Asymptotic formulae for primes represented by incomplete norm forms below the $n \ge 4k$ barrier

Maynard (Forum of Mathematics Pi 8 (2020), e3) proved that the incomplete norm form $N_{K/\mathbb{Q}}(\sum_{i=1}^{n-k} a_i \omega^{i-1})$ — the degree-$n$ norm form of $K=\mathbb{Q}(\omega)$ with the top $k$ coefficients set to zero — takes the expected asymptotic number of prime values when $n \ge 4k$, and takes infinitely many prime values, with a lower bound of the correct order of magnitude, when $K=\mathbb{Q}(\sqrt[n]{\theta})$ and $n \ge 22k/7$. The open problem is the gap between these two results: obtaining an asymptotic formula, rather than only a lower bound, for prime values of incomplete norm forms in the intermediate range $22k/7 \le n < 4k$, and any prime-infinitude result at all for general number fields below the $n \ge 4k$ barrier. A concrete first milestone is an asymptotic for $n > (2+\sqrt{2})k$ for pure-radical fields. The central technical obstacle is the absence of non-trivial bounds for the exponential sums arising from smoothed weights, equivalent to equidistribution results for highly skewed lattices.

Background

A central problem of analytic number theory is to determine which polynomials take prime values, and how often. Bunyakovsky's conjecture predicts that an irreducible integer polynomial with positive leading coefficient and no prime dividing all of its values takes infinitely many prime values; the Bateman-Horn conjecture, its standard quantitative strengthening, predicts that such a polynomial of degree $d$ represents primes about $x/(d\log x)$ times below $x$, up to an explicit Euler product over primes called the singular series; no case with $d>1$ is proved. A one-variable polynomial of degree $d>1$ has only $O(x^{1/2})$ values below $x$, thinner than any known prime-detecting method can reach, so attention turns to polynomials in several variables, where the set of values is larger.

Norm forms. Let $K=\mathbb{Q}(\omega)$ be a number field of degree $n$, where $\omega$ is a root of a monic irreducible polynomial $f \in \mathbb{Z}[X]$ of degree $n$. The field norm of the general element $X_1 + X_2\omega + \cdots + X_n\omega^{n-1}$ is a homogeneous degree-$n$ polynomial in $n$ variables with integer coefficients, called the norm form. It represents infinitely many primes: a prime ideal of the ring of integers of $K$ that is generated by a single element and has residue field of exactly $p$ elements forces that generator's norm to equal the rational prime $p$. Setting some coefficients of the norm form to zero produces an incomplete norm form; Maynard studies $N_K(\mathbf{a}) = N_{K/\mathbb{Q}}(\sum_{i=1}^{n-k} a_i\omega^{i-1})$, a degree-$n$ form in only $n-k$ variables. Vectors $\mathbf{a} \in [1,X]^{n-k}$ have norms of size about $X^n$, so the values of $N_K$ below $x$ number only about $x^{1-k/n}$ — a thin sequence of polynomial values, of density strictly below 1. Thin sequences are the hard regime: with many variables relative to the degree the Hardy-Littlewood circle method gives asymptotics (Birch's theorem, which establishes such asymptotics once the number of variables is sufficiently large compared with the degree), but for thin norm-form-type sequences only a handful of results exist. Friedlander and Iwaniec (1998) proved the expected asymptotic count of primes of the form $X^2+Y^4$ (the norm form of $\mathbb{Q}(i)$ with one variable specialized to a square; about $x^{3/4}$ values below $x$), Heath-Brown (2001) did the same for $X^3+2Y^3$ (the norm form of $\mathbb{Q}(\sqrt[3]{2})$ with the third variable set to zero; about $x^{2/3}$ values), and Heath-Brown and Li (2017) handled $a^2+p^4$. These remain essentially the only thin polynomial sequences known to capture their primes with an asymptotic formula; Merikoski (2023) has since added two further examples in the lower-bound sense.

Maynard's theorems. Maynard (2020) generalized Heath-Brown's setting to incomplete norm forms with a positive proportion of the coefficients set to zero, and proved: (Theorem 1.1) if $n \ge 4k$, then as $X \to \infty$ one has $\#\{\mathbf{a} \in [1,X]^{n-k} : N_K(\mathbf{a}) \text{ prime}\} = (S+o(1))\frac{X^{n-k}}{n\log X}$, where the singular series is $S=\prod_p (1-\nu(p)/p^{n-k})(1-1/p)^{-1}$ and $\nu(p)$ counts the solutions of $N_K(\mathbf{a}) \equiv 0 \pmod p$ in $[1,p]^{n-k}$, with all implied constants effectively computable — the count matches the Bateman-Horn prediction, since primality of a number of size $X^n$ has probability about $1/\log(X^n) = 1/(n\log X)$; and (Theorem 1.2) if $K=\mathbb{Q}(\sqrt[n]{\theta})$ with $X^n-\theta$ irreducible and $n \ge 22k/7$ (note $22/7 = 3.14\ldots < 4$), then $\#\{\mathbf{a} \in [1,X]^{n-k}: N_K(\mathbf{a}) \text{ prime}\} \gg S X^{n-k}/\log X$, that is, bounded below by a constant multiple of $S X^{n-k}/\log X$. An elementary sieve upper bound of the same order of magnitude shows this lower bound has the correct order, but no asymptotic is obtained.

