My question is about what is the problem, if there is one, to get an identity using Frullani's integral. I've in a hand the statement from MathWorld, and other statement from this site, with a nice proof, and a remark in such answer from other user.

Question.It is possible or feasible an application of Frullani's theorem (that isMathWorld) or see here in this site Giraudo's answer for Proof of Frullani's theorem, for the function $$f(x)=\sum_{n=1}^\infty \frac{e^{-\pi n^2 x}}{n^3}?$$

I know that was in the literature the function $g(x)=\sum_{n=1}^\infty e^{-n^2 \pi x}$, since there is a easy relationship with the following evaluation of a **Jacobi Theta function** $\vartheta_3(0,e^{-\pi x})$, $g(x)=\frac{1}{2}(\vartheta_3(0,e^{-\pi x})-1)$, or that $\lim_{x\to\infty}\vartheta_3(0,e^{-\pi x})=1$. But I know few facts about this kind of functions, I say about convergence questions.

**From the first reference**, if I take $a=1,b=2$, the symbolic evaluations $f(0)=\zeta(3)$ and $f(\infty)=0$, provide to me

$$\int_0^\infty \frac{1}{x}\left(\sum_{n=1}^\infty \frac{e^{-\pi n^2 x}}{n^3}-\sum_{n=1}^\infty \frac{e^{-2\pi n^2 x}}{n^3}\right)dx=\zeta(3)\log 2.$$

**But does converge the integral in LHS?** I know that there is a problem at $x=0$ (I've tried several definition for $f(x)$ to get a symbolic identities involving $\zeta(3)$ when one combine easy computations and Frullani's integral, and I don't understand if this current question the theorem provide to me a trivial identity or well I can not use it since the hypothesis of the theorem isn't right).

**From the second reference,** notice that is requiered that my $f:[0,\infty)\to\mathbb{R}$, and my proble was in my definition for $x=0$, because is required that our functions is continuously differentiable, and the finite value of the limit is not a problem by the remark, that one can redefine the function.

**Thanks in advance.**