User:Caliburn/s/ant/Truncated Perron Formula

Theorem

Let $x, c, T > 0$ be real numbers.

Let $\sequence {a_n}$ be a sequence of complex numbers such that:

$\ds \sum_{n \mathop = 1}^\infty \frac {\cmod {a_n} } {n^c}$

is convergent.

Define a real function $S : \hointr 1 \infty \to \R$ by:

$\ds \map S x = \begin{cases}\sum_{n \le x} a_n & x \not \in \Z \\ \sum_{n \le x - 1} a_n + \frac 1 2 a_x & x \in \Z \end{cases}$

Then:

$\ds \map S x = \frac 1 {2 \pi i} \int_{c - i T}^{c + i T} \paren {\sum_{n \mathop = 1}^\infty \frac {a_n} {n^s} } x^s \frac {\rd s} s + \map R x$

where $R : \hointr 1 \infty \to \R$ is a real function with:

$\ds \map R x = \map \OO {x^c \sum_{n \mathop = 1}^\infty \paren {\frac {\cmod {a_n} } {n^c} \min \set {1, \frac 1 {T \size {\map \log {x / n} } } } } }$