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I would like to know whether it is possible to obtain cancellation in the sum

$$\sum_{p \leq X} e^{{2\pi iX}/{p}}$$

where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number.

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  • $\begingroup$ what do you mean by "cancellation" ? you want the sum to equal zero? $\endgroup$ Jul 6, 2018 at 6:19
  • $\begingroup$ @CarloBeenakker I just want the absolute value of the sum, divided by the number of summands, to tend to zero as $X$ grows. $\endgroup$
    – Pablo
    Jul 6, 2018 at 6:45
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    $\begingroup$ This seems related to the sum $\sum_{p\le x} \{ x/p \}$, where $\{t\}$ denotes the fractional part of $t$; an asymptotic was found by de la Vallée–Poussin, and this paper of Eric Naslund gives a nice method for sharpening that asymptotic. $\endgroup$ Jul 6, 2018 at 7:36

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