## 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

## 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

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

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

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

- J. Sondow
- MathematicsThe American mathematical monthly
- 8 November 2002

We give analogs for Euler's constant and ln(4/Pi) of the well-known double integrals for zeta(2) and zeta(3). We also give a series for ln(4/Pi) which reveals it to be an"alternating Euler constant."

## Ramanujan Primes and Bertrand's Postulate

- J. Sondow
- MathematicsThe American mathematical monthly
- 29 July 2009

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

- J. Sondow
- MathematicsThe American mathematical monthly
- 1 August 2006

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

- J. Sondow, C. Dumitrescu
- MathematicsPeriodica Mathematica Hungarica
- 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

## Ramanujan Primes: Bounds, Runs, Twins, and Gaps

- J. Sondow, John W. Nicholson, Tony D. Noe
- Mathematics
- 1 May 2011

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then the interval $(x/2,x]$ contains at least $n$ primes. We sharpen Laishram's theorem that $R_n < p_{3n}$… Expand

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

- J. Sondow, W. Zudilin
- Mathematics
- 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

## New Vacca-Type Rational Series for Euler's Constant and Its

- J. Sondow
- Mathematics
- 1 August 2005

We recall a pair of logarithmic series that reveals ln(4 ∕ π) to be an “alternating” analog of Euler’s constant γ. Using the binary expansion of an integer, we derive linear, quadratic, and cubic… Expand

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