Calculation and applications of Epstein zeta functions

Author:
Daniel Shanks

Journal:
Math. Comp. **29** (1975), 271-287

MSC:
Primary 10C15; Secondary 10-04, 10H10

DOI:
https://doi.org/10.1090/S0025-5718-1975-0409357-2

Corrigendum:
Math. Comp. **30** (1976), 900.

Corrigendum:
Math. Comp. **30** (1976), 900.

Corrigendum:
Math. Comp. **29** (1975), 1167.

Corrigendum:
Math. Comp. **29** (1975), 1167.

MathSciNet review:
0409357

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Abstract | References | Similar Articles | Additional Information

Abstract: Rapidly convergent series are given for computing Epstein zeta functions at integer arguments. From these one may rapidly and accurately compute Dirichlet *L* functions and Dedekind zeta functions for quadratic and cubic fields of any negative discriminant. Tables of such functions computed in this way are described and numerous applications are given, including the evaluation of very slowly convergent products such as those that give constants of Landau and of Hardy-Littlewood.

**[1]**Daniel Shanks,*On the conjecture of Hardy & Littlewood concerning the number of primes of the form 𝑛²+𝑎*, Math. Comp.**14**(1960), 320–332. MR**0120203**, https://doi.org/10.1090/S0025-5718-1960-0120203-6**[2]**Daniel Shanks,*Supplementary data and remarks concerning a Hardy-Littlewood conjecture*, Math. Comp.**17**(1963), 188–193. MR**0159797**, https://doi.org/10.1090/S0025-5718-1963-0159797-6**[3]**Daniel Shanks and Larry P. Schmid,*Variations on a theorem of Landau. I*, Math. Comp.**20**(1966), 551–569. MR**0210678**, https://doi.org/10.1090/S0025-5718-1966-0210678-1**[4]**Daniel Shanks,*Generalized Euler and class numbers*, Math. Comp.**21**(1967), 689–694. MR**0223295**, https://doi.org/10.1090/S0025-5718-1967-0223295-5**[5]**Daniel Shanks,*Polylogarithms, Dirichlet series, and certain constants*, Math. Comp.**18**(1964), 322–324. MR**0175275**, https://doi.org/10.1090/S0025-5718-1964-0175275-3**[6]**Daniel Shanks and John W. Wrench Jr.,*The calculation of certain Dirichlet series*, Math. Comp.**17**(1963), 136–154. MR**0159796**, https://doi.org/10.1090/S0025-5718-1963-0159796-4**[7]**Daniel Shanks and Mohan Lal,*Bateman’s constants reconsidered and the distribution of cubic residues*, Math. Comp.**26**(1972), 265–285. MR**0302590**, https://doi.org/10.1090/S0025-5718-1972-0302590-7**[8]**P. T. Bateman and E. Grosswald,*On Epstein’s zeta function*, Acta Arith.**9**(1964), 365–373. MR**0179141**, https://doi.org/10.4064/aa-9-4-365-373**[9]**Atle Selberg and S. Chowla,*On Epstein’s zeta-function*, J. Reine Angew. Math.**227**(1967), 86–110. MR**0215797**, https://doi.org/10.1515/crll.1967.227.86**[10]**H. M. Stark,*On the zeros of Epstein’s zeta function*, Mathematika**14**(1967), 47–55. MR**0215798**, https://doi.org/10.1112/S0025579300008007**[11]**M. E. Low,*Real zeros of the Dedekind zeta function of an imaginary quadratic field*, Acta Arith**14**(1967/1968), 117–140. MR**0236127**, https://doi.org/10.4064/aa-14-2-117-140**[12]**H. WEBER,*Lehrbuch der Algebra*. Vol. III, Chelsea reprint, 1961, New York, Section 141.**[13]**MOHAN LAL & DANIEL SHANKS,*Tables of Dirichlet L Functions and Dedekind Zeta Functions*. (To appear.)**[14]**Daniel Shanks and Larry P. Schmid,*Variations on a theorem of Landau. I*, Math. Comp.**20**(1966), 551–569. MR**0210678**, https://doi.org/10.1090/S0025-5718-1966-0210678-1**[15]**N. G. W. H. Beeger,*Report on some calculations of prime numbers*, Nieuw Arch. Wiskde**20**(1939), 48–50. MR**0000393****[16]**Daniel Shanks,*Systematic examination of Littlewood’s bounds on 𝐿(1,𝜒)*, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 267–283. MR**0337827****[17]**DANIEL SHANKS, "An inductive formulation of the Riemann hypothesis,"*Abstracts of Short Communications*, International Congress of Mathematicians, 1962, Stockholm, pp. 51-52.**[18]**Emil Grosswald,*Die Werte der Riemannschen Zetafunktion an ungeraden Argumentstellen.*, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II**1970**(1970), 9–13 (German). MR**0272725****[19]**RICHARD DEDEKIND, "Über die Anzahl der Idealklassen in reinen kubischen Zahlkörpern,"*J. Reine Angew. Math.*, v. 121, 1900, pp. 40-123.**[20]**T. CALLAHAN,*The*3-*Class Groups of Non-Galois Cubic Fields*. I, Dissertation, University of Toronto, Toronto, Canada, 1974.**[21]**Daniel Shanks and Richard Serafin,*Quadratic fields with four invariants divisible by 3*, Math. Comp.**27**(1973), 183–187. MR**0330097**, https://doi.org/10.1090/S0025-5718-1973-0330097-0

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DOI:
https://doi.org/10.1090/S0025-5718-1975-0409357-2

Article copyright:
© Copyright 1975
American Mathematical Society