Conjecture about Dirichlet series.

[mhill]

Hi, i hope it is not a crack theory

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:

$$\sum_{n >1}a_{n}n^{-s}- C(s-a)^{b}$$

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

$$\sum_{n >1}n^{-s}- (s-1)^{-1}$$

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

$$\sum_{n >1}a_{n}n^{-s}-\int_{1}^{\infty}a(x)x^{-s}$$

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

 

[yasiru89]

I think you're saying that the tail integral which is characteristic of Ramanujan resummation can be put in the form c(s-a)^b always for a, b, c constants. However can it be shown that the integral will evaluate in the given form always?