On the Erd\H{o}s--Odlyzko--S\'ark\"ozy conjecture for prime-power moduli

Igor E. Shparlinski; William Banks

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On the Erd\H{o}s--Odlyzko--S\'ark\"ozy conjecture for prime-power moduli

William Banks, Igor E. Shparlinski

Number Theory · Submitted 14 Oct 2026 · 13 references

Pith's verdict: Solid, honest paper that gives the first effective subpolynomial bound for the third prime in the ternary EOS conjecture for prime-power moduli, plus a quantitative bound on missing classes in the binary case.

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arXiv ID2610.19597 (version 1)

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CitationIgor E. Shparlinski, William Banks. "On the Erd\H{o}s--Odlyzko--S\'ark\"ozy conjecture for prime-power moduli." arXiv:2610.19597 (2026). https://pith.science/paper/BVTN4TT5

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Editor's note

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This is a straightforwardly good paper. The novelties are real: for prime-power moduli they improve the Klurman–Mangerel–Teräväinen result by making the third prime subpolynomially small with an explicit exponent, and they give the first quantitative bound on the proportion of missing classes in the binary case. The conditional results for general moduli are neat but not the main draw.

The proofs are rigorous and clearly written. The key input is the wide zero-free region from the authors' own previous work (Banks–Shparlinski) combined with Iwaniec's region for smaller conductors. That reliance is legitimate: the cited theorems are published and specialized. The paper scrupulously states its assumptions (large γ, prime-power moduli, dependence on p) and marks the conditional sections. No overclaiming, no sleight of hand.

The only real limitation is the scope: it is still only for prime-power moduli, and the constants depend on the fixed prime p. The unconditional results also require γ large enough—γ₀ exists but is not computed explicitly, though the authors note it can be. These are acknowledged limitations, not flaws. I checked the arithmetic in the proof of Lemma 2.1 and the ratio of widths; the stress-test note is correct that the Iwaniec region is indeed wider for large q. The central argument holds up.

This is a paper for specialists in analytic number theory working on the EOS conjecture or character sums with prime-power moduli. It deserves a serious referee and will be accepted with minor revisions at most.

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Major objections (2)

[§2.2, Lemma 2.1] The paper relies on the zero-free region of [2, Theorem 12] which is published and specialized to prime-power moduli. The combination with Iwaniec's region and the elimination of real zeros is correctly argued. The only subtle point is that the Iwaniec region (2.5) originally excludes real zeros for real characters, but the paper handles this by noting that only finitely many real primitive characters exist for each prime p and then using that L(1,χ)≠0 to exclude zeros in R_q. This argument is sound, but the numerical bounds would be clearer if the authors stated explicitly that γ₀ is chosen so that the inequalities on lines 5-6 of p.5 hold simultaneously with the requirement that η_q ≤ 1/2. The present exposition is acceptable for a research journal.
[§3.1, proof of Theorem 1.1] The derivation of the main term from the principal character uses (2.1) and the inequality c√log x ≥ ½ η_q log q• for x = q, q•. This step is justified because η_q decays like (log q)^{-2/3}, and c√log x dominates ½ η_q log q• for sufficiently large γ (equivalently q). The constants C₁ and c₁ are chosen explicitly (C₁ = 8A₀^{-1}, c₁ = ½ A₀). The argument is correct and complete.

Minor objections (4)

[§2.2, Lemma 2.1] The constant γ₀ is not computed numerically; it is stated to exist. This is standard for such results and does not affect the correctness, but the authors could remark that γ₀ depends on p and can be made explicit if needed.
[§2.2, proof of Lemma 2.1] In the second-to-last paragraph of the proof, the phrase 'Thus every point of R_q satisfies (2.5) and so the only zeros of L(s,X) that can lie in R_q are real zeros...' is precise. However, it may be helpful to note that (2.5) does not exclude real zeros for real characters, and that the subsequent argument handles this case. The paper does this clearly.
[§3.1, p.8] Equation (3.1) writes q^{η_q/2}_• ≥ exp(½ A₀ C₁ log log q) = (log q)^4. The arithmetic is correct given C₁ = 8A₀^{-1} and η_q = A₀/((log q)^{2/3}(log log q)^{1/3}). The reader may wish to verify the exponent: ½ η_q log q• = ½ A₀/((log q)^{2/3}(log log q)^{1/3}) * C₁ (log q)^{2/3}(log log q)^{4/3} = ½ A₀ C₁ log log q. With C₁ = 8A₀^{-1}, this equals 4 log log q, giving (log q)^4. The computation is correct.
[Acknowledgments] The disclosure of AI assistance (Claude) for writing, literature searches, and proof checking is commendably transparent and conforms to current best practices.

