Sharper bounds for the Chebyshev functions and
Authors:
J. Barkley Rosser and Lowell Schoenfeld
Journal:
Math. Comp. 29 (1975), 243269
MSC:
Primary 10H05
MathSciNet review:
0457373
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Abstract: The authors demonstrate a wider zerofree 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
(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]
H.
M. Edwards, Riemann’s zeta function, Academic Press [A
subsidiary of Harcourt Brace Jovanovich, Publishers], New YorkLondon,
1974. Pure and Applied Mathematics, Vol. 58. MR 0466039
(57 #5922)
 3.
Steven
H. French, Trigonometric polynomials in prime number theory,
Illinois J. Math. 10 (1966), 240–248. MR 0214555
(35 #5404)
 [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
(91f:11064)
 [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
(34 #2546)
 [R]
R.
Sherman Lehman, On the distribution of zeros of the Riemann
zetafunction, Proc. London Math. Soc. (3) 20 (1970),
303–320. MR 0258768
(41 #3414)
 [D]
D.
H. Lehmer, On the roots of the Riemann zetafunction, Acta
Math. 95 (1956), 291–298. MR 0086082
(19,121a)
 [D]
D.
H. Lehmer, Extended computation of the Riemann zetafunction,
Mathematika 3 (1956), 102–108. MR 0086083
(19,121b)
 4.
NBS #55, see Abramowitz and Stegun of this Bibliography.
 5.
BARKLEY ROSSER, "The nth prime is greater than ," Proc. London Math. Soc. (2), v. 45, 1939, pp. 2144.
 6.
Barkley
Rosser, Explicit bounds for some functions of prime numbers,
Amer. J. Math. 63 (1941), 211–232. MR 0003018
(2,150e)
 [J]
J.
Barkley Rosser, Explicit remainder terms for some asymptotic
series, J. Rational Mech. Anal. 4 (1955),
595–626. MR 0072969
(17,360a)
 [J]
J.
Barkley Rosser, A RungeKutta for all seasons, SIAM Rev.
9 (1967), 417–452. MR 0219242
(36 #2325)
 7.
RS. 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
(25 #1139)
 [J]
J.
Barkley Rosser, J.
M. Yohe, and Lowell
Schoenfeld, Rigorous computation and the zeros of the Riemann
zetafunction. (With discussion), Information Processing 68 (Proc.
IFIP Congress, Edinburgh, 1968) NorthHolland, Amsterdam, 1969,
pp. 70–76. MR 0258245
(41 #2892)
 [S]
S.
B. Stečkin, Certain extremal properties of positive
trigonometric polynomials, Mat. Zametki 7 (1970),
411–422 (Russian). MR 0263755
(41 #8355)
 [S]
S.
B. Stečkin, The zeros of the Riemann zetafunction,
Mat. Zametki 8 (1970), 419–429 (Russian). MR 0280448
(43 #6168)
 [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
(97k:01072)
 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. 240248. 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 ," Acta Arith., v. 11, 1966, pp. 397410. MR 34 #2546. MR 0202686 (34:2546)
 [R]
 SHERMAN LEHMAN, "On the distribution of zeros of the Riemann zetafunction," Proc. London Math. Soc. (3), v. 20, 1970, pp. 303320. MR 41 #3414. MR 0258768 (41:3414)
 [D]
 H. LEHMER, "On the roots of the Riemann zetafunction," Acta Math., v. 95, 1956, pp. 291298. [Cited as 1956 A.] MR 19, 121; 1431. MR 0086082 (19:121a)
 [D]
 H. LEHMER, "Extended computation of the Riemann zetafunction," Mathematika, v. 3, 1956, pp. 102108. [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 nth prime is greater than ," Proc. London Math. Soc. (2), v. 45, 1939, pp. 2144.
 6.
 BARKLEY ROSSER, "Explicit bounds for some functions of prime numbers," Amer. J. Math., v. 63, 1941, pp. 211232. 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. 595626. MR 17, 360. MR 0072969 (17:360a)
 [J]
 BARKLEY ROSSER, "A RungeKutta for all seasons," SIAM Rev., v. 9, 1967, pp. 417452. MR 36 #2325. MR 0219242 (36:2325)
 7.
 RS. See next entry.
 [J]
 BARKLEY ROSSER & LOWELL SCHOENFELD, "Approximate formulas for some functions of prime numbers," Illinois J. Math., v. 6, 1962, pp. 6494. [Cited as RS.] 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, NorthHolland, Amsterdam, 1969, pp. 7076. 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. 411422 = Math. Notes, v. 7, 1970, pp. 248255. [Cited as 1970 A.] MR 41 #8355. MR 0263755 (41:8355)
 [S]
 B. STECHKIN (S. B. STEČKIN), "Zeros of the Riemann zetafunction," Mat. Zametki, v. 8, 1970, pp. 419429 = Math. Notes, v. 8, 1970, pp. 706711. [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)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197504573737
PII:
S 00255718(1975)04573737
Article copyright:
© Copyright 1975
American Mathematical Society
