.003
[4] Coffey, M. W.: New results concerning power series expansions of the Riemann xi function and the Li/Keiper constants, preprint (Jan. 2005); Toward verification of the Riemann Hypothesis: application of the Li criterion, Math. Phys. Anal. Geom. 8 (2005), 211–255. · Zbl 1097.11042
[5] Davenport, H.: Multiplicative Number Theory, 3rd edn., revised by H.L. Montgomery, Grad. Texts in Math. 74, Springer, New York, 2000 [chap. 16]. · Zbl 1002.11001
[6] Dingle, R. B.: Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, New York, 1973. · Zbl 0279.41030
[7] Edwards, H. M.: Riemann’s Zeta Function, Academic Press, New York, 1974 [Sect. 1.10]. · Zbl 0315.10035
[8] Erdélyi, A.: Asymptotic Expansions, Dover, 1956, [Sec. 2.5]. · Zbl 0070.29002
[9] Gradshteyn, I. S. and Ryzhik, I. M.: Table of Integrals, Series and Products, 5th edn, A., Jeffrey (ed.), Academic, 1994, [Equations (3.721(1)) p. 444 and (4.421(1)) p. 626]. · Zbl 0918.65002
[10] Hashimoto, Y.: Euler constants of Euler products, J. Ramanujan Math. Soc. 19 (2004), 1–14. · Zbl 1198.11080
[11] Israilov, M. I.: On the Laurent expansion of the Riemann zeta-function, Proc. Steklov Inst. Math. 4 (1983), 105–112, [Russian: Trudy Mat. Inst. i. Steklova 158 (1981), 98–104]. · Zbl 0524.10034
[12] Keiper, J. B.: Power series expansions of Riemann’s \(\xi\) function, Math. Comput. 58 (1992), 765–773. · Zbl 0767.11039
[13] Kurokawa, N.: Parabolic components of zeta functions, Proc. Japan. Acad. Ser. A 64 (1988), 21–24; Special values of Selberg zeta functions, In: M. R. Stein and R. Keith Dennis (eds), Algebraic K-Theory and Algebraic Number Theory, (Proceedings, Honolulu 1987), Contemp. Math. 83, Amer. Math. Soc. Providence, 1989, 133–149. · Zbl 0642.10028 · doi:10.3792/pjaa.64.21
[14] Lagarias, J. C.: Li coefficients for automorphic L-functions, Ann. Inst. Fourier, Grenoble (2006, to appear) [math.NT/0404394 v4]. · Zbl 1216.11078
[15] Landau, E.: Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale, Chelsea, New York, 1949 [Satz 173 p. 89]. · JFM 46.0242.02
[16] Lehmer, D. H.: The sum of like powers of the zeros of the Riemann zeta function, Math. Comput. 50 (1988), 265–273. · Zbl 0644.10029 · doi:10.1090/S0025-5718-1988-0917834-X
[17] Li, X.-J.: The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory 65 (1997), 325–333. · Zbl 0884.11036 · doi:10.1006/jnth.1997.2137
[18] Li, X.-J.: Explicit formulas for Dirichlet and Hecke L-functions, Illinois J. Math. 48 (2004), 491–503. · Zbl 1061.11048
[19] Li, X.-J.: An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials, J. Number Theory 113 (2005), 175–200. · Zbl 1142.11354 · doi:10.1016/j.jnt.2004.08.002
[20] Maślanka, K.: Effective method of computing Li’s coefficients and their properties, Experiment. Math. (to appear) [math.NT/0402168 v5].
[21] Maślanka, K.: An explicit formula relating Stieltjes constants and Li’s numbers, preprint [math.NT/0406312 v2].
[22] Matsuoka, Y.: A note on the relation between generalized Euler constants and the zeros of the Riemann zeta function, J. Fac. Educ. Shinshu Univ. 53 (1985), 81–82; A sequence associated with the zeros of the Riemann zeta function, Tsukuba J. Math. 10 (1986), 249–254.
[23] Oesterlé, J.: Régions sans zéros de la fonction zêta de Riemann, typescript (2000, revised 2001, uncirculated).
[24] Smith, W. D.: A ”good” problem equivalent to the Riemann Hypothesis, e-print on http://www.math.temple.edu/\(\sim\)wds/homepage/works.html (2005 version, unpublished).
[25] de la Vallée-Poussin, C. J.: Recherches analytiques sur la théorie des nombres premiers I, Ann. Soc. Sci. Brux. 20 (1896), 183–256 [p. 251].
[26] Voros, A.: Zeta functions for the Riemann zeros, Ann. Inst. Fourier (Grenoble) 53 (2003), 665–699; erratum: 54 (2004), 1139. · Zbl 1114.11077
[27] Voros, A.: Zeta functions over zeros of general zeta and L-functions, In: T. Aoki, S. Kanemitsu, M. Nakahara and Y. Ohno (eds), Zeta Functions, Topology and Quantum Physics, (Proceedings, Osaka, March 2003), Developments in Math. 14, Springer New York, (2005), pp. 171–196. · Zbl 1170.11328
[28] Voros, A.: A sharpening of Li’s criterion for the Riemann Hypothesis, preprint (Saclay-T04/040 April 2004, unpublished) [math.NT/0404213 v2].
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