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The paper

The paper claims

For prime-power moduli, every reduced residue class is a product of three primes, with a subpolynomial bound on the third prime.

The reading

The Erdős–Odlyzko–Sárközy conjecture states that every reduced residue class modulo q contains a product of two primes not exceeding q. For moduli that are prime powers q=p^γ, the authors establish strong results for representations with three primes: an unconditional asymptotic formula showing that every residue class can be expressed as ℓ1ℓ2ℓ3 with ℓ1,ℓ2 ≤ q and ℓ3 subpolynomially small. They also bound the number of classes not representable by two primes, showing it is at most φ(q) exp(−c(log q)^{1/3}(log log q)^{-1/3}. Both results rely on a wide zero-free region for Dirichlet L-functions modulo prime powers, which excludes exceptonal zeros.

What carries the argument

Lemma 2.1 provides a wide zero-free region R_q for Dirichlet L-functions L(s,χ) attached to nonprincipal characters modulo q=p^γ: no zeros in the region where σ > 1 − η_q with η_q = A_0/((log q)^{2/3}(log log q)^{1/3}) and |t| ≤ T_q = exp((log q)^{7/8}). This region has no exceptionnal zeros. It combines the authors' earlier result [2] for large conductors with Iwaniec's theorem [4] for smaller ones. From this, Proposition 2.4 derives strong bounds on ψ(x,χ) and ϑ(x,χ), which are the engine for the asymptotic formulas.

Extended reading notes

Core claim

Theorem 1.1: For any fixed prime p, there exist constants γ1, C1, c1 >0 such that for q=p^γ with γ≥γ1 and any q_• ≤ q satisfying log q_• ≥ C1 (log q)^{2/3} (log log q)^{4/3}, the weighte sum over primes ℓ1,ℓ2≤q, ℓ3≤q_• with product ≡ a (mod q) equals ($q^{2}$ q_•/φ(q))(1 + O(exp(−c1 log q_•/((log q)^{2/3}(log log q)^{1/3})))). Hence for sufficiently large γ, every a modulo q is a product ℓ1ℓ2ℓ3 with ℓ1,ℓ2≤q and ℓ3 ≤ exp(C1(log q)^{2/3}(log log q)^{4/3}) = $q^{{o(1)}}$. Theorem 1.2: The number of reduced classes not in EOS2(q) is at most φ(q) exp(−C2(log q)^{1/3}(log log q)^{-1/3}).

Load-bearing premise

The entire unconditional analysis for prime-power moduli rests on Lemma 2.1, which asserts that for q=p^γ with γ≥γ0, the Dirichlet L-functions for nonprincipal characters have no zeros in a specific wide region near σ=1. If a zero existed in that region for some nontrivial character, the bounds on ψ(x,χ) and ϑ(x,χ) in Proposition 2.4 would fail, and the asymptotic formulas would not follow.

Editorial extensions

If this is right

For prime-power moduli with exponent γ sufficiently large, the ternary Erdős–Odlyzko–Sárközy conjecture holds: every reduced residue class contains a product of three primes ≤ q.
The third prime in such a representation can be chosen as small as exp(C(log q)^{2/3}(log log q)^{4/3}), which is q^{o(1)} as γ→∞.
The binary case misses at most φ(q) exp(−C(log q)^{1/3}(log log q)^{-1/3}) classes, a nearly full coverage.
Under Q RH[q, σ0], the ternary case for general moduli requires third prime only O((log q)^{2/(1−σ0)}), and the binary case misses O(φ(q) q^{−2(1−σ0)} (log q)^3) classes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The width of the zero-free region for prime-power moduli is far larger than the classical Deuring–Heilbronn region, suggesting that character sums for such moduli might find applications in other number-theoretic problems requiring cancellation, such as the least primitive root or small prime gaps in arithmetic progressions.
The unconditional bound on the third prime in Theorem 1.1 is likely not optimal; a similar approach with a stronger zero-free region could potentially reduce the exponent (2/3) to (1/2) or lower, matching the conjectural boundaries of the Korobov–Vinogradov type bounds.
The techniques might extend to moduli that are products of a bounded number of prime powers, provided that zero-free regions can be combined across different prime factors.

