Searcharxiv⌕ Search

arXiv subjects

Jonathan Sondow

Publications and source records attributed to Jonathan Sondow.

49 records · Page 3Linked to original sources
arXiv2007

Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant

Let $T$ be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over $T$, one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler's constant. In the final section, we evaluate more general integrals -- one is a Chen (Drinfeld-Kontsevich) iterated integral -- over some polytopes that are higher-dimensional analogs of $T$. This leads to a relation between certain multiple polylogarithm values and multiple zeta values.

math.NT↗
arXiv2006

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

We define the generalized-Euler-constant function $γ(z)=\sum_{n=1}^{\infty} z^{n-1} (\frac{1}{n}-\log \frac{n+1}{n})$ when $|z|\leq 1$. Its values include both Euler's constant $γ=γ(1)$ and the "alternating Euler constant" $\log\frac{4}π=γ(-1)$. We extend Euler's two zeta-function series for $γ$ to polylogarithm series for $γ(z)$. Integrals for $γ(z)$ provide its analytic continuation to $\C-[1,\infty)$. We prove several other formulas for $γ(z)$, including two functional equations; one is an inversion relation between $γ(z)$ and $γ(1/z)$. We generalize Somos's quadratic recurrence constant and sequence to cubic and other degrees, give asymptotic estimates, and show relations to $γ(z)$ and to an infinite nested radical due to Ramanujan. We calculate $γ(z)$ and $γ'(z)$ at roots of unity; in particular, $γ'(-1)$ involves the Glaisher-Kinkelin constant $A$. Several related series, infinite products, and double integrals are evaluated. The methods used involve the Kinkelin-Bendersky hyperfactorial $K$ function, the Weierstrass products for the gamma and Barnes $G$ functions, and Jonquière's relation for the polylogarithm.

math.CA↗
arXiv2006

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 those 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 $e$, and of the second author for $π$ and $e^γ.$ 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.

math.NT↗
arXiv2006

Summation of Series Defined by Counting Blocks of Digits

We discuss the summation of certain series defined by counting blocks of digits in the $B$-ary expansion of an integer. For example, if $s_2(n)$ denotes the sum of the base-2 digits of $n$, we show that $\sum_{n \geq 1} s_2(n)/(2n(2n+1)) = (γ+ \log \frac{4}π)/2$. We recover this previous result of Sondow in math.NT/0508042 and provide several generalizations.

math.NT↗
arXiv2005

New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi)

In math.CA/0211148 we observed that $\ln(4/ π)$ is an "alternating" analog of Euler's constant $γ$. Here we use the binary expansion of an integer to give a rational series for $\ln(4/ π)$ analogous to Vacca's series for $γ$. Also, using a known generalization of Vacca's series to other bases, we accelerate Addison's series for $γ$. Along the way, we give new proofs of Vacca's and Addison's formulas.

math.NT↗
arXiv2004

Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik

In math.NT/0307308 we defined the irrationality base of an irrational number and, assuming a stronger hypothesis than the irrationality of Euler's constant, gave a conditional upper bound on its irrationality base. Here we develop the general theory of the irrationality exponent and base, giving formulas and bounds for them using continued fractions and the Fibonacci sequence. A theorem of Jarnik on Diophantine approximation yields numbers with prescribed irrationality measure. By another method we explicitly construct series with prescribed irrationality base. Many examples are given.

math.NT↗
arXiv2004

A Faster Product for Pi and a New Integral for ln(Pi/2)

From a global series for the alternating zeta function, we derive an infinite product for pi that resembles the product for $e^γ$ ($γ$ is Euler's constant) in math.CA/0306008. (An alternate derivation accelerates Wallis's product by Euler's transformation.) We account for the resemblance via an analytic continuation of the polylogarithm. An application is a 1-dim. analog for ln(pi/2) of the 2-dim. integrals for ln(4/pi) and $γ$ in math.CA/0211148.

math.NT↗
arXiv2003

An irrationality measure for Liouville numbers and conditional measures for Euler's constant

The irrationality exponent $μ(t)$ of an irrational number t, defined using the irrationality measure $1/q^μ$, distinguishes among non-Liouville numbers and is infinite for Liouville numbers. Using the irrationality measure $1/β^q$, we define the "irrationality base" $β(t)$, which distinguishes among Liouville numbers and is 1 for non-Liouville numbers. We give some properties and examples. Assuming a condition on certain linear forms in logarithms, for which we present numerical evidence supplied by P. Sebah, we prove an upper bound on the irrationality base of Euler's constant, $γ$. If $γ$ is irrational and the condition turns out to be false in a certain strong sense, we prove an upper bound on $μ(γ)$.

math.NT↗
arXiv2003

An Infinite Product for e^gamma via Hypergeometric Formulas for Euler's Constant, gamma

We derive hypergeometric formulas for Euler's constant, gamma. A "by-product" of Thomae's transformation is an infinite product for e^gamma involving the binomial coefficients. Alternate, non-hypergeometric proofs use a double integral for gamma, the beta integral, and an integral for the digamma function. (I thank P. Sebah, S. Zlobin, and W. Zudilin for valuable suggestions used in the paper.)

math.CA↗
arXiv2003

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.

math.NT↗
arXiv2003

Zeros of the alternating zeta function on the line R(s)=1

The alternating zeta function zeta*(s) = 1 - 2^{-s} + 3^{-s} - ... is related to the Riemann zeta function by the identity (1-2^{1-s})zeta(s) = zeta*(s). We deduce the vanishing of zeta*(s) at each nonreal zero of the factor 1-2^{1-s} without using the identity. Instead, we use a formula connecting the partial sums of the series for zeta*(s) to Riemann sums for the integral of x^{-s} from x=1 to x=2. We relate the proof to our earlier paper "The Riemann Hypothesis, simple zeros, and the asymptotic convergence degree of improper Riemann sums," Proc. Amer. Math. Soc. 126 (1998) 1311-1314.

math.NT↗
arXiv2002

Criteria for Irrationality of Euler's Constant

By modifying Beukers' proof of Apery's theorem that zeta(3) is irrational, we derive criteria for irrationality of Euler's constant, gamma. For n > 0, we define a double integral I(n) and a positive integer S(n), and prove that if d(n) = LCM(1,...,n), then the fractional part of logS(n) is given by {logS(n)} = d(2n)I(n), for all n sufficiently large, if and only if gamma is a rational number. A corollary is that if {logS(n)} > 1/2^n infinitely often, then gamma is irrational. Indeed, if the inequality holds for a given n (we present numerical evidence for 0 < n < 2500 and n = 10000) and gamma is rational, then its denominator does not divide the product d(2n)Binomial(2n,n). We prove a new combinatorial identity in order to show that a certain linear form in logarithms is in fact logS(n). A by-product is a rapidly converging asymptotic formula for gamma, used by P. Sebah to compute it correct to 18063 decimals.

math.NT↗