Exact & Numerical Tests of Generalised Root Identities for non-integer \mu
Richard Stone · arXiv (Cornell University) · 2011
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 -10 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.