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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 analogExpand
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 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 isExpand
The generalized-Euler-constant function γ(z) and a generalization of Somos's quadratic recurrence constant
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 “alternatingExpand
Ramanujan Primes and Bertrand's Postulate
  • J. Sondow
  • Mathematics, Computer Science
  • Am. Math. Mon.
  • 29 July 2009
TLDR
The Prime Number Theorem is used to show that $R_n$ is asymptotic to the $2n$th prime and the length of the longest string of consecutive Ramanujan primes among the first $n$ primes is estimated. Expand
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
TLDR
A new measure of irrationality for e is led to by a construction of a nested sequence of closed intervals with intersection e, where S(q)$ is the smallest positive integer such that $S(q)!$ is a multiple of $q$. Expand
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,Expand
A monotonicity property of Riemann’s xi function and a reformulation of the Riemann hypothesis
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
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 SExpand
The Parbelos, a Parabolic Analog of the Arbelos
  • J. Sondow
  • Mathematics, Computer Science
  • Am. Math. Mon.
  • 5 October 2012
TLDR
A parabolic analog of the arbelos is introduced, using theorems of Archimedes and Lambert to demonstrate seven properties of the parbelos, drawing analogies to similar properties ofThe arbelo, some of which may be new. Expand
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