A Finite Reflection Formula for a Polynomial Approximation to the Riemann Zeta Function

Stephen Crowley


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 \zeta (s) denoted by \zeta_w(N;s) which has real roots at s=-1 and s=0 is examined and an associated function \chi (N ; s) is found which solves the reflection formula \zeta_w (N ; 1 - s) = \chi (N ; s) \zeta_w (N ; s). A closed-form expression for the integral of \zeta_w (N ; s) over the interval s=-1..0 is given. The function \chi (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 \zeta_w(N ; s) and the reflection functions \chi (N ; s) are also provided. The values \zeta_w (N ; 1 - n) for integer values of n are found to be related to the Bernoulli numbers.

Full Text:



G Arfken. Mathematical Methods for Physicists, 3rd ed., chapter 5.9, Bernoulli Numbers, Euler-Maclaurin For-

mula., pages 327{338. Academic Press, 1985.

Stephen Crowley. Integral Transforms of the Harmonic Sawtooth Map, The Riemann Zeta Function, Fractal

Strings, and a Finite Re

ection Formula http://arxiv.org/abs/1210.5652, October 2012.

Stephen Crowley. Two new zeta constants: Fractal string, continued fraction, and hypergeometric aspects of the

riemann zeta function. http://arxiv.org/abs/1207.1126, July 2012.

G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins, 3 edition, 1996.


  • There are currently no refbacks.

©2017 Science Asia