Products of prime ideals in ray class groups

Abstract: We prove that every class in the narrow ray class group modulo an integral ideal $\mathfrak q$ of a fixed number field is represented by a product of three prime ideals of norm at most $( N\mathfrak q)^{{\max(1,3α,4α_0)+κ}}$ for any $κ&gt;0$, where $α$ is the exponent in short character sum bounds for general non-principal ray class characters and $α_0$ comes from a bounded-order subconvexity input for Hecke $L$-functions. Wu's subconvexity bound gives the admissible choice $α=α_0=103/256$, hence the explicit bound $(N\mathfrak q)<sup>{103/64+κ}$. This improves the previous $O_K((N\mathfrak q)<sup>3)$-scale bound of Deshouillers, Gun, Ramaré, and Sivaraman. We also prove that a positive proportion of ray classes are represented by products of two prime ideals. The proof extends the multiplicative dense-model and transference framework of Matomäki--Teräväinen to narrow ray class groups.

• The paper proves every narrow ray class is a product of three prime ideals with norms at most Q^(103/64+κ), improving the previous cubic-scale bound.
• The paper establishes that at least 2/3−ε of ray classes have binary prime-ideal representations at the same scale, rising to 11/16−ε under a larger exponent.
• The paper adapts multiplicative dense-model transference, ideal-theoretic sieve methods, Hecke L-function estimates, and quadratic-character analysis to overcome coset obstructions over number fields.

This paper by Likun Xie extends the multiplicative dense-model and transference framework of Matomäki–Teräväinen from arithmetic progressions over the integers to narrow ray class groups of a fixed number field $K$. The central object is the set $E_k(X;\mathfrak q)$ of classes in the narrow ray class group $G=\operatorname{Cl}^{(\infty)}_{\mathfrak q}$ modulo an integral ideal $\mathfrak q$ that are representable as a product of $k$ prime ideals of norm at most $X$, where $Q=N\mathfrak q$. The main theorems establish ternary representation of every ray class and positive-proportion binary representation, with exponents governed by two analytic inputs: a short character-sum bound $\mathrm{CS}(\alpha)$ for general non-principal ray class characters, and a bounded-order subconvexity input $\mathrm L^{\mathrm b}(\alpha_0)$ for Hecke $L$-functions.

Main results

The paper proves two theorems under the assumptions $E_k(X;\mathfrak q)$0 and $E_k(X;\mathfrak q)$1 with $E_k(X;\mathfrak q)$2.

Ternary representation: for any fixed $E_k(X;\mathfrak q)$3 and $E_k(X;\mathfrak q)$4 sufficiently large,

$E_k(X;\mathfrak q)$5

Binary density: at the same scale, $E_k(X;\mathfrak q)$6, and at the larger scale $E_k(X;\mathfrak q)$7, one has $E_k(X;\mathfrak q)$8.

Using Wu's Burgess-type subconvexity bound for Hecke $E_k(X;\mathfrak q)$9-functions together with the admissible Ramanujan exponent $G=\operatorname{Cl}^{(\infty)}_{\mathfrak q}$0 of Blomer–Brumley, both inputs hold with $G=\operatorname{Cl}^{(\infty)}_{\mathfrak q}$1, yielding the explicit scale

$G=\operatorname{Cl}^{(\infty)}_{\mathfrak q}$2

This improves the previous bound of Deshouillers–Gun–Ramaré–Sivaraman, who required prime ideals of norm $G=\operatorname{Cl}^{(\infty)}_{\mathfrak q}$3 and restricted to unramified degree-one primes; the present theorem allows arbitrary prime ideals, and the restriction to degree-one primes can be recovered since primes of residue degree greater than one contribute only $G=\operatorname{Cl}^{(\infty)}_{\mathfrak q}$4 up to norm $G=\operatorname{Cl}^{(\infty)}_{\mathfrak q}$5.

The separation between $G=\operatorname{Cl}^{(\infty)}_{\mathfrak q}$6 and $G=\operatorname{Cl}^{(\infty)}_{\mathfrak q}$7 is deliberate. Since a Hecke character of order dividing a fixed $G=\operatorname{Cl}^{(\infty)}_{\mathfrak q}$8 has cube-free conductor away from finitely many primes, the Weyl-type subconvexity results of Balkanova–Frolenkov–Wu suggest that $G=\operatorname{Cl}^{(\infty)}_{\mathfrak q}$9 may be attainable. The author notes this refinement is not pursued here; conditionally on it, Theorems 1 and 2(i) would hold at the scale $\mathfrak q$0, while part (ii) depends on the term $\mathfrak q$1 and would require improving the general character-sum exponent.

Method: dense model and transference

The proof adapts the Green–Tao / Matomäki–Teräväinen transference architecture. A multiplicative dense model $\mathfrak q$2 is constructed for a weighted prime-indicator function $\mathfrak q$3 on ideals coprime to $\mathfrak q$4, using Coleman's Rosser–Iwaniec linear sieve over ideals, mean-value estimates for ray class characters (a large sieve inequality with bound $\mathfrak q$5), and Halász–Montgomery-type estimates driven by $\mathfrak q$6. The model satisfies Fourier-level approximation $\mathfrak q$7 with $\mathfrak q$8, preserves coset averages for subgroups of small index, and transfers convolution lower bounds on $\mathfrak q$9 back to genuine representations by prime ideals, with exceptional sets of size $k$0 in the binary case.

