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

Authors:	Rick Kreminski

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

Abstract:	In the Laurent expansion $\begin{displaymath}\zeta (s,a)=\frac{1}{s-1}+\sum _{k=0}^{\infty }\frac{(-1)^{k} \gamma _{k}(a)}{k!} (s-1)^{k}, \text{ } 0<a\leq 1,\end{displaymath}$ of the Riemann-Hurwitz zeta function, the coefficients $\gamma _{k}(a)$ are known as Stieltjes, or generalized Euler, constants. When , $\zeta (s,1)=\zeta (s)$ (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 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 $g(k)=\sup _{0<a\leq 1}\vert\gamma _{k}(a)-\frac{\log ^{k} a}{a}\vert$ , we conjecture that is attained: for any given , $g(k)= \vert\gamma _{k}(a)-\frac{\log ^{k} a}{a}\vert$ for some (and similarly that, given $\epsilon$ and , is within $\epsilon$ of $\vert\gamma _{k}(a)-\frac{\log ^{k} a}{a}\vert$ for infinitely many ). In addition we conjecture that satisfies $\log \big (g(k)\big )]/k <\log(\log(k))$ for . 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 , we conjecture that for nonintegral , $\gamma _{r}$ is precisely $(-1)^{r}$ times the -th (Weyl) fractional derivative at of the entire function $\zeta (s)-1/(s-1)-1$ . 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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