A Banach fixed–point approach to Salem's nonhomogeneous integral equation related to the Riemann hypothesis

Salem's equivalence of the Riemann hypothesis asserts that the hypothesis is true if and only if $g(t)=0$ is the only nontrivial bounded measurable solution (for fixed $1/2<r<1$) of the following integral equation:

$${\int }_0^{\mathrm{\infty }}\frac{t^{r-1}g(t)}{e^{xt}+1}dt=0,x>0.$$

Recent works have investigated this integral equation and shown that various classes of bounded measurable functions, subject to suitable growth assumptions, cannot furnish counterexamples to Salem's criterion. These approaches typically employ Mellin inversion, Widder–Lambert–type transforms, or distribution-theoretic methods. The aim of this paper is to study the associated nonhomogeneous Salem-type equation

$$f(x)=g(x)+\lambda {\int }_0^{\mathrm{\infty }}\frac{t^{r-1}f(t)}{1+e^{xt}}dt,x>0,$$

where $g$ is a given measurable function, $r\in (1/2,1)$, and $\lambda \in \mathbb{R}$, from the viewpoint of fixed–point theory. We show that the corresponding integral operator does not map the space of bounded measurable functions into itself on $(0,\mathrm{\infty })$. We therefore introduce a suitable weighted complete function space in order to apply Banach's contraction principle. The nonhomogeneous equation is important because it is a more general form of Salem's equation. As we shall see, its analysis allows one to identify precisely why Banach's contraction principle succeeds for the nonhomogeneous equation in an appropriate weighted space, yet does not directly settle Salem's original integral equation. This paper suggests that possible future directions may include the use of generalized metric spaces and generalized contraction mappings to study Salem's integral equation.