REVIEW
Exponential sums over primes
Number Theory · Submitted 14 Aug 2026 · 9 references
Pith's verdict: First improvement of Vinogradov's 4/5 exponent in the generic range, but the headline exponent 19/24 is borrowed from an unrefereed external estimate; the internal argument looks sound.
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Referee report
The paper
The paper claims
This paper proves a sharper bound on exponential sums over primes, replacing the longstanding $N^{4/5}$ term in the classical 1937 estimate by $N^{19/24}$ in the generic range where the approximating denominator is of size about…
The reading
What carries the argument
The central object is the exponential sum $S_N(\alpha) = \sum_{n<N}\Lambda(n)e(n\alpha)$. The proof rests on four pieces of machinery: the Heath-Brown identity, which decomposes $\Lambda(n)$ into products of at most ten Dirichlet polynomials with controlled lengths (Lemma 3.3); mean value estimates for such polynomials in the character and $t$ variables (Lemma 3.4); a large-values estimate in the $q$-aspect (Lemma 3.6), taken as a black box, which bounds how many points $(\chi,t)$ can make a Dirichlet polynomial large; and a combinatorial lemma (Lemma 3.2) about the exponents of the factor lengths, which forces either an easy 'Type II' configuration or a three-part partition of the factors. In the latter case an eight-case analysis (Lemma 3.7) bounds the integral of a product of three polynomials by $\lesssim x + (qT)^{7/8}x^{29/48}$, and combining these pieces yields the exponent $19/24$.
Extended reading notes
Core claim
Theorem 1.1 asserts a bound of $N^{o(1)}(N/B^{1/2} + N^{19/24})$ for the exponential sum over primes $\sum_{n<N}\Lambda(n)e(n\alpha)$ whenever $\alpha$ is of the form $a/q + \epsilon$ with $(a,q)=1$, $q \le N^{1/2}$ and $|\epsilon| \le 1/(qN^{1/2})$; here $B = \max(q, qN|\epsilon|)$. For the generic case $q \approx N^{1/2}$ this gives $N^{19/24+o(1)}$, improving the classical $N^{4/5+o(1)}$ that had stood since 1937. The proof combines the Heath-Brown identity, a new large-values estimate for Dirichlet polynomials in the $q$-aspect, and a structural argument showing that certain bad factorisations cannot occur; the resulting bounds are assembled in Proposition 3.11 and a dyadic decomposition in Section 4.
Load-bearing premise
The $19/24$ exponent rests on a large-values estimate for Dirichlet polynomials in the $q$-aspect (Lemma 3.6) that is imported as a black box from an unpublished companion paper; if that estimate is false or substantially weaker, the proof of Theorem 1.1 collapses.
Editorial extensions
If this is right
- In the generic range $q \approx N^{1/2}$, the exponential sum bound improves from $N^{4/5+o(1)}$ to $N^{19/24+o(1)}$.
- For ranges where $B$ is small ($B \le N^{29/60}$), the paper deliberately keeps the existing small-$B$ estimate, so Theorem 1.1 is strongest exactly in the generic measure-one regime.
- The dyadic framework of Section 4 shows the new bound applies uniformly across all dyadic scales $n \sim N/2^j$ with $j$ up to about $\log N$, once each scale is re-approximated by Dirichlet.
- The proof identifies the obstruction to further improvement as the $q$-aspect large-values estimate, so any later strengthening of that estimate would automatically improve the exponent in Theorem 1.1.
Reading between the lines
- Because the proof imports the large-values estimate as a black box, any future strengthening of that estimate in the $q$-aspect should automatically lower the $19/24$ exponent; the structural lemmas (3.2, 3.7) would not need reworking, only re-optimised.
- The same eight-case framework may transfer to other sums built from three unbalanced Dirichlet polynomials, such as exponential sums over almost primes or short-interval sums over primes, where the same partition obstruction appears.
- The gain over $4/5$ is $1/120$, so the theorem is unlikely to change the numerics of any application by itself; its significance is that a barrier which had held for almost a century is not intrinsic.
- A natural next test is to see whether the exponent $19/24$ can be pushed toward $3/4$ (the GRH-consistent value) by using a fourth-moment estimate in the $q$-aspect instead of the second-moment plus large-values route.
Original abstract
Let $\alpha=a/q+\epsilon$ with $(a,q)=1$, $q\le N^{1/2}$ and $|\epsilon|\le 1/(qN^{1/2})$, and let $B:=\max(q,qN|\epsilon|)$. We show that \[ \Bigl|\sum_{n<N}\Lambda(n)e(n\alpha)\Bigr|\le N^{o(1)}\Bigl(\frac{N}{B^{1/2}}+N^{19/24}\Bigr). \] This improves on the classical bound of Vinogradov from 1937, which has $N^{4/5}$ in place of $N^{19/24}$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Let $\alpha=a/q+\epsilon$ with $(a,q)=1$, $q\le N^{1/2}$ and $|\epsilon|\le 1/(qN^{1/2})$, and let $B:=\max(q,qN|\epsilon|)$. We show that \[ \Bigl|\sum_{n<N}\Lambda(n)e(n\alpha)\Bigr|\le N^{o(1)}\Bigl(\frac{N}{B^{1/2}}+N^{19/24}\Bigr). \] This improves on the classical bound of Vinogradov from 1937, which has $N^{4/5}$ in place of $N^{19/24}$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\alpha=a/q+\epsilon$ with $(a,q)=1$, $q\le N^{1/2}$ and $|\epsilon|\le 1/(qN^{1/2})$, and let $B:=\max(q,qN|\epsilon|)$. | background 0.81 | 0.21 | 0.03 | unclear 0.82 |
| s1 | We show that \[ \Bigl|\sum_{n<N}\Lambda(n)e(n\alpha)\Bigr|\le N^{o(1)}\Bigl(\frac{N}{B^{1/2}}+N^{19/24}\Bigr). | result 0.99 | 0.96 | 0.92 | asserted 1.00 |
| s2 | \] This improves on the classical bound of Vinogradov from 1937, which has $N^{4/5}$ in place of $N^{19/24}$. | interpretation 0.71 | 0.39 | 0.14 | asserted 1.00 |
What was examined, and what was not
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References
How the literature uses it
What this paper does with the arXiv papers it cites
What it cites · 2 sentences without a confident label
Builds on 2 papers
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Then we have [equation] In particular, if $N⪅x^5/12$ and $qT⪅x^1/2$ then [equation] For $I⊆[N,2N]$ this is [ref] , extending the method of Guth–Maynard [ref] to Dirichlet characters.
New large value estimates for Dirichlet polynomials · Proof of Proposition [ref] · 0.83
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Then we have [equation] In particular, if $N⪅x^5/12$ and $qT⪅x^1/2$ then [equation] For $I⊆[N,2N]$ this is [ref] , extending the method of Guth–Maynard [ref] to Dirichlet characters.
Large Value Estimates for Dirichlet Polynomials with Characters and Zero Density of Dirichlet $L$-Functions · Proof of Proposition [ref] · 0.83
Discussion
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