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

Kreminski, Rick

doi:10.1090/S0025-5718-02-01483-7

Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants
HTML articles powered by AMS MathViewer

by Rick Kreminski PDF

Math. Comp. 72 (2003), 1379-1397 Request permission

Abstract:

In the Laurent expansion \[ \zeta (s,a)=\frac {1}{s-1}+\sum _{k=0}^{\infty }\frac {(-1)^{k} \gamma _{k}(a)}{k!} (s-1)^{k}, \text {\ \ } 0<a\leq 1,\] of the Riemann-Hurwitz zeta function, the coefficients $\gamma _{k}(a)$ are known as Stieltjes, or generalized Euler, constants. [When $a=1$, $\zeta (s,1)=\zeta (s)$ (the Riemann zeta function), and $\gamma _{k}(1)=\gamma _{k}$.] We present a new approach to high-precision approximation of $\gamma _{k}(a)$. Plots of our results reveal much structure in the growth of the generalized Euler constants. Our results when $1\leq k\leq 3200$ for $\gamma _{k}$, and when $1\leq k\leq 600$ for $\gamma _{k}(a)$ (for $a$ 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 $g(k)=\sup _{0<a\leq 1}|\gamma _{k}(a)-\frac {\log ^{k} a}{a}|$, we conjecture that $g$ is attained: for any given $k$, $g(k)= |\gamma _{k}(a)-\frac {\log ^{k} a}{a}|$ for some $a$ (and similarly that, given $\epsilon$ and $a$, $g(k)$ is within $\epsilon$ of $|\gamma _{k}(a)-\frac {\log ^{k} a}{a}|$ for infinitely many $k$). In addition we conjecture that $g$ satisfies $[\log \big (g(k)\big )]/k <\log (\log (k))$ for $k>1$. We also conjecture that $\lim _{k\rightarrow \infty }\big ( \gamma _{k}(1/2)+\gamma _{k}\big )/\gamma _{k} = 0$, a special case of a more general conjecture relating the values of $\gamma _{k}(a)$ and $\gamma _{k}(a+\frac {1}{2})$ for $0<a\leq \frac {1}{2}$. Finally, it is known that $\gamma _{k} = \lim _{n\rightarrow \infty }\{\sum _{j=2}^{n} \frac {\log ^{k} j}{j}- \frac {\log ^{k+1} n}{k+1}\}$ for $k=1,2,\dots$. Using this to define $\gamma _{r}$ for all real $r>0$, we conjecture that for nonintegral $r$, $\gamma _{r}$ is precisely $(-1)^{r}$ times the $r$-th (Weyl) fractional derivative at $s=1$ of the entire function $\zeta (s)-1/(s-1)-1$. We also conjecture that $g$, now defined for all real arguments $r>0$, 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.

References

Bruce C. Berndt, On the Hurwitz zeta-function, Rocky Mountain J. Math. 2 (1972), no. 1, 151–157. MR 289431, DOI 10.1216/RMJ-1972-2-1-151
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
Bejoy K. Choudhury, The Riemann zeta-function and its derivatives, Proc. Roy. Soc. London Ser. A 450 (1995), no. 1940, 477–499. MR 1356175, DOI 10.1098/rspa.1995.0096
Lokenath Debnath, Integral transforms and their applications, CRC Press, Boca Raton, FL, 1995. MR 1396084
Karl Dilcher, Generalized Euler constants for arithmetical progressions, Math. Comp. 59 (1992), no. 199, 259–282, S21–S24. MR 1134726, DOI 10.1090/S0025-5718-1992-1134726-5
J. B. Keiper, Power series expansions of Riemann’s $\xi$ function, Math. Comp. 58 (1992), no. 198, 765–773. MR 1122072, DOI 10.1090/S0025-5718-1992-1122072-5
Rick Kreminski, Using Simpson’s rule to approximate sums of series, College Mathematics Journal 28 (1997), 368–376.
Rick Kreminski and James Graham-Eagle, Simpson’s rule for estimating n! (and proving Stirling’s formula, almost), preprint (1998).
Vladimir Ivanovich Krylov, Approximate calculation of integrals, The Macmillan Company, New York-London, 1962, 1962. Translated by Arthur H. Stroud. MR 0144464
J.-L. Lavoie, T. J. Osler, and R. Tremblay, Fractional derivatives and special functions, SIAM Rev. 18 (1976), no. 2, 240–268. MR 409914, DOI 10.1137/1018042
Dragiša Mitrović, The signs of some constants associated with the Riemann zeta-function, Michigan Math. J. 9 (1962), 395–397. MR 164941
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
T. J. Stieltjes, Correspondance d’Hermite et de Stieltjes, volumes 1 and 2, Gauthier-Villars, Paris, 1905.
Nan Yue Zhang and Kenneth S. Williams, Some results on the generalized Stieltjes constants, Analysis 14 (1994), no. 2-3, 147–162. MR 1302533