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From this page: https://en.wikipedia.org/wiki/Chebyshev_function#Asymptotics_and_bounds

A theorem due to Erhard Schmidt states that, for some explicit positive constant $K'$, there are infinitely many natural numbers $x$ such that $$ψ(x)>x+K'√x$$

My question is: Is it possible to take the constant $K$ arbitrary large in the sense that the above inequality holds true for any $K≥K'$

and what is known about the smallest positive integer that verifying the above inequality.

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    $\begingroup$ I think the very next inequality on that page, due to Hardy and Littlewood, helps to answer both questions. $\endgroup$
    – Will Sawin
    Dec 27, 2019 at 15:19
  • $\begingroup$ @WillSawin: How this is possible. $\endgroup$
    – Safwane
    Dec 27, 2019 at 15:43
  • $\begingroup$ @Helena page 184, under section 5 of projecteuclid.org/download/pdf_1/euclid.acta/1485887467 . $\endgroup$
    – alpoge
    Dec 27, 2019 at 16:29
  • $\begingroup$ @alpoge: The result is true under RH. I want an unconditional result just like the Schmidt result. $\endgroup$
    – Safwane
    Dec 27, 2019 at 16:42
  • $\begingroup$ @Helena Sorry for the late response, but if RH is false you even gain a power. I think you read too quickly —- Hardy and Littlewood say this at exactly the same point (page 184, under section 5). $\endgroup$
    – alpoge
    Dec 28, 2019 at 20:28

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