A new bound for the smallest with

Authors:
Carter Bays and Richard H. Hudson

Journal:
Math. Comp. **69** (2000), 1285-1296

MSC (1991):
Primary 11-04, 11A15, 11M26, 11Y11, 11Y35

DOI:
https://doi.org/10.1090/S0025-5718-99-01104-7

Published electronically:
May 4, 1999

MathSciNet review:
1752093

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the number of primes and let denote the usual integral logarithm of . We prove that there are at least integer values of in the vicinity of with . This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of in four different regions, including the regions discovered by Lehman, te Riele, and the authors of this paper, and a more distant region in the vicinity of , where appears to exceed by more than . The plots strongly suggest, although upper bounds derived to date for are not sufficient for a proof, that exceeds for at least integers in the vicinity of . If it is possible to improve our bound for by finding a sign change before , our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of and find that as departs from the region in the vicinity of , the density is , and that it varies from this by no more than over the next integers. This should be compared to Rubinstein and Sarnak.

**1.**Carter Bays and Richard H. Hudson,*Zeroes of Dirichlet -functions and irregularities in the distribution of primes*, Math. Comp., posted on March 10, 1999, PII: S0025-5718(99)01105-9 (to appear in print).**2.**Carter Bays, Kevin B. Ford, Richard H. Hudson, and Michael Rubinstein,*Zeros of Dirichlet -functions near the real axis and Chebyshev's bias*, submitted for publication.**3.**M. Deléglise and J. Rivat,*Computing : The Meissel, Lehmer, Lagarias, Miller, Odlyzko method*, Math. Comp.**65**(1996), 235-245. MR**96d:11139****4.**H. M. Edwards*Riemann's zeta function*, Academic Press, New York, 1974. MR**57:5922****5.**Ian Richards,*On the incompatibility of two conjectures concerning primes*, Bull. Amer. Math. Soc.**80**(1974), 419-438. MR**49:2601****6.**Richard H. Hudson,*Averaging effects on irregularities in the distribution of primes in arithmetic progressions*, Math. Comp.**44**(1985), 561-571. MR**86h:11074****7.**A. E. Ingham,*The distribution of prime numbers*, Cambridge University Press, Cambridge, 1932; reprint, 1990. MR**91f:11064****8.**Jeff Lagarias, Victor Miller, and Andrew Odlyzko,*Computing : The Meissel-Lehmer method*, Math. Comp.**44**(1985), 537-560. MR**86h:11111****9.**R. Sherman Lehman,*On the difference*, Acta Arith.**11**(1966), 397-410. MR**34:2546****10.**J. E. Littlewood,*Sur la distribution des nombres premiers*, C. R. Acad. Sci. Paris**158**(1914), 1869-1872.**11.**J. van de Lune, J. J. te Riele, and D. T. Winter,*On the zeros of the Riemann zeta function in the critical strip*. IV, Math. Comp.**46**(1986), 667-681. MR**87e:11102****12.**H. L. Montgomery,*The zeta function and prime numbers*, Proc. Queen's Number Theory Conf., 1979 Queens Papers in Pure and Applied Mathematics, v. 54, Queen's Univ., Kingston, Ont., 1980, pp. 1-31. MR**82k:10047****13.**Herman J. J. te Riele,*On the sign of the difference*, Math. Comp.**48**(1987), 323-328. MR**88a:11135****14.**Hans Riesel,*Prime numbers and computer methods for factorization*, Birkhäuser, Boston, 1985. MR**88k:11002****15.**Michael Rubinstein and Peter Sarnak,*Chebyshev's bias*, Experimental Mathematics**3**(1994), 173-197. MR**96d:11099****16.**S. Skewes,*On the difference*, J. London Math. Soc.**8**(1933), 277-283.**17.**S. Skewes,*On the difference*, II, Proc. London Math. Soc. (3)**5**(1955), 48-70. MR**16:676c**

Retrieve articles in *Mathematics of Computation*
with MSC (1991):
11-04,
11A15,
11M26,
11Y11,
11Y35

Retrieve articles in all journals with MSC (1991): 11-04, 11A15, 11M26, 11Y11, 11Y35

Additional Information

**Carter Bays**

Affiliation:
Department of Computer Science, University of South Carolina, Columbia, South Carolina 29208

Email:
bays@cs.sc.edu

**Richard H. Hudson**

Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Email:
hudson@math.sc.edu

DOI:
https://doi.org/10.1090/S0025-5718-99-01104-7

Received by editor(s):
June 30, 1997

Received by editor(s) in revised form:
April 1, 1998, and July 7, 1998

Published electronically:
May 4, 1999

Article copyright:
© Copyright 2000
American Mathematical Society