Elliott–Halberstam conjecture

Primes spread among the possible remainders after division: apart from2, primes leave remainder1 or3 when divided by4. How far can this balanced-distribution principle extend as the divisor grows with the counting range? The precise question uses logarithmically weighted primes and prime powers, takes each divisor’s worst coprime remainder, then adds those errors over many divisors.[1]

How uniformly can primes occupy remainder classes when both the counting limit and the divisors grow? Give $n=p^j$ the von Mangoldt weight $\Lambda(n)=\log p$, and other positive integers weight zero. Define $\psi(x;q,a)=\sum_{n\leq x,\ n\equiv a\pmod q}\Lambda(n)$. For all fixed $0<\varepsilon<1$ and $A>0$, Elliott–Halberstam asks for $\sum_{q\leq x^{1-\varepsilon}}\max_{(a,q)=1}|\psi(x;q,a)-x/\varphi(q)|\ll_{A,\varepsilon}x/(\log x)^A$. The implicit constant may depend on $A,\varepsilon$; the maximum is over coprime classes separately for each $q$.[1] [2]

Known averaged estimates reach divisors roughly up to the square root of the counting range. The target extends to each fixed power below the full range. The cited 2026 implication needs two unproved assumptions; its separate twisted-weight results do not close this unconditional gap.[1] [2]

The July preprint’s Möbius-twisted diagonal/weighted work does not prove this untwisted worst-residue average. GRH alone is not the February theorem’s hypothesis.[1] [2]

Sieve arguments remove integers divisible by small primes to find numbers with few prime factors. Their guarantees depend on how reliably primes fill remainder classes. Better averaged distribution can make those guarantees stronger: the cited conditional work reduces the allowed number of factors in a prime-power-plus-almost-prime representation, rather than merely improving a counting error.[1] [3]

Distribution controls the strength of sieve conclusions, not only how well a prime count matches its leading term. Johnston–Thomas represent every sufficiently large integer as $p^k+\eta$, with $\eta>0$ having boundedly many prime factors counted with multiplicity. For each fixed $\delta>0$ and sufficiently large $k$, their admissible bound improves from $(2+\delta)k$ to $(1+\delta)k$ assuming Elliott–Halberstam. This connects improved distribution to a more tightly constrained additive representation; it does not make the remainder prime or settle Goldbach.[3]