Jonathan Sondow

DOUBLE INTEGRALS AND INFINITE PRODUCTS FOR SOME CLASSICAL CONSTANTS VIA ANALYTIC CONTINUATIONS OF LERCH'S TRANSCENDENT

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 eγ. 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.

Analytic continuation of Riemann’s zeta function and values at negative integers via Euler’s transformation of series

We prove that a series derived using Eulers transformation provides the analytic continuation of ζ(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

A Faster Product for and a New Integral for ln.

1. N. I. Akhiezer, Theory of Approximation (trans. C. J. Hyman), Frederick Ungar, New York, 1956. 2. S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul de probabilité, Proc. Math. Soc. Kharkov 13 (1912/13) 1–2. 3. M. H. Stone, The generalized Weierstrass approximation theorem, Math. Mag. 21 (1948) 167–184, 237– 254. 4. K. Weierstrass, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Berliner Berichte (1885) 633–639, 789–805.

Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper

The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values of the q-logarithm function, with natural choice of~q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main tools are Euler-type integrals and hypergeometric series.

Ramanujan Primes and Bertrand's Postulate

The

A Geometric Proof that Is Irrational and a New Measure of Its Irrationality

n

Proofs of Power Sum and Binomial Coefficient Congruences Via Pascal's Identity

th Ramanujan prime is the smallest positive integer

New Vacca-Type Rational Series for Euler’s Constant γ and Its “Alternating” Analog \ln \frac{4}{\pi }

R_n

Divisibility of power sums and the generalized Erdős–Moser equation

such that if

Robin's Theorem, Primes, and a New Elementary Reformulation of the Riemann Hypothesis

x \ge R_n