On computing Artin -functions in the critical strip

Authors:
J. C. Lagarias and A. M. Odlyzko

Journal:
Math. Comp. **33** (1979), 1081-1095

MSC:
Primary 12A70

DOI:
https://doi.org/10.1090/S0025-5718-1979-0528062-9

MathSciNet review:
528062

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Abstract: This paper gives a method for computing values of certain nonabelian Artin *L*-functions in the complex plane. These Artin *L*-functions are attached to irreducible characters of degree 2 of Galois groups of certain normal extensions *K* of Q. These fields *K* are the ones for which has an abelian subgroup *A* of index 2, whose fixed field is complex, and such that there is a for which for all . The key property proved here is that these particular Artin *L*-functions are Hecke (abelian) *L*-functions attached to ring class characters of the imaginary quadratic field and, therefore, can be expressed as linear combinations of Epstein zeta functions of positive definite binary quadratic forms. Such Epstein zeta functions have rapidly convergent expansions in terms of incomplete gamma functions.

In the special case , where is cube-free, the Artin *L*-function attached to the unique irreducible character of degree 2 of is the quotient of the Dedekind zeta function of the pure cubic field by the Riemann zeta function. For functions of this latter form, representations as linear combinations of Epstein zeta functions were worked out by Dedekind in 1879. For and 12, such representations are used to show that all of the zeroes of these *L*-functions with and are simple and lie on the critical line . These methods currently cannot be used to compute values of *L*-functions with much larger than 15, but approaches to overcome these deficiencies are discussed in the final section.

