Mostly the relationship was developed using the calculus of residues. Of course the art of using the calculus of residues is to find the appropriate contour to perform the integral around.

https://www.math.columbia.edu/~ums/pdf/abelplana.pdf

https://arxiv.org/pdf/hep-th/0002239.pdf

Truman Prevatt Thanks a lot for these two links. I am interested in to determine a structure that emerged from the development of the integral for the Riemann Zeta function, then, it is a curiosity I have regarding that.

Very useful to have this info.

regards,

Carlos

Carlos, There is a wealth of information about the Riemann Zeta function. It started out as the Dirichlet series which converged and was holomorphic for abs(z)>1. However, through the process of analytic continuation the Zeta function could be defined as a meromorphic function on the entire complex plane, holomorphic on C-{1} with a simple pole at z=1.

http://www.nhn.ou.edu/~milton/p5013/zeta.pdf

Research Analytic continuity of the Riemann Zeta function

Often times when one extends a convergent series representation to homomorphically to a larger region which is done to generate the Zeta function one finds the function is no longer single valued.

For example when considers the simple sqrt(z), then runs into the problem that it is multivalued and hence cannot be extended homomorphically to C. This gave rise to the concept of a Riemann surface, covering spaces, branch points and branch cut so to address this issue which can often arise in analytic continuation. http://www1.spms.ntu.edu.sg/~ydchong/teaching/07_branch_cuts.pdf

The Zeta function when extended to the p-adic number field, however is not single valued and has branch points.

There are many books at all levels on the zeta function. It is one of the most important special functions of mathematics and foundational to analytic number theory. Just go an Amazon and search on Riemann Zeta function in Books and see what pops up.

Truman Prevatt thanks a lot, there is a lot of literature I need to read about.

The expansion of the Hadamard product of Xi(s) and its Taylor series of Xi(s) around 1/2 has let me obtain at least two valuable expressions that related the coefficients a_2n of that Taylor series with the real par of there non trivial zeros.

Abel Plans is fascinating because let’s evaluate any s in that expression letting find a value of Zeta(s), especially in cases where s is simple like s=0 . The formula contains the attractive 1/2 in one part of the definition when it is properly.. I want to revise all the details documents you have provided me. Thanks and I will be asking later about curious results.

Have a nice Sunday