Large Sieve Inequalities for Exceptional Maass Forms and the Greatest Prime Factor of n^2+1

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dyadic_product_bound: Records the proof-stage assertion that the greatest prime factor of the dyadic product of n^2+1 is eventually at least x^1.30008.

theorem_1_1: For infinitely many positive integers n, the greatest prime factor of n^2+1 exceeds n^1.3.

theorem_1_5: Bounds the exceptional-spectrum large sieve sum for the phases e(n alpha), with a saving factor X governed by rational approximations to alpha, at least max(sqrt N, q/(a sqrt N)) uniformly in alpha.

theorem_1_7: Bounds the exceptional-spectrum large sieve sum for the smoothed dispersion-method coefficients counting h1 l1 - h2 l2 = n, for levels q >> L^2 and a saving factor X in an explicit range.

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Alexandru Pascadi, Large sieve inequalities for exceptional Maass forms and the greatest prime factor of $n^2+1$, Forum of Mathematics, Pi 14 (2026), e8, 1--54, DOI 10.1017/fmp.2026.10025.

Local artifacts. The selected file is the 54-page publisher version of record. The 51-page arXiv v3 alternate, identified by the watermark arXiv:2404.04239v3 [math.NT] 15 Jan 2026, was also read for this card and is not held. The VOR supplies the selected journal labels and pagination. The publisher version of record pascadi_2026_large_sieve_exceptional_maass_greatest_prime_factor.pdf prints on its first page "© The Author(s), 2026. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.": the Creative Commons Attribution 4.0 license. For the arXiv v3 alternate, the arXiv record names arXiv's non-exclusive distribution license (arXiv:2404.04239), every other right reserved.

VOR Theorem 1.1 on p. 3 states that, for infinitely many positive integers $n$,

$$P^+(n^2+1)>n^{1.3}.$$

This is an individual-value theorem with an infinitely-many quantifier. It is not an all-$n$ assertion.

Inside the proof, the paper also makes the stronger unnumbered dyadic-product assertion recorded in Dyadic product bound. With

$$P_x=P^+\!\left(\prod_{x\leq n\leq2x}(n^2+1)\right),$$

where the product is over integers and $x\geq1$ is real, the proof asserts

$$P_x\geq x^{1.30008}$$

for every sufficiently large $x$. In the VOR, Notation 6.7 inside Section 6.3 defines $P_x$ on p. 49; equation (6.20) and its implication are on p. 50; the final $o(1)$ estimate is on p. 51; and the exact $\bar\omega=1.30008$ endpoint is asserted on p. 52. There is no Section 6.7. The corresponding arXiv v3 locators are Theorem 1 on p. 2, Notation 23 and (6.20) on p. 47, the proof calculation on p. 49, and the endpoint on p. 50.

For the single polynomial $f(t)=t^2+1$, the dyadic assertion has an elementary compiler consequence for [[../wiki/problems/arithmetic_functions/E0976/_index|Problem 976]]. Setting $x=N/2$ for a sufficiently large integer $N$ makes the dyadic product a divisor of the initial product, so

$$F_{t^2+1}(N)\geq2^{-1.30008}N^{1.30008},$$

and hence $F_{t^2+1}(N)\gg N^{1.30008}$. Pascadi does not state this initial-product formulation. It gives no result here for an arbitrary irreducible polynomial and does not reach the degree-two target $N^2$. The author's remark that adapting the method to other irreducible quadratics should be possible is prospective only.

The endpoint $1.30008$ is the author's exact proof-stage assertion, not an independently recomputed numerical conclusion. The paper describes its supporting numerical inequality as barely true. The multidimensional integrals, their strict numerical margin, and the full analytic proof have not been independently validated in this compilation.

Source: https://doi.org/10.1017/fmp.2026.10025; alternate: https://arxiv.org/abs/2404.04239v3.

Bears on. #976, for the single polynomial $t^2+1$ only. Theorem 1.1 as printed gives $F_{t^2+1}(N)>N^{1.3}$ only along an infinite sequence of $N$; the bound $F_{t^2+1}(N)\gg N^{1.30008}$ for all large $N$ is the compiler deduction from the dyadic assertion inside the proof, described above. Neither reaches $N^2$ or concerns another polynomial. Theorems 1.5 and 1.7 bear on the problem only as inputs to Theorem 1.1.

Results.

• Theorem 1.1 (p. 3): for infinitely many positive integers $n$, $P^+(n^2+1)>n^{1.3}$.
• Dyadic product bound (pp. 49--52): the unnumbered eventual all-sufficiently-large-$x$ assertion inside the proof of Theorem 1.1 and its compiler-derived initial-product specialization for $t^2+1$.
• Theorem 1.5 (p. 5): the large sieve inequality for exceptional Maass forms with exponential phases $e(n\alpha)$, in a range of $X$ governed by rational approximations to $\alpha$.
• Theorem 1.7 (p. 5): the large sieve inequality for exceptional Maass forms with dispersion coefficients, for levels $q\gg L^2$.

The general Theorem 5.2 (p. 27), from which Theorems 1.5 and 1.7 are deduced, and the multilinear Kloosterman bounds of Section 5.3 are not transcribed.

Living verification. Needs review. The source identity, version map, displayed statements, and eventual quantifier were checked against the held VOR and the arXiv v3 copy read for this card. The elementary compiler bridge is not source-stated and was checked independently as a local deduction. The numerical integrals, their strict margin, and the full analytic proof have not been independently checked; no general-polynomial or status-transfer claim is accepted here.

Only the edition under an open license is held; the source's other editions are not, since no license on record permits their redistribution, and the card cites the edition it names above.