#zeta function

Who says you can’t do complex analysis in Excel...

Prime Counting function using the first 200 non trivial zeroes of the Zeta function (and first 5 trivial ones).  Equation used is the main sum in the top with an approximation if you want a simpler expression to try.

Main point of the famous paper by Riemann is using the zeta zeroes you can calculate the number of primes below a number, they are one in the same, interwoven in math.

A computation of Casimir energy via spectral zeta function is considered in this Chapter. The original computations deriving the Casimir energy and force consists of first taking limits of the spectral zeta function and afterwards analytically extending the result. This process of computation presents a weakness in Hendrik Casimir’s original argument since limit and analytic continuation do not commute. A case of the Laplacian on a parallelepiped box representing the space as the vacuum between two plates modelled with Dirichlet and periodic Neumann boundary conditions is constructed to address this anomaly. It involves the derivation of the regularised zeta function in terms of the Riemann zeta function on the parallelepiped. The values of the Casimir energy and Casimir force obtained from our derivation agree with those of Hendrik Casimir.

Author(s) Details

Dr. Louis Omenyi Department of Mathematics/Computer Science/Statistics/Informatics, Alex Ekwueme Federal University, Ndufu-Alike, Nigeria.