Method. Both theorems are proved with Harman's sieve — a prime-detecting sieve that circumvents the parity barrier of classical combinatorial sieves (the phenomenon that a classical sieve cannot distinguish integers with an odd number of prime factors from those with an even number, and so cannot by itself detect primes) by repeatedly applying Buchstab's identity (a recursion relating integers with no small prime factors to those with exactly one) and estimating the resulting terms through two kinds of bilinear sums: Type I sums, where the sequence is averaged against an ideal divisor of small norm, and Type II sums, bilinear forms $\sum_m \alpha(m) \sum_n \beta(n)$ with both factors of intermediate size. Maynard's Type I/Type II estimates combine a Siegel-Walfisz-type estimate for Hecke $L$-functions (the $L$-functions attached to characters of the ideals of $K$, with a possible exceptional real character coming from a Siegel zero (a real zero of an $L$-function exceptionally close to $1$) treated explicitly) with geometry-of-numbers counting of lattice points in lattices that may be highly skewed, that is, far from round. A numerical verification in the sieve decomposition — the discarded remainder terms contribute a mass $\sum I_R$, verified numerically to be below $0.99$ (computed value $0.98977$) — is what fixes the threshold $22k/7$.

The gap and the barriers. Between the two theorems there is a gap in both strength and range: in $22k/7 \le n < 4k$ only a lower bound is known, and only for the special pure-radical fields $K=\mathbb{Q}(\sqrt[n]{\theta})$; for a general $K=\mathbb{Q}(\omega)$ nothing at all is known below $n \ge 4k$. Maynard states that the thresholds $4k$ and $22k/7$ are natural barriers to any simple argument used to establish Type I and Type II estimates: they are exactly the ranges in which the error terms of the bilinear sums can be bounded by their absolute values, without proving any cancellation, and Heath-Brown's asymptotic for the sparser sequence $X^3+2Y^3$ works precisely because he treats the error terms non-trivially. Two further author remarks shape the open problem. First, restricting the variable in the Type II Cauchy-Schwarz step to a single residue class modulo a suitable $Q$ has the potential to give an asymptotic formula for primes represented by an incomplete norm form of $\mathbb{Q}(\sqrt[n]{\theta})$ in the wider range $n > (2+\sqrt{2})k \approx 3.414k$, a refinement the author intended to address in a future paper. Second, introducing a smooth weight on the sums would allow exponential-sum techniques to widen the Type II ranges, but the author has not been able to get suitable control over the resulting exponential sums; non-trivially estimating these sums for general $n$ requires equidistribution results for skewed lattices.

Problem Statement

Close the gap between Maynard's Theorem 1.1 and Theorem 1.2 on prime values of incomplete norm forms, in both strength and range.

(1) Prove an asymptotic formula, rather than only a lower bound of the correct order of magnitude, for the number of prime values of the incomplete norm form $N_{K/\mathbb{Q}}(\sum_{i=1}^{n-k} a_i\omega^{i-1})$ in at least part of the range $n/k < 4$. For the pure-radical fields $K=\mathbb{Q}(\sqrt[n]{\theta})$ (with $X^n-\theta \in \mathbb{Z}[X]$ irreducible), establish

$$\#\{\mathbf{a} \in [1,X]^{n-k} : N_{K/\mathbb{Q}}(\textstyle\sum_{i=1}^{n-k} a_i \sqrt[n]{\theta}^{\,i-1}) \text{ prime}\} = (S+o(1))\frac{X^{n-k}}{n\log X}$$

as $X \to \infty$, with $S$ the singular series of Theorem 1.1 (namely $S=\prod_p (1-\nu(p)/p^{n-k})(1-1/p)^{-1}$, where $\nu(p)$ counts the solutions of $N_K(\mathbf{a}) \equiv 0 \pmod p$ in $[1,p]^{n-k}$) and effectively computable implied constants, in a range $n > ck$ for some explicit constant $c < 4$ strictly widening the known range $n \ge 4k$ — the author's flagged first milestone being $c = 2+\sqrt{2} \approx 3.414$ — and, at full strength, throughout the range $22k/7 \le n < 4k$, where Theorem 1.2 currently gives only the lower bound $\gg S X^{n-k}/\log X$ (a constant multiple of $S X^{n-k}/\log X$).

(2) Extend the results to general fields $K=\mathbb{Q}(\omega)$ ($\omega$ a root of any monic irreducible $f \in \mathbb{Z}[X]$ of degree $n$) below the $n \ge 4k$ barrier, where at present no prime-infinitude result for the incomplete norm form is known at all: either the asymptotic above or a lower bound of the correct order of magnitude.

Both deliverables serve one scientific objective: determining the primes-representing capability of incomplete norm forms across the $n/k$ range, with asymptotic precision wherever possible. The key technical obstacle identified in the source is common to both: establishing genuine cancellation in the Type II error terms — in particular, non-trivial bounds for the exponential sums that arise from smoothed weights, equivalently equidistribution results for highly skewed lattices. Secondary sharpenings recorded in the source (a quantitative bound for the $o(1)$ term of Theorem 1.1; a sharper constant in the Theorem 1.2 lower bound by reducing the discarded remainder mass currently only bounded via $\sum I_R < 0.99$; extension of the methods to other thin sequences beyond the norm-form setting) are refinements of the same objective and do not by themselves close it.

References

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