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.
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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
