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I have seen justifications for various conjectures in analytic number theory, e.g. the (generalized) Riemann hypothesis, the Chowla conjectures, etc. justified by a heuristics in which the relevant arithmetic functions are modeled by random variables.

In the setting of primes in arithmetic progression, these heuristics say that the error term $E(x;q,a)$ for primes in particular primitive residue class should be approximately on the order of magnitude of $\frac{\sqrt{x}}{\varphi(q)}$, which is approximately GRH. If one sums this for $q$ up to $\sqrt{x}$, then the accumulated error is approximately on the order of magnitude of $x$, and the Bombieri-Vinogradov theorem actually establishes a bound of this nature.

The Elliott-Halberstam conjecture asserts that we actually get an error bound of $\frac{x}{(\log x)^A}$ for any fixed $A$ if we sum over moduli all the way up to $x^{1-\epsilon}$. My question is whether there is a convincing heuristic model that this should be the case, since the standard one that suggests GRH doesn't seem to accommodate such large moduli.

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    $\begingroup$ I am not sure your initial analysis is correct. From GRH one gets an error term in $\psi(x;q,a)$ of size $\sqrt x$ (ignoring logs), and summing over the characters gives the same size bound for $E(x;q,a)$, not $\sqrt x/\phi(q)$ as you say. But one expects cancellation over this sum, so the "proper" size is expected more like $\sqrt{x/\phi(q)}$, with logs and maybe $x^\epsilon$. Plugging this in gives Elliott-Halberstam (for the reader, it might be convenient to actually give the E-H sum: $\sum_{q\le Q} \max |E(x;q,a)|\ll x/\log^\infty x$). $\endgroup$
    – Conder
    Commented Apr 29, 2014 at 0:13

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The Elliott-Halberstam conjecture is an immediate consequence of Montgomery's (modified) conjecture, which proposes that if $\epsilon>0$, $\gcd(a,q)=1$ and $x>q^{1+\epsilon}$, then

$\Big|\displaystyle\sum_{\substack{n\leq x \\ n\equiv a\pmod{q}}}\log p -\frac{x}{\varphi(q)} \Big|\ll_{\epsilon} x^{1/2+\epsilon}q^{-1/2+\epsilon}$.

For a very nice heuristic for why one might believe this, see Heuristic for Montgomery's conjecture.

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