NewtonCotes integration for approximating Stieltjes (generalized Euler) constants
Author:
Rick Kreminski
Journal:
Math. Comp. 72 (2003), 13791397
MSC (2000):
Primary 11M06, 11M35, 11Y60; Secondary 65D32
Published electronically:
December 18, 2002
MathSciNet review:
1972742
Fulltext PDF Free Access
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Abstract: In the Laurent expansion
of the RiemannHurwitz zeta function, the coefficients are known as Stieltjes, or generalized Euler, constants. [When , (the Riemann zeta function), and .] We present a new approach to highprecision approximation of . Plots of our results reveal much structure in the growth of the generalized Euler constants. Our results when for , and when for (for such as 53/100, 1/2, etc.) suggest that published bounds on the growth of the Stieltjes constants can be much improved, and lead to several conjectures. Defining , we conjecture that is attained: for any given , for some (and similarly that, given and , is within of for infinitely many ). In addition we conjecture that satisfies for . We also conjecture that , a special case of a more general conjecture relating the values of and for . Finally, it is known that for . Using this to define for all real , we conjecture that for nonintegral , is precisely times the th (Weyl) fractional derivative at of the entire function . We also conjecture that , now defined for all real arguments , is smooth. Our numerical method uses NewtonCotes integration formulae for very highdegree interpolating polynomials; it differs in implementation from, but compares in error bounding to, EulerMaclaurin summation based methods.
 [1]
Bruce
C. Berndt, On the Hurwitz zetafunction, Rocky Mountain J.
Math. 2 (1972), no. 1, 151–157. MR 0289431
(44 #6622)
 [2]
Jan
Bohman and CarlErik
Fröberg, The Stieltjes
function—definition and properties, Math.
Comp. 51 (1988), no. 183, 281–289. MR 942155
(89i:11095), http://dx.doi.org/10.1090/S00255718198809421559
 [3]
Bejoy
K. Choudhury, The Riemann zetafunction and its derivatives,
Proc. Roy. Soc. London Ser. A 450 (1995), no. 1940,
477–499. MR 1356175
(97e:11095), http://dx.doi.org/10.1098/rspa.1995.0096
 [4]
Lokenath
Debnath, Integral transforms and their applications, CRC
Press, Boca Raton, FL, 1995. MR 1396084
(97h:44002)
 [5]
Karl
Dilcher, Generalized Euler constants for
arithmetical progressions, Math. Comp.
59 (1992), no. 199, 259–282, S21–S24. MR 1134726
(92k:11145), http://dx.doi.org/10.1090/S00255718199211347265
 [6]
J.
B. Keiper, Power series expansions of
Riemann’s 𝜉 function, Math.
Comp. 58 (1992), no. 198, 765–773. MR 1122072
(92f:11116), http://dx.doi.org/10.1090/S00255718199211220725
 [7]
Rick Kreminski, Using Simpson's rule to approximate sums of series, College Mathematics Journal 28 (1997), 368376. CMP 98:03
 [8]
Rick Kreminski and James GrahamEagle, Simpson's rule for estimating n! (and proving Stirling's formula, almost), preprint (1998).
 [9]
Vladimir
Ivanovich Krylov, Approximate calculation of integrals,
Translated by Arthur H. Stroud, The Macmillan Co., New York, 1962. MR 0144464
(26 #2008)
 [10]
J.L.
Lavoie, T.
J. Osler, and R.
Tremblay, Fractional derivatives and special functions, SIAM
Rev. 18 (1976), no. 2, 240–268. MR 0409914
(53 #13666)
 [11]
Dragiša
Mitrović, The signs of some constants associated with the
Riemann zetafunction, Michigan Math. J. 9 (1962),
395–397. MR 0164941
(29 #2232)
 [12]
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
(87e:11105)
 [13]
T. J. Stieltjes, Correspondance d'Hermite et de Stieltjes, volumes 1 and 2, GauthierVillars, Paris, 1905.
 [14]
Nan
Yue Zhang and Kenneth
S. Williams, Some results on the generalized Stieltjes
constants, Analysis 14 (1994), no. 23,
147–162. MR 1302533
(95k:11110)
 [1]
 B. C. Berndt, On the Hurwitz zetafunction, Rocky Mountain J. Math. 2 (1972), 151157. MR 44:6622
 [2]
 Jan Bohman and CarlErik Froberg, The Stieltjes function  definition and properties, Mathematics of Computation 51 (1988), 281289. MR 89i:11095
 [3]
 Bejoy K. Choudhury, The Riemann zeta function and its derivatives, Proceedings of the Royal Society of London A 450 (1995), 477499. MR 97e:11095
 [4]
 Lokenath Debnath, Integral transforms and their applications, CRC Press, Boca Raton, 1995. MR 97h:44002
 [5]
 Karl Dilcher, Generalized Euler constants for arithmetical progressions, Mathematics of Computation 59 (1992), 259282. MR 92k:11145
 [6]
 J. B. Keiper, Power series expansions of Riemann's function, Mathematics of Computation 58 (1992), 765773. MR 92f:11116
 [7]
 Rick Kreminski, Using Simpson's rule to approximate sums of series, College Mathematics Journal 28 (1997), 368376. CMP 98:03
 [8]
 Rick Kreminski and James GrahamEagle, Simpson's rule for estimating n! (and proving Stirling's formula, almost), preprint (1998).
 [9]
 V. Krylov, Approximate calculation of integrals, Macmillan, New York, 1962. MR 26:2008
 [10]
 J. L. Lavoie, T. J. Osler, and R. Tremblay, Fractional derivatives and special functions, SIAM Review 18 (1976), 240268. MR 53:13666
 [11]
 D. Mitrovic, The signs of some constants associated with the Riemann zetafunction, Michigan Mathematics Journal 9 (1962), 395397. MR 29:2232
 [12]
 Y. Matsuoka, Generalized Euler constants associated with the Riemann zeta function, Number Theory and Combinatorics, World Scientific, Singapore, 1985, pp. 279295. MR 87e:11105
 [13]
 T. J. Stieltjes, Correspondance d'Hermite et de Stieltjes, volumes 1 and 2, GauthierVillars, Paris, 1905.
 [14]
 Zhang NanYue and K. Williams, Some results on the generalized Stieltjes constants, Analysis 14 (1994), 147162. MR 95k:11110
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Additional Information
Rick Kreminski
Affiliation:
Department of Mathematics, Texas A & M UniversityCommerce, Commerce, Texas 75429
Email:
kremin@boisdarc.tamucommerce.edu
DOI:
http://dx.doi.org/10.1090/S0025571802014837
PII:
S 00255718(02)014837
Received by editor(s):
April 8, 1999
Received by editor(s) in revised form:
January 14, 2000
Published electronically:
December 18, 2002
Article copyright:
© Copyright 2002 American Mathematical Society
