ζ-function and heat kernel formulae

On Prime-Detecting Sequences From Apéry’s Recurrence Formulae for ζ(3) and ζ(2)

We consider the linear three-term recurrence formula Xn = (34(n− 1) + 51(n− 1) + 27(n− 1) + 5)Xn−1 − (n− 1)Xn−2 (n ≥ 2) corresponding to Apéry’s non-regular continued fraction for ζ(3). It is shown that integer sequences (Xn)n≥0 with 5X0 6= X1 satisfying the above relation are primedetecting, i.e., Xn 6≡ 0 (modn) if and only if n is a prime not dividing |5X0 − X1|. Similar results are given for...

Borwein and Bradley's Apérv-Like Formulae for ζ(4n + 3)

EULER’S FORMULAE FOR ζ(2n) AND PRODUCTS OF CAUCHY VARIABLES

PAUL BOURGADE Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 6 email: [email protected] TAKAHIKO FUJITA Graduate School of Commerce and management, Hitotsubashi University email: [email protected] MARC YOR Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 6 email: [email protected] Submitted 14 February 2007, accepted in final f...

Empirically Determined Apéry-Like Formulae for ζ(4n+3)

Research supported by NSERC, the Natural Sciences and Engineering Research Council of Canada. Some rapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for (4n+3) that generalizes Apéry’s series for (3), and appears to give the best possible series relations of this type, at least for n< 12. The formula reduces to a finit...

Riemann ’ s ζ - Function

We provide an overview of the Riemann ζ-function with an eye to a proof of the functional equation that is non-standard and more accessible than those proofs readily available in modern texts. This is s survey note with the goal of a more elegant and transparent presentation of the salient features surrounding the functional equation for the ζ-function and the intimate connections with the Riem...