In many treatments of Vinogradov's three prime theorem, one considers the summation $S(\alpha) = \sum_{k \leq N}\Lambda(k)e^{2\pi i\alpha k}$ in place of $T(\alpha) = \sum_{p \leq N}e^{2\pi i\alpha p}$ (where this sum runs over primes). I assume these two expressions are roughly the same, but I can't seem to be able to estimate $|S(\alpha) - T(\alpha)|$. Can anyone give an estimate?
Von Mangoldt function estimate
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analytic-number-theory
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0Do you mean $T(\alpha) = \sum_{p \leq N} \log p e^{2 \pi i \alpha p}$? – 2012-12-10
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0I do not actually, I do know that $|\sum_{p \leq N}\log p e^{2\pi i\alpha p} - S(\alpha)| \ll N^{1/2}$. – 2012-12-10
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0I see, I am then just curious about why $S$ and $T$ should be roughly the same. – 2012-12-10