Numerical computations concerning the ERH

Rumely, Robert

doi:10.1090/S0025-5718-1993-1195435-0

Numerical computations concerning the ERH
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Math. Comp. 61 (1993), 415-440 Request permission

Abstract:

This paper describes a computation which established the ERH to height 10000 for all primitive Dirichlet L-series with conductor $Q \leq 13$, and to height 2500 for all $Q \leq 72$, all composite $Q \leq 112$, and other moduli. The computations were based on Euler-Maclaurin summation. Care was taken to obtain mathematically rigorous results: the zeros were first located within ${10^{ - 12}}$, then rigorously separated using an interval arithmetic package. A generalized Turing Criterion was used to show there were no zeros off the critical line. Statistics about the spacings between zeros were compiled to test the Pair Correlation Conjecture and GUE hypothesis.

References

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, DOI 10.1090/S0025-5718-1979-0537983-2
Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931
D. Davies and C. B. Haselgrove, The evaluation of Dirichlet $L$-functions, Proc. Roy. Soc. London Ser. A 264 (1961), 122–132. MR 136052, DOI 10.1098/rspa.1961.0187
H. M. Edwards, Riemann’s zeta function, Pure and Applied Mathematics, Vol. 58, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0466039
Dennis A. Hejhal, Zeros of Epstein zeta functions and supercomputers, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 1362–1384. MR 934341, DOI 10.1215/s0012-7094-87-05514-1

iAPX 86/88, 186/188 user’s manual

Kenkichi Iwasawa, Lectures on $p$-adic $L$-functions, Annals of Mathematics Studies, No. 74, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0360526
R. Sherman Lehman, On the distribution of zeros of the Riemann zeta-function, Proc. London Math. Soc. (3) 20 (1970), 303–320. MR 258768, DOI 10.1112/plms/s3-20.2.303
D. H. Lehmer, On the roots of the Riemann zeta-function, Acta Math. 95 (1956), 291–298. MR 86082, DOI 10.1007/BF02401102
D. H. Lehmer, Extended computation of the Riemann zeta-function, Mathematika 3 (1956), 102–108. MR 86083, DOI 10.1112/S0025579300001753
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. IV, Math. Comp. 46 (1986), no. 174, 667–681. MR 829637, DOI 10.1090/S0025-5718-1986-0829637-3
Kevin S. McCurley, Explicit estimates for the error term in the prime number theorem for arithmetic progressions, Math. Comp. 42 (1984), no. 165, 265–285. MR 726004, DOI 10.1090/S0025-5718-1984-0726004-6
Kevin S. McCurley, Explicit estimates for $\theta (x;3,l)$ and $\psi (x;3,l)$, Math. Comp. 42 (1984), no. 165, 287–296. MR 726005, DOI 10.1090/S0025-5718-1984-0726005-8
M. L. Mehta and J. des Cloizeaux, The probabilities for several consecutive eigenvalues of a random matrix, Indian J. Pure Appl. Math. 3 (1972), no. 2, 329–351. MR 348823
H. L. Montgomery, The pair correlation of zeros of the zeta function, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 181–193. MR 0337821

The

th zero of the Riemann zeta function and 175 million of its neighbors

Ali E. Özlük, On the pair correlation of zeros of Dirichlet $L$-functions, Number theory (Banff, AB, 1988) de Gruyter, Berlin, 1990, pp. 471–476. MR 1106680
William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling, Numerical recipes, Cambridge University Press, Cambridge, 1986. The art of scientific computing. MR 833288
Hans Rademacher, On the Phragmén-Lindelöf theorem and some applications, Math. Z 72 (1959/1960), 192–204. MR 0117200, DOI 10.1007/BF01162949

Contribution au problème de Goldbach

tout entier supérieur a 1 est somme d’au plus 13 nombres premiers

Primes in arithmetic progressions

Carl Ludwig Siegel, Contributions to the theory of the Dirichlet $L$-series and the Epstein zeta-functions, Ann. of Math. (2) 44 (1943), 143–172. MR 7760, DOI 10.2307/1968761
Robert Spira, Calculation of Dirichlet $L$-functions, Math. Comp. 23 (1969), 489–497. MR 247742, DOI 10.1090/S0025-5718-1969-0247742-X