integral equation Given the integral equation
$$\exp(x)-1=\int_0^{\infty} \frac{\mathrm dt}{t}\operatorname{frac}\left(\frac{ \sqrt x}{\sqrt t}\right) f(t)\;,$$
where $\operatorname{frac}$ denotes the fractional part of a number, $ \operatorname{frac}(x)= x-\lfloor x\rfloor$.
My questions are:


*

*Can we deduce from this integral equation that $ f(x)= O(x^{1/4+\epsilon}) $ for some positive $\epsilon$?

*Can we solve this integral by the Hilbert-Schmidt method?

The integral equation has the form $$\int_{0}^{\infty}\frac {dt} {t} f\left(x/t\right)g\left(t\right)=h\left(x\right)$$
The left hand side is known as the Mellin convolution. Defining the Mellin transforms: 
$$F\left(s\right)=\int_{0}^{\infty}dt t^{s-1}f\left(t\right)$$
$$G\left(s\right)=\int_{0}^{\infty}dt t^{s-1}g\left(t\right)$$
$$H\left(s\right)=\int_{0}^{\infty}dt t^{s-1}h\left(t\right)$$
Your integral equation implies that 
$$F\left(s\right)G\left(s\right)=H\left(s\right)$$
See this Mellin convolution and Mellin transform. Does this help? 
