Elliott-Halberstam Conjecture

Elliott-Halberstam Conjecture

Definitions and mathematical statement

Let $\Lambda(n)=\log p$ if $n=p^m$ for a prime $p$ and $m\ge1$, and zero otherwise. Put $\psi(y;q,a)=\sum_{n\le y,\,n\equiv a\ (q)}\Lambda(n)$ and let $\varphi(q)$ be Euler's totient. The assertion is \[ \sum_{q\le x^\theta}\max_{\gcd(a,q)=1}\ \max_{2\le y\le x} \left|\psi(y;q,a)-\frac{y}{\varphi(q)}\right| \ll_{A,\theta}\frac{x}{(\log x)^A} \] for every fixed $A>0$ and $0<\theta<1$. The implicit constant is uniform in $x$, the residue class, and the modulus within the displayed range.

Formulation and concept references: [S1].

Short English statement

Are primes distributed among arithmetic progressions with the predicted strong averaged accuracy for moduli almost as large as the search range?

Research attempt

Scope and checked context.

This unresolved attempt by GPT-6 Astra Ultra was prepared on 27 September 2026. Its first task is to keep both maxima in the catalogue statement: the maximizing residue class may depend on the modulus, and the maximizing height may also depend on it. The main proved reduction shows that a full family of endpoint estimates, with room in the distribution exponent, does recover the maximum over height. A second calculation explains why an unstructured second-moment bound loses the desired distribution range.

The primary paper [S3] states the classical Bombieri--Vinogradov range and proves results beyond one half under specified factorization and uniformity conditions. Those conditions do not cover all moduli in the catalogue. The inspected abstract of [S2] assumes both GRH and a pair-correlation conjecture before deriving Elliott--Halberstam. It is a conditional route, not an unconditional theorem. No new distribution estimate is claimed here.

A useful elementary sum over moduli.

We shall need \[ \sum_{q\leq Q}\frac1{\varphi(q)}\ll\log(2Q). \tag{89.1} \] For completeness, multiplicativity gives $q/\varphi(q)=\sum_{d\mid q}\mu(d)^2/\varphi(d)$, since at a prime the factor on the right is $1+1/(p-1)=p/(p-1)$. Therefore \[ \sum_{q\leq Q}\frac1{\varphi(q)} =\sum_{d\leq Q}\frac{\mu(d)^2}{d\varphi(d)} \sum_{m\leq Q/d}\frac1m \leq(1+\log Q)\prod_p\left(1+\frac1{p(p-1)}\right). \] The last product is finite because its positive increments have summable total, bounded by a constant times $\sum_{n\geq2}n^{-2}$. This proves $89.1$ without an estimate for primes in progressions.

A monotonicity lemma for the height maximum.

Write $E(y;q,a)=\psi(y;q,a)-y/\varphi(q)$. If $u\leq y\leq v$, nonnegativity of $\Lambda$ makes $\psi$ nondecreasing and gives \[ E(u;q,a)-\frac{v-u}{\varphi(q)} \leq E(y;q,a)\leq E(v;q,a)+\frac{v-u}{\varphi(q)}. \tag{89.2} \] Hence a grid with gaps at most $h$ controls intermediate heights at an additional cost $h/\varphi(q)$, regardless of jumps at prime powers. After summing over moduli this cost is $O(h\log(2Q))$ by $89.1$. This is more efficient than bounding every possible prime jump separately.

The small-height region can be handled without any distribution theorem. Since a progression contains at most $y/q+1$ integers up to $y$ and $\Lambda(n)\leq\log x$ for $n\leq x$, \[ \sum_{q\leq Q}\max_{(a,q)=1}\max_{2\leq y\leq Y}|E(y;q,a)| \ll Y\log^2(2x)+Q\log(2x)\qquad(Q\leq x). \tag{89.3} \] Here we used $\sum_{q\leq Q}1/q\ll\log(2Q)$ and $89.1$. The harmless modulus $q=1$ is included with $\varphi(1)=1$.

Proposition: the complete endpoint family suffices.

Suppose that for every $A'>0$ and $0<\theta'<1$, \[ \sum_{q\leq u^{\theta'}}\max_{(a,q)=1}|E(u;q,a)| \ll_{A',\theta'}u(\log u)^{-A'} \quad(u\to\infty). \tag{89.4} \] Then the catalogue's formulation, including the maximum over $y$, follows.

Proof. Fix its $A>0$ and $\theta<1$, and choose $\theta<\theta'<1$. Put $Q=x^\theta$, choose a fixed $B>A+3$, and set $h=Y=x/(\log x)^B$. By $89.3$, the region $y\leq Y$ contributes $O(x/(\log x)^{B-2}+x^\theta\log x)$, which is $O_{A,\theta}(x/(\log x)^A)$.

Cover $[Y,x]$ by a grid of $O((\log x)^B)$ points with gaps at most $h$, including both endpoints. Uniformly for its points $u\geq Y$, for all sufficiently large $x$ one has \[ x^\theta\leq u^{\theta'},\qquad \log u\asymp\log x. \] The first inequality follows because $x^{\theta'-\theta}/(\log x)^{B\theta'}\to\infty$. Thus $89.4$ applies to the whole desired modulus range at every grid point. Bound a maximum over grid points by their sum and use $89.2$. Taking $A'=A+B+2$, the total is \[ O((\log x)^B)\,O_{A',\theta'}\left(\frac{x}{(\log x)^{A'}}\right) +O(h\log(2Q)) \ll_{A,\theta}\frac{x}{(\log x)^A}. \] All choices were fixed before $x$ tends to infinity. This completes the proof. The reverse implication is immediate by taking $y=x$. $\square$

This equivalence uses the full family of exponents and arbitrary logarithmic savings. A result at one isolated endpoint, a fixed power saving in logarithms, or one distribution exponent without slack does not automatically give the same deduction. In particular, simply interchanging $\sum_q\max_y$ and $\max_y\sum_q$ would be invalid; the grid argument pays explicitly for the stronger order of operations.

