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Estimating $ \pi(x)$ and related functions under partial RH assumptions

Author: Jan Büthe
Journal: Math. Comp. 85 (2016), 2483-2498
MSC (2010): Primary 11N05; Secondary 11M26
Published electronically: December 1, 2015
MathSciNet review: 3511289
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Abstract: We give a direct interpretation of the validity of the Riemann hypothesis for all zeros with $ \Im (\rho )\in (0,T]$ in terms of the prime-counting function $ \pi (x)$ by proving that Schoenfeld's explicit estimates for $ \pi (x)$ and the Chebyshov functions hold as long as $ 4.92\sqrt {x/\log (x)} \leq T$.

We also improve some of the existing bounds of Chebyshov type for the function $ \psi (x)$.

References [Enhancements On Off] (What's this?)

  • [AAR99] George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)
  • [Bar81] Klaus Barner, On A. Weil's explicit formula, J. Reine Angew. Math. 323 (1981), 139-152. MR 611448 (82i:12014),
  • [Bre79] Richard P. Brent, On the zeros of the Riemann zeta function in the critical strip, Math. Comp. 33 (1979), no. 148, 1361-1372. MR 537983 (80g:10033),
  • [Büt] J. Büthe, An improved analytic method for calculating $ \pi (x)$, arXiv:1410.7008.
  • [Büt14] J. Büthe, Untersuchung der Primzahlzählfunktion und verwandter Funktionen, Ph.D. thesis, Bonn University, March 2015.
  • [FK15] Laura Faber and Habiba Kadiri, New bounds for $ \psi (x)$, Math. Comp. 84 (2015), no. 293, 1339-1357. MR 3315511,
  • [FKBJ] J. Franke, Th. Kleinjung, J. Büthe, and A. Jost, A practical analytic method for calculating $ \pi (x)$, Math. Comp., to appear.
  • [Gou04] X. Gourdon, The $ 10^{13}$ first zeros of the Riemann Zeta function and zeros computation at very large height,, October 2004.
  • [Log88] B. F. Logan, Bounds for the tails of sharp-cutoff filter kernels, SIAM J. Math. Anal. 19 (1988), no. 2, 372-376. MR 930033 (90m:94009),
  • [Olv97] Frank W. J. Olver, Asymptotics and Special Functions, AKP Classics, A K Peters, Ltd., Wellesley, MA, 1997. Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)]. MR 1429619 (97i:41001)
  • [OS88] A. M. Odlyzko and A. Schönhage, Fast algorithms for multiple evaluations of the Riemann zeta function, Trans. Amer. Math. Soc. 309 (1988), no. 2, 797-809. MR 961614 (89j:11083),
  • [Pla15] David J. Platt, Computing $ \pi (x)$ analytically, Math. Comp. 84 (2015), no. 293, 1521-1535. MR 3315519,
  • [Ros41] Barkley Rosser, Explicit bounds for some functions of prime numbers, Amer. J. Math. 63 (1941), 211-232. MR 0003018 (2,150e)
  • [Sch76] Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $ \theta (x)$ and $ \psi (x)$. II, Math. Comp. 30 (1976), no. 134, 337-360. MR 0457374 (56 #15581b)
  • [Stu36] C. Sturm, Memoire sur les équation différentielles linéaire du second ordre, J. Math. Pure Appl. (1) 1 (1836), 106-186.
  • [vM95] H. von Mangoldt, Zu Riemanns Abhandlungen ``Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse'', J. Reine Angew. Math. 114 (1895), 255-305.
  • [Wat44] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746 (6,64a)

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Additional Information

Jan Büthe
Affiliation: Mathematisches Institut, Endenicher Allee 60, 53115 Bonn, Germany

Received by editor(s): November 13, 2014
Received by editor(s) in revised form: March 11, 2015, and March 15, 2015
Published electronically: December 1, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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