Authors: Stephen Crowley
The Riemann zeta function can be written as the Mellin transform of the unit interval map w(x)=⌊x−1⌋(x⌊x−1⌋+x−1) multiplied by s((s+1)/(s-1)). A finite-sum approximation to ζ(s) denoted by ζw(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 ζw(N;1−s)=χ(N;s)ζw(N;s). A closed-form expression for the integral of ζw(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 ζw(N;s) and the reflection functions χ(N;s) are also provided. The values ζw(N;1−n) for integer values of n are found to be related to the Bernoulli numbers.
Comments: 9 Pages.
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