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Concerning gaps between consecutive primes, Paul Erdős conjectured that:

$$\sum_{p_n < x} (p_n -p_{n-1})^2 = O(x \log x)$$

Let's call this hypothesis EH. Assuming the Riemann hypothesis (RH), Selberg showed that a result follows that differs from EH merely by a logarithmic factor.

Does EH imply RH, or RH EH? What are the state of the art results on EH?

By assuming that all the zeros of $\zeta(s)$ are simple (or other deep vertical fact tied to the critical strip), EH might also follow. Any comments are welcomed.

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    $\begingroup$ Do you have a reference for where this was conjectured by Erdős? In arxiv.org/abs/1705.10766 this conjecture (or more precisely the conjecture that the sum in question is on the order of $2x\log x$) is attributed to Heath-Brown "Gaps between primes, and the pair correlation of zeros of the zeta-function" (see matwbn.icm.edu.pl/ksiazki/aa/aa41/aa4118.pdf). $\endgroup$
    – Sam Hopkins
    Commented Mar 18 at 18:40
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    $\begingroup$ @SamHopkins See page 440 in P. Erdős: The difference of consecutive primes, Duke Math. J. 6 (1940), 438–441. $\endgroup$
    – GH from MO
    Commented Mar 18 at 21:33
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    $\begingroup$ @GHfromMO Thanks. Here is a copy I found online, in case anyone else is curious: users.renyi.hu/~p_erdos/1940-10.pdf. Note that Erdős says precisely "It might be conjectured that...", which might suggest he had some doubts. $\endgroup$
    – Sam Hopkins
    Commented Mar 18 at 21:37
  • $\begingroup$ @Sam Hopkins yes. Erdős also says, if I recall correctly, that the bound, if correct, would lie "quite deep", meaning that he saw it as important. This was actually what sparked my interest in the problem, to be honest, much more so than Selberg's paper $\endgroup$
    – Felixson
    Commented Mar 19 at 0:28

2 Answers 2

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This problem is connected with the $L^2$ average of primes in short intervals, see Selberg (1942 paper entitled “on the normal density…”). In particular, results on the integral of $\psi(x+h)-\psi(x)-h$ can be transferred on suitable estimates for $p_n-p_{n-1}$. Gallagher and Mueller were the first to show the role of the Montgomery pair correlation conjecture in estimating the $L^2$ average of primes in short intervals. Heath-Brown in 1982 (Acta Arithmetica) improved on their results. I suggest to read Selberg’s first, then Heath-Brown’s, papers. On the connection between the pair correlation conjecture and the $L^2$ average of primes in short intervals, see also the results of Goldston (Crelle’s journal 1988). See also his beautiful survey on the pair correlation conjecture entitled “Notes on pair correlation of zeros and prime numbers”, 2005.

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  • $\begingroup$ much appreciated @Alessandro Languasco. The Selberg paper I know quite well, but had no idea about Gallagher and Mueller, or about Heath-Brown and Goldston. Glancing through "The notes on the pair...", you get a nice historical sense on the natural evolution of the ideas, something often missing in dry technical papers $\endgroup$
    – Felixson
    Commented Mar 19 at 0:50
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It is not know that RH implies EH, or that EH implies RH. Let us denote $$S(x):=\sum_{p_n < x} (p_n -p_{n-1})^2.$$ Assuming the Lindelöf hypothesis, Yu (1996) proved that $S(x)\ll_\varepsilon x^{1+\varepsilon}$. Unconditionally, Stadlmann (2022) proved that $S(x)\ll_\varepsilon x^{1.23+\varepsilon}$.

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  • $\begingroup$ Thank you @GHfromMO. My impression is that by assuming the karatsuba bounds on $|\zeta(s)|$ in infinitesimally short imaginary intervals, both EH and The classical Chebyshev conjecture follow. $\endgroup$
    – Felixson
    Commented Mar 19 at 1:02

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