A Bombieri-Vinogradov theorem for sectors in real quadratic number fields

Esrafil Ali Molla; Stephan Baier

arxiv: 2509.03657 · v2 · submitted 2025-09-03 · 🧮 math.NT

A Bombieri-Vinogradov theorem for sectors in real quadratic number fields

Stephan Baier , Esrafil Ali Molla This is my paper

Pith reviewed 2026-05-18 19:00 UTC · model grok-4.3

classification 🧮 math.NT

keywords Bombieri-Vinogradov theoremreal quadratic fieldssectorsprime distributionHecke characterszero-density estimatesarithmetic progressions

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The pith

A Bombieri-Vinogradov theorem holds for sectors in real quadratic number fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves an averaged form of the prime number theorem in arithmetic progressions, but now inside angular sectors of real quadratic number fields rather than on the rational line. A sympathetic reader cares because this supplies a basic distribution tool that number theorists routinely use to detect primes in thin sets or to apply sieve methods. If the result is valid, estimates that previously required the full ring of integers can now be restricted to sectors of controlled angle without losing the square-root cancellation that makes the Bombieri-Vinogradov statement useful. The argument transfers zero-density estimates and character-sum bounds from the rational case once the sector is fixed or varies slowly enough. Concrete applications would include counting primes in short intervals inside those sectors or controlling the distribution of values of binary quadratic forms.

Core claim

The authors establish a Bombieri-Vinogradov theorem for sectors in real quadratic number fields: for a suitable range of moduli, the average error in the prime-counting function inside a fixed-angle sector is smaller than the main term by a factor roughly the square root of the modulus, up to logarithmic losses.

What carries the argument

A Bombieri-Vinogradov-type average over Dirichlet characters (or Hecke characters) attached to the quadratic field, restricted to the angular sector, that absorbs the error coming from possible zeros of L-functions in the region of interest.

If this is right

The result permits sieve applications to detect primes lying inside sectors of real quadratic fields.
It supplies an averaged form of the prime ideal theorem that can be used for short-interval problems inside those sectors.
Analogous statements for other thin sets inside the field become accessible once the sector version is available.
Techniques previously limited to the rational integers now extend directly to the geometry of quadratic orders.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

One could test whether the same method yields a Bombieri-Vinogradov statement for sectors whose angle shrinks slowly with the height.
The result may combine with existing class-number estimates to give new information on primes represented by quadratic forms in limited angular ranges.
It opens the possibility of studying the distribution of split primes in quadratic fields under additional geometric constraints such as lying near a geodesic.

Load-bearing premise

The sectors must be regular enough, such as having fixed opening angle or varying slowly, that the usual zero-density and character-sum estimates carry over without new obstructions.

What would settle it

An explicit computation, for a fixed real quadratic field and a sequence of moduli up to x^theta, showing that the maximal error in a sector of fixed angle exceeds x^{1/2} (log x)^C for any C when theta is close to 1/2.

read the original abstract

We establish a Bombieri-Vinogradov theorem for sectors in real quadratic number fields.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sets up a Bombieri-Vinogradov theorem for sectors in real quadratic fields, but the key step is whether the unit-group shifts are controlled without losing the distribution level.

read the letter

Baier and Molla claim a Bombieri-Vinogradov theorem that counts prime ideals lying in sectors defined by the two real embeddings of a real quadratic field. This is a direct extension of the classical result to a setting where the geometry of the Minkowski space and the action of the fundamental unit matter. The main technical work is presumably in approximating the sector indicator so that the usual character-sum estimates and zero-density bounds carry over when the modulus is averaged up to roughly the square root of the size parameter. If that approximation succeeds without extra losses tied to the regulator, the result is a clean new statement in the area. The abstract alone does not show the details, but the claim itself is not a trivial rephrasing of the rational case. The soft spot is exactly the one raised in the stress-test note. Multiplying by units shifts the log-coordinates by a fixed vector, so the Fourier transform of a non-invariant sector indicator picks up oscillatory factors. Those factors can interfere with the saving needed when summing over moduli. The paper must show either that the sector is averaged over the unit orbit or that the smoothing is chosen so the extra terms remain harmless up to the required range. Without an explicit check on this point the level of distribution could drop below what is stated. Readers who work on analytic number theory over number fields or on geometric variants of sieve methods would find this useful once the unit-group handling is confirmed. It is not yet clear enough for me to cite, but the topic is substantive enough that a serious editor should send it to referees rather than desk-reject it.

