A renormalization approach to the Riemann zeta function at -1, 1+2+3+... ~ -1/12

2018/06/16 by Caginalp, Gunduz
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1806.06245

Abstract

A scaling and renormalization approach to the Riemann zeta function, ζ, evaluated at -1 is presented in two ways. In the first, one takes the difference between U_n:=∑_q=1ⁿq and 4U_{\lfloor (n)/(2)\rfloor} where \lfloor (n)/(2)\rfloor is the greatest integer function. Using the Cesaro mean twice, i.e., ( C,2) , yields convergence to the appropriate value. For values of z for which the zeta function is represented by a convergent infinite sum, the double Cesaro mean also yields ζ( z) , suggesting that this could be used as an alternative method for extension from the convergent region of z. In the second approach, the difference U_n-k²U_n/k between U_n and a particular average, U_n/k, involving terms up to k

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