Zeta regularization of divergent integrals

[zetafunction]

From the model used in the zeta regularization procedure to give a meaning to divergent series in the form [tex] 1+2+3+4+... [/tex] , we propose a similar method to give a finite meaning to divergent integrals in the form [tex] \int_{0}^{\infty}dx x^{m} [/tex] for positive 'm' in terms of the negative values of the Riemann Zeta function [tex] \zeta (s) [/tex].

for the case m=-1 due to the pole of the Riemann Zeta at s=1 we can use the expression of the functional determinant [tex] detA=exp(- \zeta '(0,a)) [/tex] where the comma means differentiation with respect to variable 'a' , in terms of the Hurwitz zeta

[tex] \prod (n+a) =exp(- \partial _{a} \zeta_{H}(0,a)) [/tex] this product must be understood in the sense of Zeta regularization

we give some examples for the one dimensional case and study how this method can be applied for multi-loop integral or multiple integrals by changing to polar coordinates and making the integral over the angular variables [tex] \int d\Omega [/tex] to be replaced by a discrete sum, so in the end we have one dimensional integrals in the form

[tex] \int_{0}^{\infty}dr r^{m-1}f(r) [/tex] using Laurent series expansion we can isolate the UV behavior and the regularize these integrals in terms of the negative values of the Riemann zeta, for the case of finite (convergent) integrals , the well-known result

[tex] \int_{0}^{n}dx x^{m}= \frac{n^{m+1}}{m+1} [/tex] is inmediatly obtained

FULL PAPER: avaliable at http://vixra.org/abs/1009.0047

 

[AndyW]

Thank you for sharing your proposal for a method to give a finite meaning to divergent integrals. Your approach, using the negative values of the Riemann Zeta function, is intriguing and it seems to have potential for application in multi-loop integrals and multiple integrals.

I have read through your full paper and I have a few questions and comments. Firstly, could you please explain in more detail how the zeta regularization procedure is used to give a meaning to divergent series? I am familiar with the zeta function and its use in analytic continuation, but I am not quite sure how it can be applied to divergent series.

Secondly, in your proposal, you mention using a Laurent series expansion to isolate the UV behavior and regularize the integrals. Could you provide more information on how this is done and how it relates to your method using the negative values of the Riemann Zeta function?

Lastly, have you tested your method on any specific examples or problems? It would be helpful to see some concrete applications of your approach to better understand its effectiveness.

Overall, your proposal is interesting and I look forward to seeing further developments and applications of your method. Thank you for sharing your ideas with the scientific community.