Incidentally, the possibility of analytically continuing F(x; s) to negative values of s leads to a paradox: On the one hand, continuing (*) for a fixed positive integer value of x would apparently lead to the sum of the positive |s|th powers of integers from 1 up through x. On the other hand, taking the limit of (*) as x -> oo, this analytic continuation would lead to the zeta function evaluated at negative s (for instance, negative odd integer values of s). These two possibilities are incompatible. --Dan
On Jan 8, 2015, at 1:04 PM, Daniel Asimov <asimov@msri.org> wrote:
A formula I found in 1964 for summing at least negative powers of the "integers" from 1 to x (not necessarily an integer) is:
(*) F(x; s) = (1/Gamma(s)) Integral{0 <= t <= 1} (1 - t^x) / (1 - t) (-ln(t))^(s-1) dt = Sum_{1 <= n <= x} 1/n^s
(Letting x -> oo gives an integral formula for the product of Gamma(s) and Zeta(s). When I mentioned this to my freshman advisor, Henry McKean, he showed me was already in an old book, A Course of Modern Analysis by Whittaker and Watson -- to my great dismay.)
I never studied how far F(x; s) can be analytically continued. Maybe it comes around and works for negative s -- giving a formulas for partial sums of positive powers of integers???