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Calculation of Dirichlet  -functions
Author:
Robert Spira
Journal:
Math. Comp. 23 (1969), 489-497
MSC:
Primary 65.25; Secondary 10.00
MathSciNet review:
0247742
Full-text PDF Free Access
Abstract |
References |
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Additional Information
Abstract: A method for calculating Dirichlet  -series is presented along with the theory of residue class characters and their automatic generation. Tables are given of zeros of  series for moduli
- [1]
D.
Davies and C.
B. Haselgrove, The evaluation of Dirichlet
𝐿-functions, Proc. Roy. Soc. Ser. A 264
(1961), 122–132. MR 0136052
(24 #B2091)
- [2]
D.
Davies, An approximate functional equation for Dirichlet
𝐿-functions, Proc. Roy. Soc. Ser. A 284
(1965), 224–236. MR 0173352
(30 #3565)
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J.
Barkley Rosser, Real roots of Dirichlet
𝐿-series, Bull. Amer. Math. Soc. 55 (1949), 906–913.
MR
0032694 (11,332c), http://dx.doi.org/10.1090/S0002-9904-1949-09306-0
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J.
Barkley Rosser, Real roots of real Dirichlet 𝐿-series,
J. Research Nat. Bur. Standards 45 (1950), 505–514.
MR
0041161 (12,804i)
- [5]
William
Judson LeVeque, Topics in number theory. Vols. 1 and 2,
Addison-Wesley Publishing Co., Inc., Reading, Mass., 1956. MR 0080682
(18,283d)
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J. V. Uspensky & M. A. Heaslet, Elementary Number Theory, McGraw-Hill, New York, 1939. MR 1, 38.
- [7]
Joseph
J. Rotman, The theory of groups. An introduction, Allyn and
Bacon Inc., Boston, Mass., 1965. MR 0204499
(34 #4338)
- [8]
S. Chowla, "Note on Dirichlet's
 -functions," Acta Arith., v. 1, 1935, pp. 113-114.
- [9]
N. G. Čudakov, Introduction to the Theory of Dirichlet's
 -functions, OGIZ, Moscow, 1947. (Russian) MR 11, 234.
- [10]
R. Spira, Check Values, Zeros and Fortran Programs for the Riemann Zeta Function and its First Three Derivatives, Technical Report No. 1, University of Tennessee Computing Center, 1965.
- [11]
Harold
Davenport, Multiplicative number theory, Lectures given at the
University of Michigan, Winter Term, vol. 1966, Markham Publishing
Co., Chicago, Ill., 1967. MR 0217022
(36 #117)
- [12]
L. Schoenfeld, "A generalization of the Euler-Maclaurin sum formula." (To appear.)
- [13]
E. Landau, Vorlesungen über Zahlentheorie, Vol. I, Chelsea, New York, 1950.
- [14]
Edmund
Landau, Handbuch der Lehre von der Verteilung der Primzahlen. 2
Bände, Chelsea Publishing Co., New York, 1953 (German). 2d ed;
With an appendix by Paul T. Bateman. MR 0068565
(16,904d)
- [15]
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/S0025-5718-1963-0159796-4
- [16]
Royal Society of London, Mathematical Tables Committee. Royal Society Depository for Unpublished Mathematical Tables. Table 80. (Copy in Pennsylvania State University Library.)
- [17]
Robert
Spira, Residue class characters, Duke Math. J.
37 (1970), 633–637. MR 0269584
(42 #4479)
- [18]
RMT 72, Math. Comp., v. 17, 1963, pp. 463-464.
- [1]
- D. Davies & C. B. Haselgrove, "The evaluation of Dirichlet
 -functions," Proc. Roy Soc. Ser. A, v. 264, 1961, pp. 122-132. MR 24 #B2091. MR 0136052 (24:B2091)
- [2]
- D. Davies, "An approximate functional equation for Dirichlet
 -functions," Proc. Roy. Soc. Ser. A, v. 284, 1965, pp. 224-236. MR 30 #3565. MR 0173352 (30:3565)
- [3]
- J. B. Rosser, "Real roots of Dirichlet
 -series," Bull. Amer. Math. Soc., v. 55, 1949, pp. 906-913. MR 11, 332. MR 0032694 (11:332c)
- [4]
- J. B. Rosser, "Real roots of Dirichlet
 -series," J. Res. Nat. Bur. Standards, v. 45, 1950, pp. 505-514. MR 12, 804. MR 0041161 (12:804i)
- [5]
- W. J. LeVeque, Topics in Number Theory. Vols. I, II, Addison-Wesley, Reading, Mass., 1956. MR 18, 283. MR 0080682 (18:283d)
- [6]
- J. V. Uspensky & M. A. Heaslet, Elementary Number Theory, McGraw-Hill, New York, 1939. MR 1, 38.
- [7]
- J. J. Rotman, The Theory of Groups. An Introduction, Allyn & Bacon, Boston, Mass., 1965. MR 34 #4338. MR 0204499 (34:4338)
- [8]
- S. Chowla, "Note on Dirichlet's -functions," Acta Arith., v. 1, 1935, pp. 113-114.
- [9]
- N. G. Čudakov, Introduction to the Theory of Dirichlet's -functions, OGIZ, Moscow, 1947. (Russian) MR 11, 234.
- [10]
- R. Spira, Check Values, Zeros and Fortran Programs for the Riemann Zeta Function and its First Three Derivatives, Technical Report No. 1, University of Tennessee Computing Center, 1965.
- [11]
- H. Davenport, Multiplicative Number Theory, Markham, Chicago, Ill., 1967. MR 36 #117. MR 0217022 (36:117)
- [12]
- L. Schoenfeld, "A generalization of the Euler-Maclaurin sum formula." (To appear.)
- [13]
- E. Landau, Vorlesungen über Zahlentheorie, Vol. I, Chelsea, New York, 1950.
- [14]
- E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen. 2 Bände, 2nd ed., Chelsea, New York, 1953. MR 16, 904. MR 0068565 (16:904d)
- [15]
- D. Shanks & J. W. Wrench, Jr., "The calculation of certain Dirichlet series," Math. Comp., v. 17, 1963, pp. 136-154. MR 28 #3012. MR 0159796 (28:3012)
- [16]
- Royal Society of London, Mathematical Tables Committee. Royal Society Depository for Unpublished Mathematical Tables. Table 80. (Copy in Pennsylvania State University Library.)
- [17]
- R. Spira, "Residue class characters." (To appear.) MR 0269584 (42:4479)
- [18]
- RMT 72, Math. Comp., v. 17, 1963, pp. 463-464.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025-5718-1969-0247742-X
PII:
S 0025-5718(1969)0247742-X
Article copyright:
© Copyright 1969 American Mathematical Society
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