**[1]***Handbook of mathematical functions, with formulas, graphs, and mathematical tables*, Edited by Milton Abramowitz and Irene A. Stegun. Third printing, with corrections. National Bureau of Standards Applied Mathematics Series, vol. 55, Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1965. MR**0177136****[2]**J. V. ARMITAGE, "Zeta functions with a zero at ,"*Invent. Math.*, v. 15, 1972, pp. 199-205.**[3]**E. ARTIN, "Über eine neue Art von*L*-Reihen,"*Abh. Math. Sem. Univ. Hamburg*, v. 3, 1923, pp. 89-108;*Collected Papers*, pp. 105-124.**[4]**E. ARTIN, "Zur Theorie der*L*-Reihen mit allgemeinen Gruppencharakteren,"*Abh. Math. Sem. Univ. Hamburg*, v. 8, 1930, pp. 292-306;*Collected Papers*, pp. 165-179.**[5]**P. T. Bateman and E. Grosswald,*On Epstein’s zeta function*, Acta Arith.**9**(1964), 365–373. MR**179141**, https://doi.org/10.4064/aa-9-4-365-373**[6]**Gottfried Bruckner,*Charakterisierung der galoisschen Zahlkörper, deren zerlegte Primzahlen durch binäre quadratische Formen gegeben sind*, Math. Nachr.**32**(1966), 317–326 (German). MR**217043**, https://doi.org/10.1002/mana.19660320604**[7]**S. Chowla,*\cal𝐿-series and elliptic curves*, Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976), Springer, Berlin, 1977, pp. 1–42. Lecture Notes in Math., Vol. 626. MR**0466075****[8]**Atle Selberg and S. Chowla,*On Epstein’s zeta-function*, J. Reine Angew. Math.**227**(1967), 86–110. MR**215797**, https://doi.org/10.1515/crll.1967.227.86**[9]**Harvey Cohn,*A numerical study of Dedekind’s cubic class number formula*, J. Res. Nat. Bur. Standards**59**(1957), 265–271. MR**0091308**, https://doi.org/10.6028/jres.059.031**[10]**Charles W. Curtis and Irving Reiner,*Representation theory of finite groups and associative algebras*, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR**0144979****[11]**H. Davenport and H. Heilbronn,*On indefinite quadratic forms in five variables*, J. London Math. Soc.**21**(1946), 185–193. MR**20578**, https://doi.org/10.1112/jlms/s1-21.3.185**[12]**D. Davies,*The computation of the zeros of Hecke zeta functions in the Gaussian field*, Proc. Roy. Soc. London Ser. A**264**(1961), 496–502. MR**132731**, https://doi.org/10.1098/rspa.1961.0213**[13]**D. Davies and C. B. Haselgrove,*The evaluation of Dirichlet 𝐿-functions*, Proc. Roy. Soc. London Ser. A**264**(1961), 122–132. MR**136052**, https://doi.org/10.1098/rspa.1961.0187**[14]**R. DEDEKIND, "Über die Anzahl der Idealklassen in reinen kubischen Zahlkörpern,"*J. Reine Angew. Math.*, v. 121, 1900, pp. 40-123.**[15]**A. R. Di DONATO & R. K. HAGEMAN,*Computation of the Incomplete Gamma Function*, Report TR-3492, Naval Surface Weapons Center, April 1976.**[16]**H. M. EDWARDS,*Riemann's Zeta Function*, Academic Press, New York, 1974.**[17]**H. Heilbronn,*Zeta-functions and 𝐿-functions*, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 204–230. MR**0218327****[18]**R. Sherman Lehman,*On the distribution of zeros of the Riemann zeta-function*, Proc. London Math. Soc. (3)**20**(1970), 303–320. MR**0258768**, https://doi.org/10.1112/plms/s3-20.2.303**[19]**J. Martinet,*Character theory and Artin 𝐿-functions*, Algebraic number fields: 𝐿-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 1–87. MR**0447187****[20]**H. S. A. POTTER & E. C. TITCHMARSH, "The zeroes of Epstein's zeta functions,"*Proc. London Math. Soc.*, v. 39, 1935, pp. 372-384.**[21]**G. PURDY, A. TERRAS, R. TERRAS & H. C. WILLIAMS, "Graphing*L*-functions of Kronecker symbols in the real part of the critical strip." (Preprint.)**[22]**J. Barkley Rosser, J. M. Yohe, and Lowell Schoenfeld,*Rigorous computation and the zeros of the Riemann zeta-function. (With discussion)*, Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) North-Holland, Amsterdam, 1969, pp. 70–76. MR**0258245****[23]**Daniel Shanks,*Calculation and applications of Epstein zeta functions*, Math. Comp.**29**(1975), 271–287. MR**409357**, https://doi.org/10.1090/S0025-5718-1975-0409357-2**[24]**Daniel Shanks,*Class number, a theory of factorization, and genera*, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 415–440. MR**0316385****[25]**Daniel Shanks,*Five number-theoretic algorithms*, Proceedings of the Second Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972) Utilitas Math., Winnipeg, Man., 1973, pp. 51–70. Congressus Numerantium, No. VII. MR**0371855****[26]**Daniel Shanks,*Table errata: “Über die Lösung einiger unbestimmten Gleichungen vierten Grades” (Avh. Norske Vid. Akad. Oslo. I. 1934, no. 14. 1–35) by W. Ljunggren*, Math. Comp.**31**(1977), no. 140, 1049–1050. MR**441858**, https://doi.org/10.1090/S0025-5718-1977-0441858-5**[27]**D. SHANKS, "Dedekind zeta functions that have sums of Epstein zeta functions as factors." (Preprint.)**[28]**Robert Spira,*Calculation of Dirichlet 𝐿-functions*, Math. Comp.**23**(1969), 489–497. MR**247742**, https://doi.org/10.1090/S0025-5718-1969-0247742-X**[29]**H. M. Stark,*On the zeros of Epstein’s zeta function*, Mathematika**14**(1967), 47–55. MR**215798**, https://doi.org/10.1112/S0025579300008007**[30]**H. M. Stark,*Values of 𝐿-functions at 𝑠=1. I. 𝐿-functions for quadratic forms*, Advances in Math.**7**(1971), 301–343 (1971). MR**289429**, https://doi.org/10.1016/S0001-8708(71)80009-9**[31]**Audrey A. Terras,*Bessel series expansions of the Epstein zeta function and the functional equation*, Trans. Amer. Math. Soc.**183**(1973), 477–486. MR**323735**, https://doi.org/10.1090/S0002-9947-1973-0323735-6**[32]**A. TERRAS, "On a relation between the size of the minima of quadratic forms and the behavior of Epstein and Dedekind zeta functions in the real part of the critical strip,"*J. Number Theory*. (To appear.)**[33]**R. TERRAS, "Determination of incomplete gamma functions by analytic integration." (Preprint.)**[34]**A. I. Vinogradov,*Artin’s 𝐿-series and the adele group*, Trudy Mat. Inst. Steklov.**112**(1971), 105–122, 387 (Russian). Collection of articles dedicated to Academician Ivan Matveevič Vinogradov on his eightieth birthday, I. MR**0342494****[35]**Peter J. Weinberger,*On small zeros of Dirichlet 𝐿-functions*, Math. Comp.**29**(1975), 319–328. MR**376564**, https://doi.org/10.1090/S0025-5718-1975-0376564-7**[36]**Lenard Weinstein,*The zeros of the Artin 𝐿-series of a cubic field on the critical line*, J. Number Theory**11**(1979), no. 2, 279–284. MR**535398**, https://doi.org/10.1016/0022-314X(79)90046-5

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DOI:
https://doi.org/10.1090/S0025-5718-1979-0528062-9

Article copyright:
© Copyright 1979
American Mathematical Society