Counting primes by circle method $$\int_{0}^{1}\sum_{n_0=1}^{N}e^{2\pi ian_0}\sum_{n=1}^{N}\Lambda (n)e^{-2\pi ian}da$$ I tried to find the main term by looking at major arcs, but the singular series doesn't seem to behave in a way that gives the main term as in second Chebyshev function $\Psi(N)=N+o(N)$. Is it possible to get the main term by considering major arcs? Or at least where it can be located and what to consider?
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1$\begingroup$ Are the two sums supposed to be independent of each other or are they nested? In the latter case, the summation indices should not be the same in them. $\endgroup$– WojowuCommented Feb 4, 2022 at 15:08
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2$\begingroup$ This will eventually work - it is just the major arc near the origin which is relevant (with the bulk of the contribution coming from the region $a = O(1/N)$ or $a = 1-O(1/N)$), but it is rather circular, since to estimate $\sum_{n=1}^N \Lambda(n) e^{-2\pi i an}$ for such values of $a$ requires a version of the prime number theorem with reasonably good error terms in the first place. $\endgroup$– Terry TaoCommented Feb 4, 2022 at 22:37
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