The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and of the second author for Euler's constant γ and its alternating analog ln(4/π), and on the other hand the infinite products of the first author for e, of the second author for π, and of Ser for e γ . We obtain new double integral and infinite product representations of many classical constants, as well as a generalization to Lerch's transcendent of Hadjicostas's double integral formula for the Riemann zeta function, and logarithmic series for the digamma and Euler beta functions. The main tools are analytic continuations of Lerch's function, including Hasse's series. We also use Ramanujan's polylogarithm formula for the sum of a particular series involving harmonic numbers, and his relations between certain dilogarithm values.

Abstract.We prove that a series derived using Euler's transformation provides the analytic continuation of ((s) for all complex s ^ 1 . At negative integers the series becomes a finite sum whose value is given by an explicit formula for Bernoulli numbers.

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Motivated by our previous work on hypergeometric functions and the parbelos constant, we perform a deeper investigation on the interplay among generalized complete elliptic integrals, Fourier-Legendre (FL) series expansions, and p F q series. We produce new hypergeometric transformations and closed-form evaluations for new series involving harmonic numbers, through the use of the integration method outlined as follows: Letting K denote the complete elliptic integral of the first kind, for a suitable function g we evaluate integrals such as 1 0 * Corresponding author

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Abstract. By modifying Beukers' proof of Apéry's theorem that ζ(3) is irrational, we derive criteria for irrationality of Euler's constant, γ. For n > 0, we define a double integral In and a positive integer Sn, and prove that with dn = LCM(1, . . . , n) the following are equivalent:1. The fractional part of log Sn is given by {log Sn} = d 2n In for some n. 2. The formula holds for all sufficiently large n. 3. Euler's constant is a rational number. A corollary is that if {log Sn} ≥ 2 −n infinitely often, then γ is irrational. Indeed, if the inequality holds for a given n (we present numerical evidence for 1 ≤ n ≤ 2500) and γ is rational, then its denominator does not divide d 2n 2n n . We prove a new combinatorial identity in order to show that a certain linear form in logarithms is in fact log Sn. A by-product is a rapidly converging asymptotic formula for γ, used by P. Sebah to compute γ correct to 18063 decimals.

In a two-page paper [8; 9, pp. 208-209] published one year before his death in 1920 at the age of 32, the Indian mathematical genius Srinivasa Ramanujan wrote: Landau in his Handbuch [5], pp. 89-92, gives a proof of a theorem the truth of which was conjectured by Bertrand: namely that there is at least one prime p such that x < p ≤ 2x , if x ≥ 1. Landau's proof is substantially the same as that given by Tschebyschef. The following is a much simpler one.Ramanujan then ``uses simple properties of the Γ-function'' (P. Ribenboim [10, p. 188]) to prove the theorem, which is known as Bertrand's postulate or Tschebyschef's theorem.(For an exposition of Ramanujan's proof, see Shapiro [14, Section 9.3B].) 2. RAMANUJAN PRIMES. In [6, p. 178] W. J. LeVeque explains that the theorem is called Bertrand's ``postulate'' rather than ``conjecture'' because he took it as a working tool in his study of a problem in group theory. This must have seemed entirely safe, considering the actual density of primes in the tables. There is not merely one prime between 500,000 and l,000,000, say, there are 36,960 of them!