## A Finite Reflection Formula For A Polynomial Approximation To The Riemann Zeta Function

**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.

**Download:** **PDF**

### Submission history

[v1] 2012-05-18 16:52:12

[v2] 2012-06-06 15:10:00

[v3] 2012-06-14 18:12:25

[v4] 2012-06-23 13:52:48

**Unique-IP document downloads:** 155 times

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*