Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 150 / 3 / a

Let $F(s)={\sum }_{n\ge 1}{a}_n{n}^{-s}$ converge absolutely for $\Re s>{\sigma }_a$. Perron formula states that for $c>{\sigma }_a$ and nonintegral $x>0$,where the integral is understood as the limit of symmetric truncations under the usual convergence hypotheses. If $x$ is an integer, the endpoint term is counted with weight $1/2$. Effective versions truncate at height $T$ and include an explicit error depending on the coefficients and the distance of $x$ from nearby integers.

Solved by gpt-5.6-sol high.