Quantitative averages of function-field prime-tuple singular series

Definitions and mathematical statement

The AIM item asks whether averages of Hardy--Littlewood constants can be computed in the function-field setting, followed by the word "Yes". It specifies no family, normalization or limit. The supplied report explicitly reconstructs one reading, without asserting it is the only intended one.

Fix a prime power $q$ and an integer tuple size $k\geq1$. Let $B_m=\{h\in{\mathbb{F}}_q[T]:\deg h<m\}$, including zero, with $N_m=q^m$. For an ordered pairwise-distinct tuple $\mathbf h=(h_1,\ldots,h_k)$, let $\nu_P(\mathbf h)$ be the number of its distinct residue classes modulo a monic irreducible $P$, and put $Q_P=q^{\deg P}$. The prime-tuple singular series in this normalization is \[ \mathfrak S(\mathbf h)=\prod_P\frac{1-\nu_P(\mathbf h)/Q_P}{(1-1/Q_P)^k}. \] A tuple is admissible when $\nu_P<Q_P$ for every $P$; otherwise its series is zero. Pairwise distinctness matters for the full product.

The report claims the quantitative degree-box mean \[ q^{-mk}\sum_{\substack{\mathbf h\in B_m^k\\h_i\ne h_j\ (i\ne j)}}\mathfrak S(\mathbf h) =1+O_{q,k}\left(1/\log(m+2)\right), \] and the same estimate when normalized instead by $(N_m)_k=N_m(N_m-1)\cdots(N_m-k+1)$. These are fixed-$q$, fixed-$k$, large-$m$ statements for the full series, not just a finite truncation.

For a finite set of primes, independent uniform residue tuples give an exact truncated mean of one by CRT. The associated local occupancy formula, for an integer moment $a\geq1$ and $Q=Q_P$, is \[ {\mathbb{E}}\left[\left(\frac{1-\nu_P/Q}{(1-1/Q)^k}\right)^a\right] =(1-1/Q)^{-ka}\sum_{j=1}^{\min(k,Q)} \frac{(Q)_j\left\{\begin{smallmatrix}k\\j\end{smallmatrix}\right\}}{Q^k}(1-j/Q)^a, \] where the braces are Stirling numbers of the second kind. This residue model permits repeated residues; exact CRT averaging does not by itself prove the distinct-tuple full-tail estimate.

The report also gives an $L^2$ profinite residue-product model with mean one. In that model the admissibility probability is the finite product \[ p_{\mathrm{adm}}(q,k)=\prod_{Q_P\leq k} \left(1-\frac{Q_P!\left\{\begin{smallmatrix}k\\Q_P\end{smallmatrix}\right\}}{Q_P^k}\right)>0. \] It claims the corresponding admissibility-conditioned limiting mean is $1/p_{\mathrm{adm}}$, not one. Finally it reports exponential convergence of the pair average, without supplying its exact rate or formula. The finite CRT identity, repeated-tuple treatment and full infinite-product tail are separate steps; detailed proofs and authoritative references are absent.

Short English statement

Compute averages of function-field Hardy--Littlewood constants after specifying a family. The report chooses fixed-field additive degree boxes and distinct ordered shifts, claiming full singular-series mean one with a quantitative error. It separates exact finite CRT moments from infinite-tail estimates and gives a different mean after conditioning on admissibility.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent ID $20000408$, source code catalogue entry; original item, explicit-reconstruction qualification and supplied research summary. Statement status reconstructed and unverified.
• AIM workshop "Arithmetic statistics over finite fields and function fields", Question 30, attributed as "Rubenstein" in the original: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf. Canonical location aim-algebraic-number-theory-notes.json, notes[59]; original remarks and literature field empty. No PDF was opened here.
• No full authoritative reference is supplied for the degree-box mean, occupancy/profinite construction or pair-average rate. The clean-statement field also contains the variant spelling "Rubinstein"; no identity expansion is guessed.
• Supplied machine-generated research attempt 1: partial result; problem solved at the run and in the registry, classification SOLVED-IN-LITERATURE; rigor metadata "proved", novelty confidence low. The non-peer-reviewed theorem package requires independent verification before citation.