Sharper bounds for the Chebyshev functions and

Authors:
J. Barkley Rosser and Lowell Schoenfeld

Journal:
Math. Comp. **29** (1975), 243-269

MSC:
Primary 10H05

MathSciNet review:
0457373

Full-text PDF Free Access

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.

**1.**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****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]**H. M. Edwards,*Riemann’s zeta function*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Pure and Applied Mathematics, Vol. 58. MR**0466039****3.**Steven H. French,*Trigonometric polynomials in prime number theory*, Illinois J. Math.**10**(1966), 240–248. MR**0214555****[A]**A. E. Ingham,*The distribution of prime numbers*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint of the 1932 original; With a foreword by R. C. Vaughan. MR**1074573****[E]**LANDAU,*Handbuch der Lehre von der Verteilung der Primzahlen*, 2 vols., Teubner, Leipzig, 1909; reprint, Chelsea, New York, 1953.**[R]**R. Sherman Lehman,*On the difference 𝜋(𝑥)-𝑙𝑖(𝑥)*, Acta Arith.**11**(1966), 397–410. MR**0202686****[R]**R. Sherman Lehman,*On the distribution of zeros of the Riemann zeta-function*, Proc. London Math. Soc. (3)**20**(1970), 303–320. MR**0258768****[D]**D. H. Lehmer,*On the roots of the Riemann zeta-function*, Acta Math.**95**(1956), 291–298. MR**0086082****[D]**D. H. Lehmer,*Extended computation of the Riemann zeta-function*, Mathematika**3**(1956), 102–108. MR**0086083****4.**NBS #55, see Abramowitz and Stegun of this Bibliography.**5.**BARKLEY ROSSER, "The*n*-th prime is greater than ,"*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.**63**(1941), 211–232. MR**0003018****[J]**J. Barkley Rosser,*Explicit remainder terms for some asymptotic series*, J. Rational Mech. Anal.**4**(1955), 595–626. MR**0072969****[J]**J. Barkley Rosser,*A Runge-Kutta for all seasons*, SIAM Rev.**9**(1967), 417–452. MR**0219242****7.**R--S. See next entry.**[J]**J. Barkley Rosser and Lowell Schoenfeld,*Approximate formulas for some functions of prime numbers*, Illinois J. Math.**6**(1962), 64–94. MR**0137689****[J]**J. Barkley Rosser, J. M. Yohe, and Lowell Schoenfeld,*Rigorous computation and the zeros of the Riemann zeta-function. (With discussion)*, Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) North-Holland, Amsterdam, 1969, pp. 70–76. MR**0258245****[S]**S. B. Stečkin,*Certain extremal properties of positive trigonometric polynomials*, Mat. Zametki**7**(1970), 411–422 (Russian). MR**0263755****[S]**S. B. Stečkin,*The zeros of the Riemann zeta-function*, Mat. Zametki**8**(1970), 419–429 (Russian). MR**0280448****[E]**E. T. Whittaker and G. N. Watson,*A course of modern analysis*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR**1424469**

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