# Gram series for more general integrals

given the Gram series

$$\pi (x) \sim \sum_{n=1}^{\infty} \frac{1}{n}\frac{log^{n}(x)}{n!\zeta (n+1)}$$

can this result be generalized to more integral transforms ?

$$g(x)= \int_{0}^{\infty}dyK(xy)f(y)dy$$

for exapmle in Gramm series for the prime counting function the integral equation is

$$log\zeta(s)= s\int_{0}^{\infty}\frac{1}{e^{xy}-1}\pi(e^{y})dy$$

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