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on Elliott-Halberstam level-of-distribution exponent ($C_{66}$)

The statement

Let $\Lambda$ denote the von Mangoldt function. For coprime positive integers $a,q$ , define

\psi(x;q,a)\ :=\ \sum_{n\le x,\ n\equiv a\ (\mathrm{mod}\ q)} \Lambda(n).

[Ked2007-def-psi]

The Bombieri-Vinogradov theorem gives (for every fixed $A>0$ ) a bound of the form, where $\phi(q)$ denotes Euler's totient function:

\sum_{q\le Q}\max_{(a,q)=1}\left\lvert\psi(x;q,a)-\frac{x}{\phi(q)}\right\rvert \ \ll_A\ x(\log x)^{-A} \qquad \text{for }Q=x^{1/2}(\log x)^{-B(A)}.

[Ked2007-BV]

For $\vartheta\in[0,1]$ , call $\vartheta$ an admissible level of distribution if (for every fixed $A>0$ ) there is some $B=B(A,\vartheta)$ such that

\sum_{q\le Q}\max_{(a,q)=1}\left\lvert\psi(x;q,a)-\frac{x}{\phi(q)}\right\rvert \ \ll_A\ x(\log x)^{-A} \qquad \text{for all }Q\le x^{\vartheta}(\log x)^{-B}.

The level-of-distribution optimization problem is to determine the largest admissible $\vartheta$ .

We define

C_{66}\ :=\ \vartheta_{\mathrm{EH}},

where $\vartheta_{\mathrm{EH}}$ is the supremum of admissible levels $\vartheta$ . [Ked2007-BV]

The Bombieri-Vinogradov theorem implies

\vartheta_{\mathrm{EH}}\ \ge\ \frac{1}{2}.

[Ked2007-BV]

The Elliott-Halberstam conjecture predicts the optimal value

\vartheta_{\mathrm{EH}}\ =\ 1.

[Ked2007-EH]

The best established range currently is

\frac12\ \le\ C_{66}\ \le\ 1.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.