Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants

Author:
Rick Kreminski

Journal:
Math. Comp. **72** (2003), 1379-1397

MSC (2000):
Primary 11M06, 11M35, 11Y60; Secondary 65D32

Published electronically:
December 18, 2002

MathSciNet review:
1972742

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Abstract | References | Similar Articles | Additional Information

Abstract: In the Laurent expansion

of the Riemann-Hurwitz zeta function, the coefficients are known as Stieltjes, or generalized Euler, constants. [When , (the Riemann zeta function), and .] We present a new approach to high-precision 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 Newton-Cotes integration formulae for very high-degree interpolating polynomials; it differs in implementation from, but compares in error bounding to, Euler-Maclaurin summation based methods.

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Additional Information

**Rick Kreminski**

Affiliation:
Department of Mathematics, Texas A & M University-Commerce, Commerce, Texas 75429

Email:
kremin@boisdarc.tamu-commerce.edu

DOI:
https://doi.org/10.1090/S0025-5718-02-01483-7

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