Calculation and applications of Epstein zeta functions
Author:
Daniel Shanks
Journal:
Math. Comp. 29 (1975), 271287
MSC:
Primary 10C15; Secondary 1004, 10H10
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
Fulltext PDF Free Access
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 HardyLittlewood.
 [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 (22 #10960), http://dx.doi.org/10.1090/S00255718196001202036
 [2]
Daniel
Shanks, Supplementary data and remarks
concerning a HardyLittlewood conjecture, Math.
Comp. 17 (1963),
188–193. MR 0159797
(28 #3013), http://dx.doi.org/10.1090/S00255718196301597976
 [3]
Daniel
Shanks and Larry
P. Schmid, Variations on a theorem of Landau.
I, Math. Comp. 20 (1966), 551–569. MR 0210678
(35 #1564), http://dx.doi.org/10.1090/S00255718196602106781
 [4]
Daniel
Shanks, Generalized Euler and class
numbers, Math. Comp. 21 (1967), 689–694. MR 0223295
(36 #6343), http://dx.doi.org/10.1090/S00255718196702232955
 [5]
Daniel
Shanks, Polylogarithms, Dirichlet series, and
certain constants, Math. Comp. 18 (1964), 322–324. MR 0175275
(30 #5460), http://dx.doi.org/10.1090/S00255718196401752753
 [6]
Daniel
Shanks and John
W. Wrench Jr., The calculation of certain Dirichlet
series, Math. Comp. 17 (1963), 136–154. MR 0159796
(28 #3012), http://dx.doi.org/10.1090/S00255718196301597964
 [7]
Daniel
Shanks and Mohan
Lal, Bateman’s constants reconsidered
and the distribution of cubic residues, Math.
Comp. 26 (1972),
265–285. MR 0302590
(46 #1734), http://dx.doi.org/10.1090/S00255718197203025907
 [8]
P.
T. Bateman and E.
Grosswald, On Epstein’s zeta function, Acta Arith.
9 (1964), 365–373. MR 0179141
(31 #3392)
 [9]
Atle
Selberg and S.
Chowla, On Epstein’s zetafunction, J. Reine Angew.
Math. 227 (1967), 86–110. MR 0215797
(35 #6632)
 [10]
H.
M. Stark, On the zeros of Epstein’s zeta function,
Mathematika 14 (1967), 47–55. MR 0215798
(35 #6633)
 [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
(38 #4425)
 [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
(35 #1564), http://dx.doi.org/10.1090/S00255718196602106781
 [15]
N.
G. W. H. Beeger, Report on some calculations of prime numbers,
Nieuw Arch. Wiskde 20 (1939), 48–50. MR 0000393
(1,65g)
 [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
(49 #2596)
 [17]
DANIEL SHANKS, "An inductive formulation of the Riemann hypothesis," Abstracts of Short Communications, International Congress of Mathematicians, 1962, Stockholm, pp. 5152.
 [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
(42 #7606)
 [19]
RICHARD DEDEKIND, "Über die Anzahl der Idealklassen in reinen kubischen Zahlkörpern," J. Reine Angew. Math., v. 121, 1900, pp. 40123.
