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Original Articles

Discretized Keiper/Li Approach to the Riemann Hypothesis

References

  • [Arias de Reyna 11] J. Arias de Reyna. “Asymptotics of Keiper–Li Coefficients.” Funct. Approx. Comment. Math. 45 (2011), 7–21.
  • [Ayoub 63] R. Ayoub. An Introduction to the Analytic Theory of Numbers, Mathematical Surveys 10. Providence, RI: American Mathematical Society, 1963.
  • [Báez-Duarte 05] L. Báez-Duarte. “A Sequential Riesz-like Criterion for the Riemann Hypothesis.” Int. J. Math. Math. Sci. 21 (2005), 3527–3537.
  • [Balanzario and Sánchez-Ortiz 07] E. P. Balanzario and J. Sánchez-Ortiz. “Zeros of the Davenport–Heilbronn counterexample.” Math. Comput. 76 (2007), 2045–2049.
  • [Beliakov and Matiyasevich 15] G. Beliakov and Y. Matiyasevich. “Approximation of Riemann’s zeta Function by Finite Dirichlet Series: A Multiprecision Numerical Approach.” Exp. Math. 24 (2015), 150–161.
  • [Biane et al. 01] P. Biane, J. Pitman, and M. Yor. “Probability Laws Related to the Jacobi Theta and Riemann zeta Functions.” Bull. Amer. Math. Soc. 38 (2001), 435–465.
  • [Bombieri and Ghosh 11] E. Bombieri and A. Ghosh. “Around the Davenport–Heilbronn Function.” Uspekhi Mat. Nauk 66 (2011), 15–66, Russian Math. Surveys 66 (2011), 221–270.
  • [Bombieri and Lagarias 99] E. Bombieri and J. C. Lagarias. “Complements to Li’s Criterion for the Riemann Hypothesis.” J. Number Theory 77 (1999), 274–287.
  • [Coffey 05] M. W. Coffey. “Toward Verification of the Riemann Hypothesis: Application of the Li Criterion.” Math. Phys. Anal. Geom. 8 (2005), 211–255.
  • [Coffey 08] M. W. Coffey. “New Results Concerning Power Series Expansions of the Riemann xi Function and the Li/Keiper Constants.” Proc. R. Soc. Lond. A 464 (2008), 711–731, and refs. therein.
  • [Davenport 00] H. Davenport. Multiplicative Number Theory (Graduate Texts in Mathematics), vol. 74, 3rd ed. (revised by H.L. Montgomery). New York: Springer, 2000, and refs. therein.
  • [Davenport and Heilbronn 36] H. Davenport and H. Heilbronn. “On the Zeros of Certain Dirichlet Series I, II.” J. London Math. Soc. 11 (1936), 181–185, 307–312.
  • [Dingle 73] R. B. Dingle. Asymptotic Expansions: Their Derivation and Interpretation. London: Academic Press, 1973.
  • [Erdélyi 56] A. Erdélyi. Asymptotic Expansions. New York: Dover, 1956.
  • [Flajolet and Vepstas 08] Ph. Flajolet and L. Vepstas. “On Differences of Zeta Values.” J. Comput. Appl. Math. 220 (2008), 58–73, and refs. therein.
  • [Gourdon 04] X. Gourdon. The 1013 First zeros of the Riemann Zeta Function, and Zeros Computation at Very Large Height, preprint. Available online http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf, 2004.
  • [Gradshteyn and Ryzhik 94] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series and Products, 5th ed., edited by A. Jeffrey. London: Academic Press, 1994.
  • [Johansson 15] F. Johansson. “Rigorous High-Precision Computation of the Hurwitz zeta Function and its Derivatives.” Numer. Algor. 69 (2015), 253–270.
  • [Keiper 92] J. B. Keiper. “Power Series Expansions of Riemann’s ξ Function.” Math. Comput. 58 (1992), 765–773.
