# Reference for explicit formula for $\sum_n \Lambda(n) \chi(n)$ with smooth weights

Let $\Lambda$ be the von Mangoldt function and $\chi$ a primitive character mod $q$, then we have the explicit formula $$\sum_{n \leq X} \Lambda(n) \chi(n) = \delta_{\chi} X - \sum_{ |Im \ \rho| \leq T} \frac{X^{\rho}}{\rho} + O((\frac{X}{T}+1) \log^2(qXT)),$$ where $\delta_{\chi}$ is $1$ if $\chi$ is the principal character and $0$ otherwise, and $\rho$'s are the non-trivial zeros of the $L$ function $L(s, \chi)$. I was interested in the explicit formula of $$\sum_{n \leq X} \Lambda(n) \chi(n) \phi(n)$$ where $\phi$ is a smooth function. I would greatly appreciate a reference for this. Thank you very much.

• I recall Iwaniec-Kowalski's book has some asymptotics on this kind of sums. Commented Oct 7, 2017 at 9:39
• @Wojowu That was one of the sources I looked into, but I couldn't find it. Could you possibly let me know the page number if I had missed it? Commented Oct 7, 2017 at 9:41
• I was thinking of results of section 5.7, but they all seem to be using a smoothing factor $\phi(n/X)$, so are probably not of use for you. Commented Oct 7, 2017 at 9:54
• $\Lambda(n)\chi(n)$ is just the von Mangoldt function $\Lambda_f(n)$ associated to the Dirichlet L-series $L(\chi,s)$. Commented Oct 7, 2017 at 11:28
• There's also a discussion in Section 5.1 of Montgomery/Vaughan's "Multiplicative Number Theory. I" (smooth weighted sums of general arithmetic functions). Commented Oct 7, 2017 at 16:54

Also, the answer is addressed in Theorem 5.11 in Iwaniec and Kowalski. Their answer may look slightly different from your first centered equation above because they do not "push the contour all the way to the left". Instead, they push the contour to the line $$\mathrm{Re}(s)=-c$$ for some fixed $$c>0$$; then, they use the functional equation. However, one can indeed "push the contour all the way to the left" and achieve what is perhaps a more classical-looking result (like your first centered equation). But all of the key ingredients are indeed present in the setup of Section 5.5 in Iwaniec and Kowalski.