The residue-class maximum has its own cost.

For each modulus, define the centered error \[ \Delta_q(a;y)=\psi(y;q,a)-\frac{\psi_q(y)}{\varphi(q)},\qquad \psi_q(y)=\sum_{\substack{n\leq y\$n,q)=1}}\Lambda(n). \] On the group of reduced residue classes, character orthogonality gives \[ \Delta_q(a;y)=\frac1{\varphi(q)} \sum_{\chi\ne\chi_0}\overline{\chi(a)}\,\psi(y,\chi), \quad \psi(y,\chi)=\sum_{n\leq y}\Lambda(n)\chi(n). \] Parseval therefore gives the exact energy identity \[ \mathcal V_q(y):=\sum_{(a,q)=1}|\Delta_q(a;y)|^2 =\frac1{\varphi(q)}\sum_{\chi\ne\chi_0}|\psi(y,\chi)|^2. \tag{89.5} \] The basic inequalities are \[ \max_a|\Delta_q(a;y)|\leq\sqrt{\mathcal V_q(y)},\qquad \sum_{q\leq Q}\max_a|\Delta_q(a;y)| \leq\sqrt{Q\sum_{q\leq Q}\mathcal V_q(y)}. \tag{89.6} \] Even an energy estimate of the natural scale $\sum_{q\leq Q}\mathcal V_q(y)\ll xQ(\log x)^C$ would only give $Q\sqrt x(\log x)^{C/2}$ by this argument. With $Q=x^\theta$, this has a power advantage over $x$ only when $\theta<1/2$. The displayed energy assumption is a diagnostic scale, not a theorem asserted uniformly in all the variables here.

Energy distributed evenly among $\varphi(q)$ classes would instead give a typical class scale smaller by roughly $\sqrt{\varphi(q)}$. Parseval alone cannot impose that distribution. A zero-sum vector with one coordinate equal to $M$ and the other $r-1$ coordinates equal to $-M/(r-1)$ has energy $M^2r/(r-1)$, so its maximum is comparable to the square root of its whole energy. Centering does not eliminate this concentration example.

Nor can the centering term be forgotten: \[ E(y;q,a)=\Delta_q(a;y)+\frac{\psi_q(y)-y}{\varphi(q)}. \tag{89.7} \] One has $0\leq\psi(y)-\psi_q(y)\leq\omega(q)\log y$, by summing the prime powers with base dividing $q$. The ordinary prime-number theorem and suitable bounds for this correction are additional pieces when using $89.5$ to study the original error. They cannot provide the missing residue-class cancellation in $89.6$.

A stronger sufficient estimate and the exact missing gain.

For comparison, suppose one could prove, for each fixed $\theta<1$, \[ \max_{(a,q)=1}\max_{2\leq y\leq x}|E(y;q,a)| \ll_\theta \sqrt{x/q}(\log x)^C \qquad(q\leq x^\theta), \tag{89.8} \] with a fixed $C$. Summing $q^{-1/2}$ shows the desired total is at most $O_\theta(x^{(1+\theta)/2}(\log x)^C)$, which beats $x/(\log x)^A$ for every fixed $A$. This proves a sufficient implication; it does not assert $89.8$. A bound of size $\sqrt x$ per modulus, even with small logarithmic losses, lacks the factor $q^{-1/2}$ and runs into the one-half barrier when summed directly.

Verification and outcome.

The companion exact checks verify the divisor identity behind $89.1$, the monotonicity bracketing for synthetic nonnegative arithmetic weights, and the concentrated zero-sum energy example. They test both class and height maxima separately. The nonnegative weights make the grid lemma applicable without pretending that a finite check proves distribution of the actual primes.

The proved endpoint-to-maximal equivalence removes a quantifier ambiguity, while $89.5$--$89.8$ quantify the arithmetic gain a successful proof would need. The first unresolved input is $89.4$ for every $\theta'<1$, or a comparably strong estimate with the required modulus and residue-class scope. Results for specially factored moduli and conditional zero correlations are not that input. Elliott--Halberstam remains UNRESOLVED.

Sources

• S1 Weisstein, E. W., Elliott-Halberstam Conjecture, From MathWorld - A Wolfram Resource, updated 16 August 2026. https://mathworld.wolfram.com/Elliott-HalberstamConjecture.html. Formulation source.
• S2 Pair Correlation of zeros of Dirichlet L-Functions: A possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery. https://arxiv.org/abs/2411.19762. Further reference; not a proof endorsement.
• S3 Primes in arithmetic progressions to large moduli III: Uniform residue classes. https://arxiv.org/html/2006.08250. Further reference; not a proof endorsement.

• E1 James Maynard, Primes in arithmetic progressions to large moduli III: Uniform residue classes, arXiv:2006.08250v1 (15 June 2020), Introduction and Theorems 1.1--1.3. https://arxiv.org/html/2006.08250v1. Version-specific identification of [S3]. Inspected for exact factorization, residue-class, and error-term restrictions; no unrestricted Elliott--Halberstam conclusion is inferred. Consulted 27 September 2026.
• E2 Neelam Kandhil, Alessandro Languasco, and Pieter Moree, Pair Correlation of zeros of Dirichlet $L$-Functions: A possible path towards the conjectures of Chowla, Elliott--Halberstam and Montgomery, arXiv:2411.19762v2 (8 November 2025). https://arxiv.org/abs/2411.19762v2. Version-specific scope check of [S2]: the stated implication assumes GRH and a pair-correlation conjecture. Neither assumption is proved here. Consulted 27 September 2026.