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I am looking for a bound of type

$$\sum_{x<n\leq x+y} \Lambda(n) \leq \frac{\log(x+y)}{\log y} \cdot 2y$$

(or better). Of course such a bound has to exist: the idea of the proof of Brun-Titchmarsh (which is more general) still applies when one considers prime powers. What I am looking for is a clean, explicit result (as clean and explicit as Brun-Titchmarsh, in the well-known Montgomery-Vaughan version) available right off the shelf.

(In particular, I do not want an error term proportional to $\sqrt{x+y}$.)

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  • $\begingroup$ How small is your $y$ compared to $x$? $\endgroup$
    – Daniel Weber
    Commented Dec 31, 2023 at 4:52
  • $\begingroup$ It would be best if $y$ could be allowed to be arbitrarily small (just as in Brun-Titchmarsh). $\endgroup$
    – H A Helfgott
    Commented Dec 31, 2023 at 4:56
  • $\begingroup$ mathoverflow.net/questions/235463/… seems relevant. Does partial summation introduce too much error? $\endgroup$
    – Daniel Weber
    Commented Dec 31, 2023 at 5:04
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    $\begingroup$ From the comments on that answer there's a known $\pi(x+y) - \pi(x) \leq 2\frac{y}{\log y}$, is that strong enough? There's some error introduced by prime powers, but it might be negligible. $\endgroup$
    – Daniel Weber
    Commented Dec 31, 2023 at 9:18
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    $\begingroup$ I've said "Brun-Titchmarsh". Of course that has $\pi(x+y) - \pi(x) \leq 2 \frac{y}{\log y}$ as a special case! $\endgroup$
    – H A Helfgott
    Commented Dec 31, 2023 at 10:39

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