Original abstract

Erd\H{o}s, Odlyzko and S\'ark\"ozy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$. O. Klurman, A. P. Mangerel and J. Ter\"av\"ainen (2023) have shown that three primes suffice when $q$ is sufficiently smooth. We study the moduli $q=p^\gamma$ that are powers of a fixed prime $p$ and we show that the third prime can be taken subpolynomially small. For any $q_\bullet\le q$ for which $q_\bullet\ge\exp\bigl(C(\log q)^{2/3}(\log\log q)^{4/3}\bigr)$, we give an asymptotic formula for the weighted number of representations of each reduced class as a product of two primes not exceeding $q$ and a third not exceeding~$q_\bullet$. We also show that the conjecture holds for all but a proportion $\exp\bigl(-c(\log q)^{1/3}(\log\log q)^{-1/3}\bigr)$ of the reduced classes. Here $C$ and $c$ are positive constants that depend only on $p$. Both results are unconditional and they rely on a wide zero-free region with no exceptional zeros for the Dirichlet $L$-functions attached to characters modulo $p^\gamma$.We also present conditional versions of these results for arbitrary $q$ under a quasi-Riemann hypothesis.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.79 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.90 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Erd\H{o}s, Odlyzko and S\'ark\"ozy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$. O. Klurman, A. P. Mangerel and J. Ter\"av\"ainen (2023) have shown that three primes suffice when $q$ is sufficiently smooth. We study the moduli $q=p^\gamma$ that are powers of a fixed prime $p$ and we show that the third prime can be taken subpolynomially small. For any $q_\bullet\le q$ for which $q_\bullet\ge\exp\bigl(C(\log q)^{2/3}(\log\log q)^{4/3}\bigr)$, we give an asymptotic formula for the weighted number of representations of each reduced class as a product of two primes not exceeding $q$ and a third not exceeding~$q_\bullet$. We also show that the conjecture holds for all but a proportion $\exp\bigl(-c(\log q)^{1/3}(\log\log q)^{-1/3}\bigr)$ of the reduced classes. Here $C$ and $c$ are positive constants that depend only on $p$. Both results are unconditional and they rely on a wide zero-free region with no exceptional zeros for the Dirichlet $L$-functions attached to characters modulo $p^\gamma$.We also present conditional versions of these results for arbitrary $q$ under a quasi-Riemann hypothesis.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Erd\H{o}s, Odlyzko and S\'ark\"ozy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$. O. Klurman, A. P.	background 1.00	0.04	0.03	speculative 0.73
s1	Mangerel and J. Ter\"av\"ainen (2023) have shown that three primes suffice when $q$ is sufficiently smooth.	background 1.00	0.03	0.04	asserted 1.00
s2	We study the moduli $q=p^\gamma$ that are powers of a fixed prime $p$ and we show that the third prime can be taken subpolynomially small.	contribution 0.79	0.96	0.66	asserted 1.00
s3	For any $q_\bullet\le q$ for which $q_\bullet\ge\exp\bigl(C(\log q)^{2/3}(\log\log q)^{4/3}\bigr)$, we give an asymptotic formula for the weighted number of representations of each reduced class as a product of two primes not exceeding $q$ and a third not exceeding~$q_\bullet$.	result 0.95	0.95	0.39	asserted 1.00
s4	We also show that the conjecture holds for all but a proportion $\exp\bigl(-c(\log q)^{1/3}(\log\log q)^{-1/3}\bigr)$ of the reduced classes.	result 1.00	0.95	0.27	asserted 1.00
s5	Here $C$ and $c$ are positive constants that depend only on $p$.	background 0.32	0.46	0.03	asserted 0.99
s6	Both results are unconditional and they rely on a wide zero-free region with no exceptional zeros for the Dirichlet $L$-functions attached to characters modulo $p^\gamma$.We also present conditional versions of these results for arbitrary $q$ under a quasi-Riemann hypothesis.	method 0.88	0.89	0.14	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

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full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (13)

[1]

W. D. Banks and I. E. Shparlinski, Bounds on short character sums andL-functions with characters to a powerful modulus.J. Anal. Math.139 (2019), 239–263. (p. 2)

View on Pith 2019
[2]

W. D. Banks and I. E. Shparlinski, Sums with the M¨ obius function twisted by characters with powerful moduli.Trans. Amer. Math. Soc.373 (2020), 249–272. (pp. 2 and 4)

View on Pith 2020
[3]

P. Erd˝ os, A. Odlyzko, and A. S´ ark¨ ozy, On the residues of products of prime numbers.Period. Math. Hungar.18 (1987), 229–239. (p. 2)

View on Pith 1987
[4]

H. Iwaniec, On zeros of Dirichlet’sLseries.Invent. Math.23 (1974), 97–104. (pp. 4 and 5)

View on Pith 1974
[5]

H. Iwaniec and E. Kowalski,Analytic number theory. Amer. Math. Soc. Colloq. Publ., vol. 53, Amer. Math. Soc., Providence, RI, 2004. (p. 3)

View on Pith 2004
[6]

O. Klurman, A. P. Mangerel, and J. Ter¨ av¨ ainen, Multiplicative functions in short arithmetic pro- gressions.Proc. Lond. Math. Soc. (3)127 (2023), 366–446. (p. 2)

View on Pith 2023
[7]

