[Paper Review] A conjecture-generalization of Sondow's formula

Petros Hadjicostas|ArXiv.org|May 21, 2004

Advanced Mathematical Identities12 references3 citations

TL;DR

This paper generalizes Sondow's integral formula for Euler's constant by conjecturing a complex-variable extension involving the Riemann zeta function and Gamma function. Using generalized Beukers-type integrals and fractional calculus, it proposes a double integral representation for $\Gamma(z+2)\left[\zeta(z+2) - \frac{1}{z+1}\right]$ valid for $\Re(z) > -2$, which reduces to Sondow's formula in the limit $z \to -1$. The conjecture unifies and extends known results on zeta values and Euler's constant.

ABSTRACT

An easy generalization of Beukers' integrals allows us to conjecture a double integral formula involving the zeta and the gamma functions. A special case of this formula is Sondow's double integral formula for Euler's constant gamma.

Motivation & Objective

• To generalize Sondow's double integral formula for Euler's constant to a broader class of zeta and Gamma function expressions.
• To extend known Beukers-type integral evaluations of zeta values to complex parameters via analytic continuation.
• To propose a unifying integral representation that interpolates between rational zeta values and Euler's constant.
• To provide a framework for proving the irrationality of zeta values at odd integers through generalized integral identities.
• To explore connections between multiple integrals, special functions, and transcendental number theory.

Proposed method

• Generalizing Beukers' arguments on rational approximations to zeta values using double integrals over the unit square.
• Using the identity $\int_0^1 \int_0^1 \frac{[-\ln(xy)]^n}{1-xy} (1-x) \, dx\,dy = \Gamma(n+2)\left[\zeta(n+2) - \frac{1}{n+1}\right]$ for integer $n$ as a foundation.
• Extending the formula to complex $z$ via analytic continuation, assuming convergence for $\Re(z) > -2$.
• Applying techniques from fractional calculus to handle non-integer powers of logarithmic terms in the integrand.
• Deriving the conjectured formula by taking the limit $z \to -1$ to recover Sondow's original result.
• Using known results on zeta function values and factorial moments to validate the structure of the conjectured identity.

Experimental results

Research questions

• RQ1 Can Sondow's double integral formula for Euler's constant be generalized to complex parameters using analytic continuation?
• RQ2 What is the structure of the double integral $\int_0^1 \int_0^1 \frac{[-\ln(xy)]^z}{1-xy}(1-x)\,dx\,dy$ for $\Re(z) > -2$?
• RQ3 Does the conjectured identity $\int_0^1 \int_0^1 \frac{[-\ln(xy)]^z}{1-xy}(1-x)\,dx\,dy = \Gamma(z+2)\left[\zeta(z+2) - \frac{1}{z+1}\right]$ hold for all complex $z$ with $\Re(z) > -2$?
• RQ4 How does this generalized formula relate to known Beukers-type integrals and their evaluations at integer $n$?
• RQ5 Can this conjecture be proven using fractional calculus and generalized integral identities?

Key findings

• For integer $n \geq 0$, the double integral $\int_0^1 \int_0^1 \frac{[-\ln(xy)]^n}{1-xy}(1-x)\,dx\,dy$ evaluates exactly to $\Gamma(n+2)\left[\zeta(n+2) - \frac{1}{n+1}\right]$.
• The conjectured formula reduces to Sondow's original formula for Euler's constant when $z \to -1$, confirming consistency with known results.
• The integrand's structure ensures that the result is a meromorphic function in $z$, with poles at $z = -1$ and $z = -2$.
• The conjecture is supported by the fact that the integer case matches known evaluations and rationality bounds via $d_r^{n+2}$-denominator control.
• The use of fractional calculus suggests a pathway to proving the conjecture, though a full proof remains open.
• The identity unifies discrete and continuous representations of zeta values and Euler's constant through a single analytic formula.