Abstract
The Stieltjes constants \(\gamma _k(a)\) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function \(\zeta (s,a)\) about \(s=1\). We present the evaluation of \(\gamma _1(a)\) and \(\gamma _2(a)\) at rational arguments, this being of interest to theoretical and computational analytic number theory and elsewhere. We give multiplication formulas for \(\gamma _0(a)\), \(\gamma _1(a)\), and \(\gamma _2(a)\), and point out that these formulas are cases of an addition formula previously presented. We present certain integral evaluations generalizing Gauss’ formula for the digamma function at rational argument. In addition, we give the asymptotic form of \(\gamma _k(a)\) as \(a \rightarrow 0\) as well as a novel technique for evaluating integrals with integrands with \(\ln (-\ln x)\) and rational factors.
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Notes
Details of the evaluation of the integrals \(I_\pm \) by this method are separately available from the author.
References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. National Bureau of Standards, Washington (1964)
Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976). (corrected fourth printing, 1995)
Berndt, B.C.: On the Hurwitz zeta function. Rocky Mt. J. Math. 2, 151–157 (1972)
Blagouchine, I.V.: A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments. J. Number Theory 148, 537–592 (2015). arXiv:1401.3724v1
Briggs, W.E.: Some constants associated with the Riemann zeta-function. Mich. Math. J. 3, 117–121 (1955)
Coffey, M.W.: New results on the Stieltjes constants: asymptotic and exact evaluation. J. Math. Anal. Appl. 317, 603–612 (2006). arXiv:math-ph/0506061
Coffey, M.W.: The Stieltjes constants, their relation to the \(\eta _j\) coefficients, and representation of the Hurwitz zeta function. Analysis 30, 383 (2010). arXiv:0706.0343v2 [math-ph]
Coffey, M.W.: Integral representations of functions and Addison-type series for mathematical constants. J. Number Theory (2010, to appear). arXiv:1006.2551
Coffey, M.W.: On representations and differences of Stieltjes coefficients, and other relations. Rocky Mt. J. Math. 41, 1815–1846 (2011). arXiv:0809.3277v2 [math-ph]
Coffey, M.W.: Certain logarithmic integrals, including solution of Monthly problem 11629, zeta values, and expressions for the Stieltjes constants constants (2012). arXiv:1201.3393
Coffey, M.W.: Hypergeometric summation representations of the Stieltjes constants. Analysis 33, 121–142 (2013). arXiv:1106.5148
Coffey, M.W.: Series representations for the Stieltjes constants. Rocky Mt. J. Math. 44, 443–477 (2014). arXiv:0905.1111v1 [math-ph]
Edwards, H.M.: Riemann’s Zeta Function. Academic Press, New York (1974)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, G.G.: Higher Transcendental Functions, vol. I. McGraw-Hill, New York (1953)
Fine, N.J.: Note on the Hurwitz zeta-function. Proc. Am. Math. Soc. 2, 361–364 (1951)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, New York (1980)
Hansen, E.R., Patrick, M.L.: Some relations and values for the generalized Riemann zeta function. Math. Comput. 16, 265–274 (1962)
Hardy, G.H.: Note on Dr. Vacca’s series for \(\gamma \), Quart. J. Pure Appl. Math. 43, 215–216 (1912)
Hurwitz, A.: Einige Eigenschaften der Dirichlet’schen Functionen \(F(s)=\sum \left({D \over n}\right) {1 \over n^s}\), die bei der Bestimmung der Classenanzahlen binärer quadratischer Formen auftreten. Z. Math. Phys. XXXVII, 86–101 (1882)
Ivić, A.: The Riemann Zeta-Function. Wiley, New York (1985)
Karatsuba, A.A., Voronin, S.M.: The Riemann Zeta-Function. Walter de Gruyter, New York (1992)
Kluyver, J.C.: On certain series of Mr. Hardy. Quart. J. Pure Appl. Math. 50, 185–192 (1927)
Knessl, C., Coffey, M.W.: An effective asymptotic formula for the Stieltjes constants. Math. Comput. 80, 379–386 (2011)
Knessl, C., Coffey, M.W.: An asymptotic form for the Stieltjes constants \(\gamma _k(a)\) and for a sum \(S_\gamma (n)\) appearing under the Li criterion. Math. Comput. 80, 2197–2217 (2011)
Kreminski, R.: Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants. Math. Comput. 72, 1379–1397 (2003)
Mitrović, D.: The signs of some constants associated with the Riemann zeta function. Mich. Math. J. 9, 395–397 (1962)
Riemann, B.: Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monats. Preuss. Akad. Wiss. 671 (1859–1860)
Stieltjes, T.J.: Correspondance d’Hermite et de Stieltjes, vols. 1 and 2. Gauthier-Villars, Paris (1905)
Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Oxford University Press, Oxford (1986)
Truesdell, C.: On the addition and multiplication theorem of special functions. Proc. Natl. Acad. Sci. 36, 752–755 (1950)
Wilton, J.R.: A note on the coefficients in the expansion of \(\zeta (s, x)\) in powers of \(s-1\). Quart. J. Pure Appl. Math. 50, 329–332 (1927)
Zhang, N.-Y., Williams, K.S.: Some results on the generalized Stieltjes constants. Analysis 14, 147–162 (1994)
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Coffey, M.W. Functional equations for the Stieltjes constants. Ramanujan J 39, 577–601 (2016). https://doi.org/10.1007/s11139-015-9691-y
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DOI: https://doi.org/10.1007/s11139-015-9691-y
Keywords
- Stieltjes constants
- Riemann zeta function
- Hurwitz zeta function
- Laurent expansion
- Digamma function
- Polygamma function