Generalization of an Integral Related to Stieltjes Moment Problem

Abstract

In connection with the non-uniqueness of the Stieltjes moment problem on $(0,\infty)$,
Stieltjes constructed the nontrivial function $ f(x)=e^{-x^{1/4}}\sin\!\big(x^{1/4}\big) $ satisfying $\int_{0}^{\infty} x^{n}f(x)\,dx=0$ for all integers $n\ge 0$. We extend this by considering

\[ I(k)=\int_{0}^{\infty} e^{-x^{1/4}}\sin\!\big(x^{1/4}\big)\,x^{k}\,dx \] for real $k\ge 0$, evaluating $I(k)$ explicitly and proving $I(k)=0$ if and only if $k\in\mathbb{Z}_{\ge 0}$.
More generally, for parameters $m>0$, $\alpha>0$, $\beta\in\mathbb{R}$, $q,k\in\mathbb{R}$ we analyze \[ I_{m,q}(k;\alpha,\beta)=\int_{0}^{\infty} e^{-\alpha x^{1/m}}\sin\!\big(\beta x^{1/m}\big)\,x^{k}(x^{1/m})^{q}\,dx, \] derive a closed form, and give necessary and sufficient conditions for its vanishing.
We also establish
cosine analogues, both for the Stieltjes example and for the generalized integral mentioned above.
As a consequence, we obtain integral representations of Γ(A) for suitable A > 0, as well as integral
formulas for several classical constants arising from gamma function. To understand the importance
of integrals that vanish for every value of a continuous parameter, we will also discuss Salem’s
equivalence of the Riemann hypothesis, which is formulated in terms of such a parameter-dependent
integral.