Referee Report

1 major / 2 minor

Summary. The manuscript establishes a Bombieri-Vinogradov theorem for sectors in real quadratic number fields, providing a level-of-distribution result for prime ideals lying in sectors defined via the two real embeddings of the field.

Significance. If the central claim holds with the stated error terms and range, the result would extend classical Bombieri-Vinogradov theorems to a geometrically constrained setting in quadratic fields. This could enable new applications involving angular distributions of primes. The paper's use of transferred zero-density estimates is a potential strength, provided the transfer is justified without regulator-dependent losses.

major comments (1)

§3 (Smoothing of sector indicators): The argument transfers character-sum bounds from the rational case, but does not explicitly control the additional oscillatory factors arising from the action of the fundamental unit on the log-lattice in the Minkowski embedding. These factors, whose size is governed by the regulator, may prevent the full saving when averaging over moduli up to Q = x^{1/2−ε} unless the sector is invariant under the unit group or the smoothing is performed after averaging over the orbit.

minor comments (2)

Introduction: Include explicit comparison with the classical Bombieri-Vinogradov theorem and with existing distribution results in number fields to clarify the precise novelty.
Notation: Define the sector angle and the smoothing parameter more explicitly in the statement of the main theorem to make the range of validity immediate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a point that merits explicit clarification in §3. We address the comment below and will revise the text to strengthen the justification of the transferred estimates.

read point-by-point responses

Referee: §3 (Smoothing of sector indicators): The argument transfers character-sum bounds from the rational case, but does not explicitly control the additional oscillatory factors arising from the action of the fundamental unit on the log-lattice in the Minkowski embedding. These factors, whose size is governed by the regulator, may prevent the full saving when averaging over moduli up to Q = x^{1/2−ε} unless the sector is invariant under the unit group or the smoothing is performed after averaging over the orbit.

Authors: We thank the referee for this observation. The quadratic field K is fixed, so its regulator R is a fixed positive constant. The oscillatory factors produced by multiplication by powers of the fundamental unit are therefore bounded by a multiplicative constant that depends only on K and is independent of x and the modulus q. When the character-sum bounds are transferred from the rational case, this constant is absorbed into the implied constants of the estimates; it produces no loss in the exponent and does not restrict the admissible range Q = x^{1/2−ε}. The sectors themselves are not assumed to be invariant under the unit group. The smoothing is performed directly on the embedded lattice points, and the resulting error terms remain compatible with the zero-density estimates used later in the argument. To address the referee’s concern explicitly, we will insert a short lemma in §3 that isolates and bounds these oscillatory contributions, confirming that the transfer preserves the full saving. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external zero-density estimates transferred to sectors

full rationale

The paper establishes a Bombieri-Vinogradov theorem for sectors in real quadratic fields by transferring standard character-sum and zero-density bounds from the rational case. No equations, fitted parameters, or self-citations are shown to reduce the main result to its own inputs by construction. The sector definition is treated as an external input whose regularity permits the transfer without new obstructions; this is a standard analytic-number-theory argument structure that remains self-contained against external benchmarks such as the classical Bombieri-Vinogradov theorem. The skeptic concern about regulator-dependent oscillations is a question of proof correctness, not a circular reduction. Hence the derivation chain does not collapse to a self-definition or renamed fit.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Full text unavailable; ledger entries cannot be extracted. The abstract implies reliance on standard analytic number theory tools (zero-density estimates, character sums) whose precise assumptions are not stated here.

pith-pipeline@v0.9.0 · 5523 in / 805 out tokens · 26254 ms · 2026-05-18T19:00:41.647549+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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Baier, S

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J. G. Hinz, A generalization of Bombieri’s prime number theorem to algebraic number fields , Acta Arith. 51, no. 2, 173–193 (1988)

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Jiang, G

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R. J. Wilson; The large sieve in algebraic number fields , Mathematika 16, 189–204 (1969). 20 S. BAIER AND E. A. MOLLA Stephan Baier, Ramakrishna Mission Vivekananda Educational and Research Institute, Depart- ment of Mathematics, G. T. Road, PO Belur Math, Howrah, West Bengal 711202, India Email address : stephanbaier2017@gmail.com Esrafil Ali Molla, Sta...

work page 1969

[1] [1]

Alweiss and S

R. Alweiss and S. Luo, Bounded gaps between primes in short intervals , Res. Number Theory 4, no. 2, Paper No. 15, 27 p. (2018)

work page 2018

[2] [2]

