I try to understand a number theoretical identity used by
Jan-Christoph Schlage-Puchta in this [answer](https://mathoverflow.net/questions/161947/what-keeps-asymptotic-goldbachs-conjecture-out-of-reach-of-current-technology/162085#162085).


He defined the function


$$S(\alpha)=\sum\_{n\leq N}\Lambda(n) e(n\alpha)$$


where $\Lambda(n)$ is the
[Mangoldt function](https://en.wikipedia.org/wiki/Von_Mangoldt_function) and $e(x)$ the exponential $e(x)=e^{2\pi i x}$.


Assume $\alpha=\frac{p}{q}$ is rational, and $p:= p\_1 p\_2...p\_n$ and
$q:= q\_1 ...p\_m$ are *coprime* positive integers where $p\_i$ and
$q\_j$ are primes such that every pair $p\_i, q\_j$ is pair wise different.


Why the following identity is true:


$$\sum\_{n\leq N}\Lambda(n) e(n\alpha) = 
\sum\_{(a,q)=1} e(\frac{ap}{q})\underset{n\equiv a\pmod{q}}{\sum\_{n\leq N}} \Lambda(n)$$



A number theoretical identity of exponential sum
$$\sum\_{n \leq N} e(n\alpha) \Lambda(n) =\left(\sum\_{p \leq N} \log(p) \sum\_{r=1}^{a(p): p^{a(p)} \leq N} e(p^r\alpha)\right)$$


But, the right hand side is gives,
$$\left(\sum\_{(a,q)=1 ,a<q} e(a\frac{p}{q})\sum\_{\substack{n \leq N \\ n \equiv a (\text{modulo q})}} \Lambda(n) \right)=\sum\_{\substack{p \leq N \\ p \nmid q}} \log(p) \sum\_{r=0}^{a(p): p^{a(p)} \leq N} e(p^r\alpha)$$.


[ This is because $q\_i \nmid n, \text{where} n \equiv a (\text{modulo q}), (a,q)=1$ ].


$$\chi(q)=\sum\_{\substack q\_i} \log(q\_i) \sum\_{r=0}^{b\_i: {q\_i}^{b\_i} \leq N} e({q\_i}^r\alpha)$$.


($q\_i$ s are prime factors of $q$)


This, $\chi(q)$ obviously depends on the $N$ and isn't zero generally.


