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Double integrals and infinite products for some classical constants via analytic continuations of Lerch’s transcendent

- Jesús Guillera, J. Sondow
- Mathematics
- 16 June 2005

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… Expand

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

- J. Sondow
- Mathematics
- 1 February 1994

We prove that a series derived using Euler's 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… Expand

The generalized-Euler-constant function γ(z) and a generalization of Somos's quadratic recurrence constant

- J. Sondow, P. Hadjicostas
- Mathematics
- 16 October 2006

Abstract We define the generalized-Euler-constant function γ ( z ) = ∑ n = 1 ∞ z n − 1 ( 1 n − log n + 1 n ) when | z | ⩽ 1 . Its values include both Euler's constant γ = γ ( 1 ) and the “alternating… Expand

Ramanujan Primes and Bertrand's Postulate

- J. Sondow
- Mathematics, Computer Science
- Am. Math. Mon.
- 29 July 2009

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Double Integrals for Euler's Constant and In and an Analog of Hadjicostas's Formula

- J. Sondow
- Mathematics, Computer Science
- Am. Math. Mon.
- 1 January 2005

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

- J. Sondow
- Computer Science, Mathematics
- Am. Math. Mon.
- 1 August 2006

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

- J. Sondow, W. Zudilin
- Mathematics, Engineering
- 2 April 2003

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,… Expand

A monotonicity property of Riemann’s xi function and a reformulation of the Riemann hypothesis

- J. Sondow, C. Dumitrescu
- Mathematics, Computer Science
- Period. Math. Hung.
- 18 March 2010

We prove that Riemann’s xi function is strictly increasing (respectively, strictly decreasing) in modulus along every horizontal half-line in any zero-free, open right (respectively, left)… Expand

Criteria for irrationality of Euler’s constant

- J. Sondow
- Mathematics
- 6 September 2002

By modifying Beukers' proof of Apery'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 S… Expand

The Parbelos, a Parabolic Analog of the Arbelos

- J. Sondow
- Mathematics, Computer Science
- Am. Math. Mon.
- 5 October 2012

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