I believe that I.M. Vinogradov's proof of the ternary Goldbach conjecture used the observation that the number of ways $n$ can be written as a sum of three primes equals $$ \int_0^1 \sum_{p , q , r \leq n} e^{2 \pi i t (p+q+r-n)} dt = \int_0^1 E_n^3 (t) e^{-2 \pi i n t} dt $$ where $p, q, r$ are primes and $$ E_n(t) = \sum_{p \leq n} e^{2 \pi i p t} . $$ I believe that this observation had been made earlier and would like to track down where it had appeared before.
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4$\begingroup$ Vinogradov did not prove ternary Goldbach, he only proved an "asymptotic" version of it. I'm fairly sure Hardy and Littlewood, who have some 20 years earlier proven ternary Goldbach assuming GRH, have used the same ideas. $\endgroup$– WojowuCommented Apr 24, 2022 at 21:49
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5$\begingroup$ Indeed, this expression appears in their 1923 paper here. $\endgroup$– WojowuCommented Apr 24, 2022 at 21:55
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6$\begingroup$ Let me add the obvious: the proof of the ternary Goldbach conjecture is due to fellow MO user Harald Helfgott. $\endgroup$– GH from MOCommented Apr 24, 2022 at 21:57
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2$\begingroup$ Yes, thanks. I was aware of Helfgott (2013) and other references and had phrased my question too quickly. I was (and remain) focused on trying to track down the originator(s) of the mentioned `observation'. In Hardy-Littlewood (1923) that observation is rather heavily camouflaged. Any other/earlier references would be appreciated. $\endgroup$– AndreyFCommented Apr 24, 2022 at 23:19
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3$\begingroup$ Well, Hardy and Littlewood invented the Hardy-Littlewood circle method, so... $\endgroup$– GH from MOCommented Apr 25, 2022 at 7:20
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