  • [Lagarias 07] J. C. Lagarias. “Li Coefficients for Automorphic L-functions.” Ann. Inst. Fourier, Grenoble 57 (2007), 1689–1740.
  • [Li 97] X.-J. Li. “The Positivity of a Sequence of Numbers and the Riemann Hypothesis.” J. Number Theory 65 (1997), 325–333.
  • [Maślanka 04a] K. Maślanka. “Li’s Criterion for the Riemann Hypothesis — Numerical Approach.” Opuscula Math. 24 (2004a), 103–114.
  • [Maślanka 04b] K. Maślanka. “Effective Method of Computing Li’s Coefficients and their Properties.” preprint (2004b), arXiv:math.NT/0402168 v5.
  • [Maślanka 06] K. Maślanka. “Báez-Duarte’s Criterion for the Riemann Hypothesis and Rice’s Integrals.” preprint (2006), arXiv:math/0603713 v2 [math.NT].
  • [Misguich 17] G. Misguich. Calculations for n > 20000, using http://www.mpfr.org/, 2017.
  • [Oesterlé 00] J. Oesterlé. “Régions Sans zéros de la Fonction zêta de Riemann.” typescript (2000, revised 2001, uncirculated).
  • [Riemann 59] B. Riemann. “Über die Anzahl der Primzahlen unter Einer Gegebenen Grösse.” Monatsb. Preuss. Akad. Wiss. (1859), 671–680 [English translation by R. Baker, Ch. Christenson, and H. Orde: Bernhard Riemann: Collected Papers, paper VII, pp. 135–143. Heber City, UT: Kendrick Press, 2004].
  • [Rudin 66] W. Rudin. Real and Complex Analysis, New York: McGraw–Hill, 1966.
  • [Sekatskii 14] S. K. Sekatskii. “Generalized Bombieri–Lagarias’ Theorem and Generalized Li’s Criterion with its Arithmetic Interpretation.” Ukr. Mat. Zh. 66 (2014), 371–383, Ukr. Math. J. 66 (2014), 415–431; and Asymptotic of the Generalized Li’s Sums Which Non-negativity is Equivalent to the Riemann Hypothesis. arXiv:1403.4484 [math.NT] (2014).
  • [Sekatskii 16] S. K. Sekatskii. “Analysis of Voros Criterion Equivalent to Riemann Hypothesis.” Anal. Geom. Number Theory 1 (2016), 95–102. arXiv:1407.5758 [math.NT].
  • [Spira 94] R. Spira. “Some Zeros of the Titchmarsh Counterexample.” Math. Comput. 63 (1994), 747–748.
  • [Titchmarsh 86] E. C. Titchmarsh. The Theory of the Riemann Zeta-Function, 2nd ed. (revised by D. R. Heath-Brown). Oxford: Oxford Univ. Press, 1986.
  • [Voros 06] A. Voros. A Sharpening of Li’s Criterion for the Riemann Hypothesis. (Preprint Saclay-T04/040, 2004), arXiv: math.NT/0404213 v2, and “Sharpenings of Li’s criterion for the Riemann Hypothesis.” Math. Phys. Anal. Geom. 9 (2006), 53–63.
  • [Voros 10] A. Voros. Zeta Functions over Zeros of Zeta Functions (Lecture Notes of the Unione Matematica Italiana), vol. 8. Berlin: Springer-Verlag, 2010.
  • [Voros 14] A. Voros. “Zeta Functions over zeros of Zeta Functions and an Exponential-Asymptotic view of the Riemann Hypothesis.” In Exponential Analysis of Differential Equations and Related Topics (Proceedings, Kyoto, Oct. 2013, ed. Y. Takei), pp. 147–164. RIMS Kôkyûroku Bessatsu B52, 2014.
  • [Voros 15] A. Voros. An Asymptotic Criterion in an Explicit Sequence, (Preprint IPhT15/106, 2015), HAL archive: cea-01166324, and Simplifications of the Keiper/Li Approach to the Riemann Hypothesis (Preprint IPhT16/011, 2016), arXiv:1602.03292 [math.NT].
  • [Wolfram 96] S. Wolfram. Mathematica, 3rd ed. New York: Wolfram Media/Cambridge University Press, 1996.

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