Truncated Perron formula for coefficients of von Mangoldt size

Truncated Perron formula. There is an absolute constant $K>0$ with the following property. Let $a:\mathbb{N}\to\mathbb{C}$ satisfy $|a(n)|\le\Lambda(n)$ for every $n$, where $\Lambda$ is the von Mangoldt function, and let $G$ agree with the Dirichlet series $\sum_{n\ge1}a(n)n^{-s}$ on the half-plane $\operatorname{Re}s>1$. Then for every integer $N\ge2$ and every real $T\ge2$, writing $\sigma_0=1+1/\log N$,

The constant is uniform: it depends on none of $a$, $G$, $N$, $T$.

This is the entry point of the classical contour argument. It replaces the partial sum, an arithmetic object, by an integral of $G$ along a vertical line to the right of the abscissa of absolute convergence, at the cost of an error decreasing in the height $T$. The abscissa $\sigma_0=1+1/\log N$ is the standard choice: it makes $N^{\sigma_0}=eN$ while keeping $\sum_{n}|a(n)|n^{-\sigma_0}\ll\log N$.

The second error term $\log N$ is not an artefact of the formalisation. The sum runs over $n<N$ while the integral is the symmetric Perron integral at $x=N$; because $x$ is here an integer the two differ by about $\tfrac12 a(N)$, of size up to $\Lambda(N)\le\log N$, and that discrepancy does not decay as $T\to\infty$. Any version of the statement without it would be false for large $T$.

Context. The theorem this lemma was introduced to decompose, Davenport.perron_of_region_bound, was proved independently — via the smoothed Perron route (Davenport.perron_smoothed_eq_integral and its siblings), which replaces the sharp cutoff by a smooth weight and so avoids the truncation error entirely. The statement here belongs to the classical sharp-cutoff route of Davenport Sections 17-18 and is left standing as a self-contained, reusable statement in its own right; it is not on the critical path of any current mission.