Product-set analysis and coset obstructions

On the finite abelian group side, Kneser's theorem and a popular-products lemma reduce matters to either a direct density conclusion or a structural case: the dense sets concentrate in $k$1 cosets of a subgroup $k$2 of index $k$3 with $k$4, and $k$5 equals the square of that coset union. The indices $k$6 and $k$7 are ruled out analytically via weighted prime-sum estimates: an explicit-formula argument (following Zaman's number-field version of Heath-Brown's inequality) shows that for bounded-order characters the real part of the weighted prime sum is at most $k$8, where $k$9 is the total weight with $X$0; elementary trigonometric computations in cyclic groups of orders $X$1 and $X$2 then force a positive proportion ($X$3) of prime mass outside the obstructing block of cosets, giving a dyadic escape interval used in the transference criterion.

The index-$X$4 obstruction is genuinely exceptional and requires an ideal-theoretic analogue of Matomäki–Teräväinen's quadratic argument. For a non-principal quadratic Hecke character $X$5 modulo $X$6, the paper develops:

• an asymptotic for $X$7 valid down to $X$8, together with a uniform divisibility version;
• a beta-sieve treatment showing the associated multiplicative function has sieve dimension $X$9, with singular series $Q=N\mathfrak q$0;
• a dichotomy lemma: either many primes satisfy $Q=N\mathfrak q$1 below $Q=N\mathfrak q$2, or there exists a dyadic block $Q=N\mathfrak q$3 with $Q=N\mathfrak q$4 such primes;
• a sharp Halász–Montgomery inequality weighted by $Q=N\mathfrak q$5, exploiting that off-diagonal characters $Q=N\mathfrak q$6 give twisted sums that cancel via a hyperbola estimate.

These feed into a quadratic transference proposition converting the dichotomy into representations $Q=N\mathfrak q$7 with $Q=N\mathfrak q$8 in the dense set, thereby covering the remaining coset $Q=N\mathfrak q$9 in the ternary theorem.

Binary representation

For the two-prime theorem, the key new ingredient is a multiplicative energy estimate: for suitable prime sets $\mathrm{CS}(\alpha)$0 (in a dyadic interval, concentrated in one coset) and $\mathrm{CS}(\alpha)$1 (large primes in another coset),

$\mathrm{CS}(\alpha)$2

proved by combining the linear sieve upper bound with the ray class large sieve and higher-moment bounds on prime sums. In the structural case, Cauchy–Schwarz applied against this energy bound yields new elements of $\mathrm{CS}(\alpha)$3 outside $\mathrm{CS}(\alpha)$4, in proportions $\mathrm{CS}(\alpha)$5 with $\mathrm{CS}(\alpha)$6, $\mathrm{CS}(\alpha)$7, $\mathrm{CS}(\alpha)$8. Since $\mathrm{CS}(\alpha)$9 and also exceeds $\mathrm L^{\mathrm b}(\alpha_0)$0 for all $\mathrm L^{\mathrm b}(\alpha_0)$1, the binary densities follow in both the direct and structural cases.

Limitations and open questions

Several caveats are stated explicitly. The constants depend ineffectively on $\mathrm L^{\mathrm b}(\alpha_0)$2 through Siegel-type lower bounds for $\mathrm L^{\mathrm b}(\alpha_0)$3 used in the quadratic arguments. The exponent $\mathrm L^{\mathrm b}(\alpha_0)$4 is limited by the current best general subconvexity and Ramanujan inputs; improving the scale of Theorem 2(ii) specifically requires a better general character-sum exponent $\mathrm L^{\mathrm b}(\alpha_0)$5, not merely better bounded-order subconvexity. The conjectural bounded-order input $\mathrm L^{\mathrm b}(\alpha_0)$6 — which would bring parts of the theorems to scale $\mathrm L^{\mathrm b}(\alpha_0)$7 — remains unverified, and the author leaves open whether a variant of the Balkanova–Frolenkov–Wu method yields it. Finally, full binary representation of every ray class (the number-field analogue of Erdős's original two-prime conjecture) is not addressed; only the proportion lower bounds $\mathrm L^{\mathrm b}(\alpha_0)$8 and $\mathrm L^{\mathrm b}(\alpha_0)$9 are obtained.

Conclusion

The paper establishes that every narrow ray class modulo $L$0 over a fixed number field is a product of three prime ideals of norm at most $L$1, improving the previous cubic-scale bound, and that a positive proportion of classes are products of two primes at comparable scales. Technically, its contribution is a complete port of the multiplicative dense-model transference machinery to the ray class group setting, including the ideal-theoretic sieve inputs, the trigonometric elimination of index-$L$2 and index-$L$3 coset obstructions via bounded-order subconvexity, and a self-contained treatment of the exceptional quadratic case. The framework is modular in the analytic inputs, so further progress on Hecke $L$4-function subconvexity translates directly into smaller exponents in the representation theorems.