On the zeros of the Riemann zeta function in the critical strip. IV

Authors:
J. van de Lune, H. J. J. te Riele and D. T. Winter

Journal:
Math. Comp. **46** (1986), 667-681

MSC:
Primary 11M26; Secondary 11-04, 11Y35, 30C15

MathSciNet review:
829637

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Abstract: Very extensive computations are reported which extend and, partly, check previous computations concerning the location of the complex zeros of the Riemann zeta function. The results imply the truth of the Riemann hypothesis for the first 1,500,000,001 zeros of the form in the critical strip with , i.e., all these zeros have real part . Moreover, all these zeros are simple. Various tables are given with statistical data concerning the numbers and first occurrences of Gram blocks of various types; the numbers of Gram intervals containing *m* zeros, for and 4; and the numbers of exceptions to "Rosser's rule" of various types (including some formerly unobserved types). Graphs of the function are given near five rarely occurring exceptions to Rosser's rule, near the first Gram block of length 9, near the closest observed pair of zeros of the Riemann zeta function, and near the largest (positive and negative) found values of at Gram points. Finally, a number of references are given to various number-theoretical implications.

**[1]**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**, 10.1090/S0025-5718-1979-0537983-2**[2]**R. P. Brent, J. van de Lune, H. J. J. te Riele, and D. T. Winter,*On the zeros of the Riemann zeta function in the critical strip. II*, Math. Comp.**39**(1982), no. 160, 681–688. MR**669660**, 10.1090/S0025-5718-1982-0669660-1**[3]**E. Karkoschka and P. Werner,*Einige Ausnahmen zur Rosserschen Regel in der Theorie der Riemannschen Zetafunktion*, Computing**27**(1981), no. 1, 57–69 (German, with English summary). MR**623176**, 10.1007/BF02243438**[4]**I. Kátai,*On oscillations of number-theoretic functions*, Acta Arith.**13**(1967/1968), 107–122. MR**0219496****[5]**R. Sherman Lehman,*On the difference 𝜋(𝑥)-𝑙𝑖(𝑥)*, Acta Arith.**11**(1966), 397–410. MR**0202686****[6]**J. van de Lune, H. J. J. te Riele & D. T. Winter,*Rigorous High Speed Separation of Zeros of Riemann's Zeta Function*, Report NW 113/81, Mathematical Centre, Amsterdam, October, 1981.**[7]**J. van de Lune and H. J. J. te Riele,*On the zeros of the Riemann zeta function in the critical strip. III*, Math. Comp.**41**(1983), no. 164, 759–767. MR**717719**, 10.1090/S0025-5718-1983-0717719-3**[8]**János Pintz,*On the sign changes of 𝑀(𝑥)=∑_{𝑛≤𝑥}𝜇(𝑛)*, Analysis**1**(1981), no. 3, 191–195. MR**660714**, 10.1524/anly.1981.1.3.191**[9]**H. J. J. te Riele, D. T. Winter & J. van de Lune,*Numerical Verification of the Riemann Hypothesis on the*CYBER 205, in:*Proc. International Supercomputer Applications Symp.*(A. Emmen, ed.), North-Holland, Amsterdam, 1985, pp. 33-38.**[10]**J. Barkley Rosser and Lowell Schoenfeld,*Approximate formulas for some functions of prime numbers*, Illinois J. Math.**6**(1962), 64–94. MR**0137689****[11]**J. Barkley Rosser and Lowell Schoenfeld,*Sharper bounds for the Chebyshev functions 𝜃(𝑥) and 𝜓(𝑥)*, Math. Comp.**29**(1975), 243–269. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. MR**0457373**, 10.1090/S0025-5718-1975-0457373-7**[12]**Lowell Schoenfeld,*Sharper bounds for the Chebyshev functions 𝜃(𝑥) and 𝜓(𝑥). II*, Math. Comp.**30**(1976), no. 134, 337–360. MR**0457374**, 10.1090/S0025-5718-1976-0457374-X**[13]**Lowell Schoenfeld,*An improved estimate for the summatory function of the Möbius function*, Acta Arith.**15**(1968/1969), 221–233. MR**0241376****[14]**D. T. Winter & H. J. J. te Riele, "Optimization of a program for the verification of the Riemann hypothesis,"*Supercomputer*, v. 5, 1985, pp. 29-32.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1986-0829637-3

Keywords:
Riemann hypothesis,
Riemann zeta function,
Gram blocks,
Rosser's rule

Article copyright:
© Copyright 1986
American Mathematical Society