- #1

mhill

- 189

- 1

i had the idea when reading Ramanujan resummation, i believe that for a Dirichlet series which converges for Re (s) > a , with a a positive real number then we can obtain a regularized sum of the divergent series in the form:

[tex] \sum_{n >1}a_{n}n^{-s}- C(s-a)^{b} [/tex]

with C, a and b real numbers , in case a_n are all 1 we recover the result

[tex] \sum_{n >1}n^{-s}- (s-1)^{-1} [/tex]

however i believe that 'Ramanujan resummation of series' given by

[tex] \sum_{n >1}a_{n}n^{-s}-\int_{1}^{\infty}a(x)x^{-s} [/tex]

must be satisfied no matter what kind of numerical series is.