A Finite Reflection Formula For A Polynomial Approximation To The Riemann Zeta Function
Crowley, Stephen · arXiv (Cornell University) · 2012
The Riemann zeta function can be written as the Mellin transform of the unit interval map w(x) = floor(1/x)*(-1+x*floor(1/x)+x) multiplied by s((s+1)/(s-1)). A finite-sum approximation to ζ(s) denoted by ζ_w(N;s) which has real roots at s=-1 and s=0 is examined and an associated function χ(N ; s) is found which solves the reflection formula ζ_w (N ; 1 - s) = χ(N ; s) ζ_w (N ; s). A closed-form expression for the integral of ζ_w (N ; s) over the interval s=-1..0 is given. The function χ(N ; s) is singular at s=0 and the residue at this point changes sign from negative to positive between the values of N=176 and N=177. Some rather elegant graphs of ζ_w(N ; s) and the reflection functions χ(N ; s) are also provided. The values ζ_w (N ; 1 - n) for integer values of n are found to be related to the Bernoulli numbers.