 [20]
T. CALLAHAN, The 3Class Groups of NonGalois 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
(48 #8436a), http://dx.doi.org/10.1090/S00255718197303300970
 [1]
 DANIEL SHANKS, "On the conjecture of Hardy & Littlewood concerning the number of primes of the form ," Math. Comp., v. 14, 1960, pp. 321332. MR 22 #10960. MR 0120203 (22:10960)
 [2]
 DANIEL SHANKS, "Supplementary data and remarks concerning a HardyLittlewood conjecture," Math. Comp., v. 17, 1963, pp. 188193. MR 28 #3013. MR 0159797 (28:3013)
 [3]
 DANIEL SHANKS & LARRY P. SCHMID, "Variations on a theorem of Landau. I," Math. Comp., v. 20, 1966, pp. 551569. MR 35 #1564. MR 0210678 (35:1564)
 [4]
 DANIEL SHANKS, "Generalized Euler and class numbers," Math. Comp., v. 21, 1967, pp. 689694; "Corrigenda," ibid., v. 22, 1968, p. 699. MR 36 #6343; 37 #2678. MR 0223295 (36:6343)
 [5]
 DANIEL SHANKS, "Polylogarithms, Dirichlet series, and certain constants," Math. Comp., v. 18, 1964, pp. 322324. MR 30 #5460. MR 0175275 (30:5460)
 [6]
 DANIEL SHANKS & JOHN W. WRENCH, JR., "The calculation of certain Dirichlet series," Math. Comp., v. 17, 1963, pp. 135154; "Corrigenda," ibid., v. 17, pp. 488, 699; v. 22, p. 246. MR 28 #3012. MR 0159796 (28:3012)
 [7]
 DANIEL SHANKS & MOHAN LAL, "Bateman's constants reconsidered and the distribution of cubic residues," Math. Comp., v. 26, 1972, pp. 265285. MR 46 #1734. MR 0302590 (46:1734)
 [8]
 P. T. BATEMAN & E. GROSSWALD, "On Epstein's zeta function," Acta Arith., v. 9, 1964, pp. 365373. MR 31 #3392. MR 0179141 (31:3392)
 [9]
 ATLE SELBERG & S. CHOWLA, "On Epstein's zetafunction," J. Reine Angew. Math., v. 227, 1967, pp. 86110. MR 35 #6632. MR 0215797 (35:6632)
 [10]
 H. M. STARK, "On the zeros of Epstein's zeta function," Mathematika, v. 14, 1967, pp. 4755. MR 35 #6633. MR 0215798 (35:6633)
 [11]
 M. E. LOW, "Real zeros of the Dedekind zeta function of an imaginary quadratic field," Acta Arith., v. 14, 1967/68, pp. 117140. MR 38 #4425. MR 0236127 (38:4425)
 [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 & LARRY P. SCHMID, "Variations on a theorem of Landau. II." (To appear.) MR 0210678 (35:1564)
 [15]
 N. G. W. H. BEEGER, "Report on some calculations of prime numbers," Nieuw Arch. Wisk., v. 20, 1939, pp. 4850. MR 1, 65. MR 0000393 (1:65g)
 [16]
 DANIEL SHANKS, "Systematic examination of Littlewood's bounds on ," Proc. Sympos. Pure Math., vol. 24, Amer. Math. Soc., Providence, R.I., 1973, pp. 267283. MR 0337827 (49:2596)
 [17]
 DANIEL SHANKS, "An inductive formulation of the Riemann hypothesis," Abstracts of Short Communications, International Congress of Mathematicians, 1962, Stockholm, pp. 5152.
 [18]
 EMIL GROSSWALD, "Die Werte der Riemannschen Zetafunktion an ungeraden Argumentstellen," Nachr. Akad. Wiss. Göttingen Math.Phys. Kl. II, 1970, pp. 913. MR 42 #7606. MR 0272725 (42:7606)
 [19]
 RICHARD DEDEKIND, "Über die Anzahl der Idealklassen in reinen kubischen Zahlkörpern," J. Reine Angew. Math., v. 121, 1900, pp. 40123.
 [20]
 T. CALLAHAN, The 3Class Groups of NonGalois Cubic Fields. I, Dissertation, University of Toronto, Toronto, Canada, 1974.
 [21]
 DANIEL SHANKS & RICHARD SERAFIN, "Quadratic fields with four invariants divisible by 3," Math. Comp., v. 27, 1973, pp. 183187; "Corrigenda," ibid., p. 1012. MR 0330097 (48:8436a)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
10C15,
1004,
10H10
Retrieve articles in all journals
with MSC:
10C15,
1004,
10H10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197504093572
PII:
S 00255718(1975)04093572
Article copyright:
© Copyright 1975
American Mathematical Society