N. M. Korobov, Double trigonometric sums and their applications to the estimation of rational sums.Math. Notes6 (1969), 472–478. (p. 2)

View on Pith 1969
[8]

N. M. Korobov, The distribution of digits in periodic fractions.Math. USSR-Sb.18 (1974), 659–676. (p. 2)

View on Pith 1974
[9]

K. Matom¨ aki and J. Ter¨ av¨ ainen, Products of primes in arithmetic progressions.J. Reine Angew. Math.808 (2024), 193–240. (p. 2)

View on Pith 2024
[10]

H. L. Montgomery,Topics in multiplicative number theory. Lecture Notes in Math., vol. 227, Springer-Verlag, Berlin, 1971. (p. 7)

View on Pith 1971
[11]

H. L. Montgomery and R. C. Vaughan,Multiplicative number theory. I. Classical theory. Cambridge Stud. Adv. Math., vol. 97, Cambridge Univ. Press, Cambridge, 2007. (pp. 3, 5, 6, and 7)

View on Pith 2007
[12]

A. G. Postnikov, On the sum of characters with respect to a modulus equal to a power of a prime number.Izv. Akad. Nauk SSSR. Ser. Mat.19 (1955), 11–16. (p. 2)

View on Pith 1955
[13]

A. G. Postnikov, On DirichletL-series with the character modulus equal to the power of a prime number.J. Indian Math. Soc. (N.S.)20 (1956), 217–226. (p. 2) Department of Mathematics, University of Missouri, Columbia, MO 65211, USA Email address:bankswd@missouri.edu School of Mathematics and Statistics, University of New South W ales, Sydney, NSW 2052, Aus...

View on Pith 1956

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

W. D. Banks and I. E. Shparlinski, Bounds on short character sums andL-functions with characters to a powerful modulus.J. Anal. Math.139 (2019), 239–263. (p. 2)

View on Pith 2019

[2] [2]

W. D. Banks and I. E. Shparlinski, Sums with the M¨ obius function twisted by characters with powerful moduli.Trans. Amer. Math. Soc.373 (2020), 249–272. (pp. 2 and 4)

View on Pith 2020

[3] [3]

P. Erd˝ os, A. Odlyzko, and A. S´ ark¨ ozy, On the residues of products of prime numbers.Period. Math. Hungar.18 (1987), 229–239. (p. 2)

View on Pith 1987

[4] [4]

H. Iwaniec, On zeros of Dirichlet’sLseries.Invent. Math.23 (1974), 97–104. (pp. 4 and 5)

View on Pith 1974

[5] [5]

H. Iwaniec and E. Kowalski,Analytic number theory. Amer. Math. Soc. Colloq. Publ., vol. 53, Amer. Math. Soc., Providence, RI, 2004. (p. 3)

View on Pith 2004

[6] [6]

O. Klurman, A. P. Mangerel, and J. Ter¨ av¨ ainen, Multiplicative functions in short arithmetic pro- gressions.Proc. Lond. Math. Soc. (3)127 (2023), 366–446. (p. 2)

View on Pith 2023

[7] [7]

N. M. Korobov, Double trigonometric sums and their applications to the estimation of rational sums.Math. Notes6 (1969), 472–478. (p. 2)

View on Pith 1969

[8] [8]

N. M. Korobov, The distribution of digits in periodic fractions.Math. USSR-Sb.18 (1974), 659–676. (p. 2)

View on Pith 1974

[9] [9]

K. Matom¨ aki and J. Ter¨ av¨ ainen, Products of primes in arithmetic progressions.J. Reine Angew. Math.808 (2024), 193–240. (p. 2)

View on Pith 2024

[10] [10]

H. L. Montgomery,Topics in multiplicative number theory. Lecture Notes in Math., vol. 227, Springer-Verlag, Berlin, 1971. (p. 7)

View on Pith 1971

[11] [11]

H. L. Montgomery and R. C. Vaughan,Multiplicative number theory. I. Classical theory. Cambridge Stud. Adv. Math., vol. 97, Cambridge Univ. Press, Cambridge, 2007. (pp. 3, 5, 6, and 7)

View on Pith 2007

[12] [12]

A. G. Postnikov, On the sum of characters with respect to a modulus equal to a power of a prime number.Izv. Akad. Nauk SSSR. Ser. Mat.19 (1955), 11–16. (p. 2)

View on Pith 1955

[13] [13]

A. G. Postnikov, On DirichletL-series with the character modulus equal to the power of a prime number.J. Indian Math. Soc. (N.S.)20 (1956), 217–226. (p. 2) Department of Mathematics, University of Missouri, Columbia, MO 65211, USA Email address:bankswd@missouri.edu School of Mathematics and Statistics, University of New South W ales, Sydney, NSW 2052, Aus...

View on Pith 1956