Elliott-Halberstam level-of-distribution exponent
Description of constant
Let $\Lambda$ denote the von Mangoldt function. For coprime positive integers $a,q$, define
\[\psi(x;q,a)\ :=\ \sum_{n\le x,\ n\equiv a\ (\mathrm{mod}\ q)} \Lambda(n).\]The Bombieri-Vinogradov theorem gives (for every fixed $A>0$) a bound of the form, where $\phi(q)$ denotes Euler’s totient function:
\[\sum_{q\le Q}\max_{(a,q)=1}\left\lvert\psi(x;q,a)-\frac{x}{\phi(q)}\right\rvert \ \ll_A\ x(\log x)^{-A} \qquad \text{for }Q=x^{1/2}(\log x)^{-B(A)}.\]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
\[\vartheta_{\mathrm{EH}}\ \ge\ \frac{1}{2}.\]The Elliott-Halberstam conjecture predicts the optimal value
\[\vartheta_{\mathrm{EH}}\ =\ 1.\]The best established range currently is
\[\frac12\ \le\ C_{66}\ \le\ 1.\]Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $1$ | Trivial ceiling in the standard level-of-distribution formulation. |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $1/2$ | [Ked2007] | Bombieri-Vinogradov range $Q=x^{1/2}(\log x)^{-B}$. [Ked2007-BV] |
Additional comments and links
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Conjectural endpoint. Elliott-Halberstam asks for the same type of estimate up to $Q=x^{1-\epsilon}$ for each fixed $\epsilon>0$. [Ked2007-EH]
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Original conjecture source. The original paper reference is Elliott-Halberstam, A conjecture in prime number theory. [EH1970-original]
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Classical background source. See also Vinogradov’s density-hypothesis paper for Dirichlet $L$-series. [Vin1965]
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Status note. The same source remarks that this conjecture appears extremely hard. [Ked2007-hard]
References
- [Ked2007] Kedlaya, Kiran S. 18.785 Analytic Number Theory (MIT): The Bombieri-Vinogradov theorem (statement). Course notes (2007). PDF: https://kskedlaya.org/18.785/bombieri.pdf. Publisher page: https://kskedlaya.org/18.785/. Google Scholar
- [Ked2007-def-psi] loc: bombieri.pdf p.1, section “1 Statement of the theorem” quote: “For $m, N$ coprime positive integers, put $\psi(x; N, m)=\sum_{n\le x,\ n\equiv m\ (\mathrm{mod}\ N)} \Lambda(n)$.”
- [Ked2007-BV] loc: bombieri.pdf p.1, Theorem 1 (Bombieri-Vinogradov) quote: “For any fixed $A>0$, there exist constants $c=c(A)$ and $B=B(A)$ such that $\sum_{N\le Q}\max_{m\in (\mathbb{Z}/N\mathbb{Z})^\ast}\left\lvert\psi(x;N,m)-\frac{x}{\phi(N)}\right\rvert \le c x(\log x)^{-A}$ for $Q=x^{1/2}(\log x)^{-B}$.”
- [Ked2007-EH] loc: bombieri.pdf p.1, Conjecture 2 (Elliott-Halberstam) quote: “For any fixed $A>0$ and $\epsilon>0$, there exists $c>0$ such that $\sum_{N\le Q}\max_{m\in(\mathbb{Z}/N\mathbb{Z})^\ast}\left\lvert\psi(x;N,m)-\frac{x}{\phi(N)}\right\rvert \le c x(\log x)^{-A}$ for $Q=x^{1-\epsilon}$.”
- [Ked2007-hard] loc: bombieri.pdf p.1, paragraph below Conjecture 2 quote: “This conjecture appears to be extremely hard; for instance, it is not known to follow from GRH.”
- [EH1970] Elliott, P. D. T. A.; Halberstam, H. A conjecture in prime number theory. In Symposia Mathematica, Vol. IV (Teoria dei numeri, Roma 1968; Algebra, Roma 1969), 59-72 (1970). Publisher page: https://zbmath.org/?q=an%3A0238.10030. Google Scholar
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[Bom1965] Bombieri, Enrico. On the large sieve. Mathematika 12 (1965), 201-225. DOI: https://doi.org/10.1112/S0025579300005313. Google Scholar
- [Vin1965] Vinogradov, Askold Ivanovich. The density hypothesis for Dirichlet L-series. Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), no. 4, 903-934 (in Russian). MR: 0197414. Corrigendum: Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 719-720 (in Russian). Google Scholar
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.