A conjecture about Dirichlet series.

by tpm
Tags: conjecture, dirichlet, series
tpm is offline
Mar27-07, 03:20 PM
P: 76
if [tex] g(s)= \sum_{n=1}^{\infty} a(n) n^{-s} [/tex]

Where g(s) has a single pole at s=1 with residue C, then my question/conjecture is if for s >0 (real part of s bigger than 0) we can write

[tex] g(s)= C(\frac{1}{s-1}+1)-s\int_{0}^{\infty}dx(Cx-A(x))x^{-s-1} [/tex]

of course [tex] A(x)=\sum_{n \le x}a(n) [/tex]

the question is if the series converge for s >1 with a pole there is a method to 'substract' this singularity (pole) at s=1 to give meaning for the series at any positive s.

I think that the 'Ramanujan resummation' may help to give the result:

[tex] \sum_{ n >1}^{[R]}a(n)n^{-s} = g(s)-C(s-1)^{-1} [/tex] valid even for s=1 or s>0 (??)
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
DeadWolfe is offline
Mar27-07, 04:39 PM
P: 461
Jose, how's it been?

Register to reply

Related Discussions
conjecture about Dirichlet series. Linear & Abstract Algebra 1
Dirichlet series Linear & Abstract Algebra 5
Dirichlet series.... Linear & Abstract Algebra 1
Dirichlet series inversion.. Linear & Abstract Algebra 10
Dirichlet series inversion?... Linear & Abstract Algebra 0