Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Sharper bounds for the Chebyshev functions $ \theta (x)$ and $ \psi (x)$


Authors: J. Barkley Rosser and Lowell Schoenfeld
Journal: Math. Comp. 29 (1975), 243-269
MSC: Primary 10H05
DOI: https://doi.org/10.1090/S0025-5718-1975-0457373-7
MathSciNet review: 0457373
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The authors demonstrate a wider zero-free region for the Riemann zeta function than has been given before. They give improved methods for using this and a recent determination that the first 3,502,500 zeros lie on the critical line to develop better bounds for functions of primes.


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

  • 1. MILTON ABRAMOWITZ & IRENE A. STEGUN (Editors), Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables, Nat. Bur. Standards Appl. Math. Series, 55, Superintendent of Documents, U. S. Government Printing Office, Washington, D. C., 1964; reprinted, Dover, New York, 1964. [Cited as NBS #55.] MR 29 #4914; 34 #8606. MR 0167642 (29:4914)
  • 2. KENNETH I. APPEL & J. BARKLEY ROSSER, Table for Estimating Functions of Primes, Communications Research Division Technical Report No. 4, Institute for Defense Analyses, Princeton, N. J., 1961.
  • [H] M. EDWARDS, Riemann's Zeta Function, Academic Press, New York and London, 1974. MR 0466039 (57:5922)
  • 3. STEVEN H. FRENCH, "Trigonometric polynomials in prime number theory," Illinois J. Math., v. 10, 1966, pp. 240-248. MR 35 #5404. MR 0214555 (35:5404)
  • [A] E. INGHAM, The Distribution of Prime Numbers, Cambridge Tracts in Math. and Math. Phys., no. 30, Cambridge Univ. Press, London, 1932. MR 1074573 (91f:11064)
  • [E] LANDAU, Handbuch der Lehre von der Verteilung der Primzahlen, 2 vols., Teubner, Leipzig, 1909; reprint, Chelsea, New York, 1953.
  • [R] SHERMAN LEHMAN, "On the difference $ \pi (x) = {\operatorname{li}}(x)$," Acta Arith., v. 11, 1966, pp. 397-410. MR 34 #2546. MR 0202686 (34:2546)
  • [R] SHERMAN LEHMAN, "On the distribution of zeros of the Riemann zeta-function," Proc. London Math. Soc. (3), v. 20, 1970, pp. 303-320. MR 41 #3414. MR 0258768 (41:3414)
  • [D] H. LEHMER, "On the roots of the Riemann zeta-function," Acta Math., v. 95, 1956, pp. 291-298. [Cited as 1956 A.] MR 19, 121; 1431. MR 0086082 (19:121a)
  • [D] H. LEHMER, "Extended computation of the Riemann zeta-function," Mathematika, v. 3, 1956, pp. 102-108. [Cited as 1956 B.] MR 19, 121; 1431. MR 0086083 (19:121b)
  • 4. NBS #55, see Abramowitz and Stegun of this Bibliography.
  • 5. BARKLEY ROSSER, "The n-th prime is greater than $ n \log n$," Proc. London Math. Soc. (2), v. 45, 1939, pp. 21-44.
  • 6. BARKLEY ROSSER, "Explicit bounds for some functions of prime numbers," Amer. J. Math., v. 63, 1941, pp. 211-232. MR 2, 150. MR 0003018 (2:150e)
  • [J] BARKLEY ROSSER, "Explicit remainder terms for some asymptotic series," J. Rational Mech. Anal., v. 4, 1955, pp. 595-626. MR 17, 360. MR 0072969 (17:360a)
  • [J] BARKLEY ROSSER, "A Runge-Kutta for all seasons," SIAM Rev., v. 9, 1967, pp. 417-452. MR 36 #2325. MR 0219242 (36:2325)
  • 7. R--S. See next entry.
  • [J] BARKLEY ROSSER & LOWELL SCHOENFELD, "Approximate formulas for some functions of prime numbers," Illinois J. Math., v. 6, 1962, pp. 64-94. [Cited as R--S.] MR 25 #1139. MR 0137689 (25:1139)
  • [J] BARKLEY ROSSER, J. M. YOHE & LOWELL SCHOENFELD, "Rigorous computation and the zeros of the Riemann zeta function," Proc. IFIP (Edinburgh), Vol. I: Mathematics, Software, North-Holland, Amsterdam, 1969, pp. 70-76. MR 41 #2892. MR 0258245 (41:2892)
  • [S] B. STECHKIN (S. B. STEČKIN), "Some extremal properties of positive trigonometric polynomials," Mat. Zametki, v. 7, 1970, pp. 411-422 = Math. Notes, v. 7, 1970, pp. 248-255. [Cited as 1970 A.] MR 41 #8355. MR 0263755 (41:8355)
  • [S] B. STECHKIN (S. B. STEČKIN), "Zeros of the Riemann zeta-function," Mat. Zametki, v. 8, 1970, pp. 419-429 = Math. Notes, v. 8, 1970, pp. 706-711. [Cited as 1970 B.] MR 43 #6168. MR 0280448 (43:6168)
  • [E] T. WHITTAKER & G. N. WATSON, A Course of Modern Analysis, 4th ed., Cambridge Univ. Press, New York, 1962. MR 31 #2375. MR 1424469 (97k:01072)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10H05

Retrieve articles in all journals with MSC: 10H05


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1975-0457373-7
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society