Research proof index

The series ∑ f(n)/n · 𝓕ψ(log(n/x)/(2π)) is summable for x ≥ 1.

A boundary Fourier identity for a Dirichlet series with a simple pole. Suppose $f\ge0$, the Dirichlet series $L(f,s)$ differs from $A/(s-1)$ by a function $G$ that extends continuously to $\operatorname{Re}s\ge1$, and $\psi$ is a compactly supported $C^2$ test function with nonnegative real Fourier transform. For $x>0$, $\sum_{n\ge1}\frac{f(n)}{n}\widehat\psi\!\left(\frac{1}{2\pi}\log\frac{n}{x}\right)-A\int_{-\log x}^{\infty}\widehat\psi\!\left(\frac{u}{2\pi}\right)du=\int_{\mathbb R}G(1+it)\psi(t)x^{it}\,dt.$