Generalized Euler constants for arithmetical progressions
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- by Karl Dilcher PDF
- Math. Comp. 59 (1992), 259-282 Request permission
Abstract:
The work of Lehmer and Briggs on Euler constants in arithmetical progressions is extended to the generalized Euler constants that arise in the Laurent expansion of $\zeta (s)$ about $s = 1$. The results are applied to the summation of several classes of slowly converging series. A table of the constants is provided.References
-
M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover, New York, 1965.
- Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
- Bruce C. Berndt, Ramanujan’s notebooks. Part I, Springer-Verlag, New York, 1985. With a foreword by S. Chandrasekhar. MR 781125, DOI 10.1007/978-1-4612-1088-7
- Jan Bohman and Carl-Erik Fröberg, The Stieltjes function—definition and properties, Math. Comp. 51 (1988), no. 183, 281–289. MR 942155, DOI 10.1090/S0025-5718-1988-0942155-9
- A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
- W. E. Briggs, The irrationality of $\gamma$ or of sets of similar constants, Norske Vid. Selsk. Forh. (Trondheim) 34 (1961), 25–28. MR 139579
- Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128
- Christopher Deninger, On the analogue of the formula of Chowla and Selberg for real quadratic fields, J. Reine Angew. Math. 351 (1984), 171–191. MR 749681, DOI 10.1515/crll.1984.351.171 K. Dilcher, On a generalized gamma function related to the Laurent coefficients of the Riemann zeta function (in preparation). E. R. Hansen, A table of series and products, Prentice-Hall, Englewood Cliffs, NJ, 1975.
- M. I. Israilov, The Laurent expansion of the Riemann zeta function, Trudy Mat. Inst. Steklov. 158 (1981), 98–104, 229 (Russian). Analytic number theory, mathematical analysis and their applications. MR 662837 E. Jacobsthal, Über die Eulersche Konstante, K. Norske Vid. Selsk. Skrifter (Trondheim), 1967. J. C. Kluyver, On a certain series of Mr. Hardy, Quart. J. Pure Appl. Math. 50 (1927), 185-192.
- J. Knopfmacher, Generalised Euler constants, Proc. Edinburgh Math. Soc. (2) 21 (1978/79), no. 1, 25–32. MR 472742, DOI 10.1017/S0013091500015844
- D. H. Lehmer, Euler constants for arithmetical progressions, Acta Arith. 27 (1975), 125–142. MR 369233, DOI 10.4064/aa-27-1-125-142
- J. J. Y. Liang and John Todd, The Stieltjes constants, J. Res. Nat. Bur. Standards Sect. B 76B (1972), 161–178. MR 326974
- Y. Matsuoka, Generalized Euler constants associated with the Riemann zeta function, Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) World Sci. Publishing, Singapore, 1985, pp. 279–295. MR 827790
- Yasushi Matsuoka, On the power series coefficients of the Riemann zeta function, Tokyo J. Math. 12 (1989), no. 1, 49–58. MR 1001731, DOI 10.3836/tjm/1270133547
- È. P. Stankus, A remark on the coefficients of Laurent series of the Riemann zeta function, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 121 (1983), 103–107 (Russian). Studies in number theory, 8. MR 711377
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Math. Comp. 59 (1992), 259-282
- MSC: Primary 11Y60; Secondary 11M20, 65B10, 65B15
- DOI: https://doi.org/10.1090/S0025-5718-1992-1134726-5
- MathSciNet review: 1134726