Math Conjectures

Elliott–Halberstam Conjecture

OPENLandmarkConjectureProposed 1968–1970 · Standard version

Canonical statement

Let Λ(n)=log⁡p\Lambda(n)=\log p if n=pmn=p^m for a prime pp and an integer m≥1m\ge1, and let Λ(n)=0\Lambda(n)=0 otherwise. Put
ψ(y;q,a)=∑n≤yn≡a(modq)Λ(n)andφ(q)=#(Z/qZ)×. \psi(y;q,a)=\sum_{\substack{n\le y\\ n\equiv a\pmod q}}\Lambda(n) \quad\text{and}\quad \varphi(q)=\#(\mathbb Z/q\mathbb Z)^\times.
For every 0<θ<10<\theta<1 and every A>0A>0,
∑q≤xθmax⁡1≤a≤q(a,q)=1max⁡2≤y≤x∣ψ(y;q,a)−yφ(q)∣≪θ,Ax(log⁡x)A(x→∞). \sum_{q\le x^\theta} \max_{\substack{1\le a\le q\\(a,q)=1}} \max_{2\le y\le x} \left|\psi(y;q,a)-\frac{y}{\varphi(q)}\right| \ll_{\theta,A}\frac{x}{(\log x)^A} \qquad(x\to\infty).
View source LaTeX
Let \(\Lambda(n)=\log p\) if \(n=p^m\) for a prime \(p\) and an integer \(m\ge1\), and let \(\Lambda(n)=0\) otherwise. Put
\[
  \psi(y;q,a)=\sum_{\substack{n\le y\\ n\equiv a\pmod q}}\Lambda(n)
  \quad\text{and}\quad
  \varphi(q)=\#(\mathbb Z/q\mathbb Z)^\times.
\]
For every \(0<\theta<1\) and every \(A>0\),
\[
  \sum_{q\le x^\theta}
    \max_{\substack{1\le a\le q\\(a,q)=1}}
    \max_{2\le y\le x}
    \left|\psi(y;q,a)-\frac{y}{\varphi(q)}\right|
  \ll_{\theta,A}\frac{x}{(\log x)^A}
  \qquad(x\to\infty).
\]
NT-DIST-032:[ElliottHalberstam1970][FriedlanderGranville1989][Maynard2015SmallGaps]
Number theoryAnalysis and functional analysis

Notes

The Bombieri–Vinogradov theorem gives this estimate for every θ<1/2\theta<1/2, while improvements beyond 1/21/2 require restricted moduli, weights, or other averaging. The unrestricted estimate for every θ<1\theta<1 remains open.
Some early formulations allowed moduli almost as large as xx with an arbitrarily strong logarithmic saving; Friedlander and Granville showed that such overstrong formulations are false. The fixed-power range q≤xθq\le x^\theta for every θ<1\theta<1 is the standard corrected level-one conjecture recorded here.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.