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NOTES RELATING TO NEWTON SERIES FOR THE RIEMANN ZETA FUNCTION

This paper consists of the extended working notes and observations made during the development of a joint paper[?] with Philippe Flajolet on the Riemann zeta function. Most of the core ideas of that paper, of which a majority are due to Flajolet, are reproduced here; however, the choice of wording used here, and all errors and omissions are my own fault. This set of notes contains considerably more content, although is looser and sloppier, and is an exploration of tangents, dead-ends, and ideas shooting off in uncertain directions. The finite differences or Newton series of certain expressions involving the Riemann zeta function are explored. These series may be given an asymptotic expansion by converting them to Norlund-Rice integrals and applying saddle-point integration techniques. Numerical evaluation is used to confirm the appropriateness of the asymptotic expansion. The results extend on previous results for such series, and a general form for Dirichlet L-functions is given. Curiously, because successive terms in the asymptotic expansion are exponentially small, these series lead to simple near-identities. A resemblance to similar near-identities arising in complex multiplication is noted.