Exact & Numerical Tests of Generalised Root Identities for non-integer \mu

TL;DR

This paper investigates the validity of generalized root identities for functions like mma and , extending previous work to non-integer , and provides numerical evidence supporting their applicability.

Contribution

It extends the validation of generalized root identities to non-integer  for , , and , with new asymptotic formulas and numerical verifications.

Findings

Root identities hold for arbitrary real  for simple functions.

Numerical evidence shows  and  satisfy these identities for non-integer .

Ce9saro averaging confirms identities for <0.

Abstract

We consider the generalised root identities introduced in [1] for simple functions, and also for \Gamma(z+1) and \zeta(s). In this paper, unlike [1], we focus on the case of noninteger \mu. For the simplest function f(z)=z, and hence for arbitrary polynomials, we show that they are satisfied for arbitrary real {\mu} (and hence for arbitrary complex {\mu} by analytic continuation). Using this, we then develop an asymptotic formula for the derivative side of the root identities for \Gamma(z+1) at arbitrary real \mu, from which we are able to demonstrate numerically that \Gamma(z+1) also satisfies the generalised root identities for arbitrary \mu, not just integer values. Finally we examine the generalised root identites for {\zeta} also for non-integer values of \mu. Having shown in [1] that {\zeta} satisfies these identities exactly for integer \mu>1 (and also for \mu=1 after removal of an obstruction), in this paper we present strong numerical evidence first that {\zeta} satisfies them for arbitrary \mu>1 where the root side is classically convergent, and then that this continues to be true also for -1<\mu<1 where C\'esaro divergences must be removed and C\'esaro averaging of the residual partial-sum functions is required (when \mu<0). Careful consideration of a neighbourhood of \mu=0 also sheds light on the appearance of the 2d ln-divergence that was handled heuristically in [1] and why the assignment of 2d C\'esaro limit 0 to this in [1] is justified. The numerical calculations for \mu>0 are bundled in portable R-code; the code for the case -1<\mu<0, including the C\'esaro averaging required when \mu<0, is in VBA. Both the R-scripts and XL spreadsheet are made available with this paper, along with supporting files, and can be readily used to further verify these claims.