Character-phase evidence for Elliott--Halberstam and a sparse twisted-psi diagonal

Definitions and mathematical statement

Let $\Lambda$ be the von Mangoldt function and define \[ \psi(x;q,a)=\sum_{n\leq x,\ n\equiv a\ (q)}\Lambda(n),\qquad \psi(x,\chi)=\sum_{n\leq x}\Lambda(n)\chi(n). \] For $(a,q)=1$, character orthogonality gives the exact source identity \[ \psi(x;q,a)=\frac1{\varphi(q)}\sum_{\chi\bmod q}\overline{\chi(a)}\psi(x,\chi).\tag{1} \] One usual Elliott--Halberstam (EH) formulation asks, for every fixed $\delta>0$ and $A>0$, \[ \sum_{q\leq x^{1-\delta}}\max_{(a,q)=1} \left|\psi(x;q,a)-\frac{x}{\varphi(q)}\right| \ll_{A,\delta}\frac{x}{(\log x)^A}. \]

Workshop Problem 1: evidence for EH. J.B. Conrey asks what evidence supports EH and whether finding an "easy" case could yield bounded prime gaps through GPY; G. Harcos makes similar comments. R.C. Vaughan observes consistency with Barban--Davenport--Halberstam, and A. Granville notes support from the Cramér model. These are the December 2005 workshop's assessments and motivation.

Vaughan proposes exploiting cancellation among the phases $\overline{\chi(a)}$ in (1). Conrey asks whether asymptotics for \[ \sum_{\chi\bmod q}^{*}\chi(a)|L(1/2,\chi)|^2 \] cast doubt on such cancellation. K. Soundararajan replies that the central-$L$ phenomenon is explained by its Euler product and need not occur for $\psi(x,\chi)$; the source therefore leaves Vaughan's proposal viable. The star denotes primitive characters.

Reported random-phase evidence. The attached aid states that Barban--Davenport--Halberstam character energy yields an EH-strength bound with high probability after independent randomization of nonprincipal character phases. A model expression is \[ \frac1{\varphi(q)}\sum_{\chi\ne\chi_0}\xi_{q,\chi}\overline{\chi(a)}\psi(x,\chi), \] where $\xi_{q,\chi}$ are independent uniform unit-circle phases. The summary does not supply the concentration inequality, probability or parameter ranges, so a precise quantitative theorem cannot be recovered here. Randomized phase cancellation is evidence about a model; it is not cancellation of the actual deterministic phases.

Reported exact twisted-$\psi$ diagonal. Put $\varphi^*(q)=\#\{\text{primitive characters modulo }q\}$. For $2\leq a<q$, $(a,q)=1$, expand \[ \sum_{\chi\bmod q}^{*}\chi(a)|\psi(x,\chi)|^2 =\sum_{\substack{m,n\leq x\$mn,q)=1}}\Lambda(m)\Lambda(n) \sum_{d\mid q,\ d\mid(am-n)}\mu(q/d)\varphi(d), \] where $\mu$ is the Möbius function. The ordinary equality line $n=am$, not merely the congruence class, contributes exactly \[ \begin{cases} \varphi^*(q)(\log p)^2\max\{0,\lfloor\log_p x\rfloor-k\},&a=p^k\text{ is a prime power},\\ 0,&a\text{ is not a prime power}. \end{cases} \] Here $k\geq1$ and $(a,q)=1$ ensures $p\nmid q$. The aid describes this sparse von-Mangoldt diagonal as an explanation of why the long Euler-product diagonal in central-$L$ moments does not transfer to $\psi$. It does not estimate the remaining shifted terms or prove EH. The unrestricted EH conjecture remains open in the supplied research assessment; the historical bounded-gap motivation is not being restated as a current unknown.

Short English statement

The source asks for evidence that character-phase cancellation can support Elliott--Halberstam. The aid reports a random-phase model and an exact primitive-character calculation showing that the unshifted twisted-$\psi$ diagonal vanishes except for prime-power twists. Neither result controls the actual remaining correlations or establishes EH.

Sources

• Local UnsolvedMath registry, permanent ID $20000739$, catalogue entry; canonical aim-analytic-number-theory-notes.json, notes[102].
• Angel Kumchev's notes from the December 2005 ARCC workshop "Gaps Between Primes," Problem 1: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf. Supplied verification says the original PDF was inspected and only ligatures and vertically split sums need repair; no independent inspection is claimed here.
• Source discussion attributes comments to J.B. Conrey, G. Harcos, R.C. Vaughan, A. Granville and K. Soundararajan, and names GPY, Bombieri--Vinogradov, the large sieve, Barban--Davenport--Halberstam and the Cramér model. No complete references are supplied.
• Research attempt 1, partial_result, run status open, internal rigor proved, low novelty confidence; current registry partially solved. Explicitly machine-generated, not peer reviewed, and to be independently verified before citation. Original literature field empty.