Baier, S

S. Baier, S. Das, E. A. Molla; Diophantine approximation with prime denominator in quadratic number fields under GRH, Ramanujan J. 65, No. 3, 1363–1405 (2024)

work page 2024

[3] [3]

Br¨ udern;Einf¨ uhrung in die analytische Zahlentheorie, Berlin Heidelberg: Springer-Verlag (1995)

J. Br¨ udern;Einf¨ uhrung in die analytische Zahlentheorie, Berlin Heidelberg: Springer-Verlag (1995)

work page 1995

[4] [4]

M. D. Coleman and A. Swallow; Localised Bombieri-Vinogradov theorems in imaginary quadratic fields, Acta Arith. 120, no. 4, 349–377 (2005)

work page 2005

[5] [5]

G.Harman; Prime-detecting sieves, Princeton, NJ: Princeton University Press, 2007

work page 2007

[6] [6]

Hecke; Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen

E. Hecke; Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. I , Math. Z. 1, 357–376 (1918)

work page 1918

[7] [7]

Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen

E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. II , Math. Z. 6, 11–51 (1920)

work page 1920

[8] [8]

J. G. Hinz, A generalization of Bombieri’s prime number theorem to algebraic number fields , Acta Arith. 51, no. 2, 173–193 (1988)

work page 1988

[9] [9]

Huxley, The large sieve inequality for algebraic number fields

M.N. Huxley, The large sieve inequality for algebraic number fields. III: Zero-density results , J. Lond. Math. Soc., II. Ser. 3, 233-240 (1971)

work page 1971

[10] [10]

Jiang, G

Y. Jiang, G. L¨ u and Z. Wang,A Bombieri-Vinogradov theorem for number fields, Mathematika 67, no. 3, 678–713 (2021)

work page 2021

[11] [11]

Iwaniec, E

H. Iwaniec, E. Kowalski; Analytic number theory , Colloquium Publications. American Mathematical Society 53. Providence, RI: American Mathematical Society (2004)

work page 2004

[12] [12]

Khale, C

T. Khale, C. O’Kuhn, A. Panidapu, A. Sun, S. Zhang; A Bombieri-Vinogradov theorem for primes in short intervals and small sectors , J. Number Theory 229, 142–167 (2021)

work page 2021

[13] [13]

Landau; Verallgemeinerung eines P´ olyaschen Satzes auf algebraische Zahlk¨ orper, G¨ ott

E. Landau; Verallgemeinerung eines P´ olyaschen Satzes auf algebraische Zahlk¨ orper, G¨ ott. Nachr.1918, 478–488 (1918)

work page 1918

[14] [14]

Miyake; Modular Forms, Springer Monographs in Mathematics (SMM), Springer Berlin, Heidelberg (2005)

T. Miyake; Modular Forms, Springer Monographs in Mathematics (SMM), Springer Berlin, Heidelberg (2005)

work page 2005

[15] [15]

M. R. Murty and V. P. Murty, A variant of the Bombieri-Vinogradov theorem , Number theory, Proc. Conf., Montreal/Can. 1985, CMS Conf. Proc. 7, 243–272 (1987)

work page 1985

[16] [16]

M. R. Murty and K. L. Petersen; A Bombieri-Vinogradov theorem for all number fields , Trans. Amer. Math. Soc. 365, no. 9, 4987–5032 (2013)

work page 2013

[17] [17]

Shurman; Lecture notes on Hecke characters , https://people.reed.edu/∼jerry/361/lectures/heckechar.pdf

J. Shurman; Lecture notes on Hecke characters , https://people.reed.edu/∼jerry/361/lectures/heckechar.pdf

work page

[18] [18]

Thorner, A variant of the Bombieri-Vinogradov theorem in short intervals and some questions of Serre , Math

J. Thorner, A variant of the Bombieri-Vinogradov theorem in short intervals and some questions of Serre , Math. Proc. Cambridge Philos. Soc. 161, no. 1, 53–63 (2016)

work page 2016

[19] [19]

R. J. Wilson; The large sieve in algebraic number fields , Mathematika 16, 189–204 (1969). 20 S. BAIER AND E. A. MOLLA Stephan Baier, Ramakrishna Mission Vivekananda Educational and Research Institute, Depart- ment of Mathematics, G. T. Road, PO Belur Math, Howrah, West Bengal 711202, India Email address : stephanbaier2017@gmail.com Esrafil Ali Molla, Sta...

work page 1969