Abstract
This entry provides a proof of Perron's Formula, which states that for a Dirichlet series $f(s) = \sum_{n=1}^\infty a_n n^{-s}$ with abscissa of convergence $\sigma_c$ we have for any $c, z, x$ with $c > 0$, $x > 0$, $\text{Re}(z) > \sigma_c - c$:
\[\sum\limits_{n\leq x}\kern-1pt^{'} a_n n^{-z} = \frac{1}{2\pi i}\, \lim_{T\to\infty} \int\limits_{c-iT}^{c+iT}\hskip-4pt f(z + w)\, x^s\kern1pt w\,\frac{\text{d}w}{w} \]Additionally, various explicit bounds for the remainder (i.e.\ when using some finite value of $T$ instead of the limit) are shown.
As an interesting nontrivial auxiliary result, asymptotic bounds for a Dirichlet series near $\pm i\infty$ are also included, namely that if $a\in[0,1)$ then $f(\sigma + it) \in o(t^{1-a})$ as $t\to\pm\infty$, uniformly for all $\sigma\geq \sigma_c + a + \varepsilon$.
The proofs mainly follow Tenenbaum's Introduction to Analytic and Probabilistic Number Theory and Titchmarsh's Theory of Functions.