Linnik's large sieve and the $L^{1}$ norm of exponential sums

Andrew Ledoan; Brian Tobin; Emily Eckels; Steven Jin

REVIEW 2 major objections 5 minor 15 references

Linnik's large sieve and the $L^{1}$ norm of exponential sums

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Linnik's large sieve gives a sqrt-log improvement for square-free sums and new prime-sum bounds.

desk verdict The main theorems are plausible but the proof as written has a false displayed identity at (17) that is load-bearing, plus a second gap in Lemma 10; both are likely repairable but need fixing before the results stand. read the letter →

arxiv 1908.06946 v1 pith:W27GHCLA submitted 2019-08-19 math.NT

Emily Eckels , Steven Jin , Andrew Ledoan , Brian Tobin This is my paper · ORCID

classification math.NT MSC 11L0311L0711L2011N3642A05

keywords MöbiusfunctionvonMangoldtprimenumbersquare-freeintegerLinnik'slargesieveboundforL1normofexponentialsumRamanujan

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Working in analytic number theory, this paper studies how large the average of an exponential sum can be forced to be when its coefficients are supported on a sparse set. Its central claim is that coefficients supported on square-free integers (integers divisible by no square larger than 1) satisfy $\int_0^1 |\sum_{n=1}^N b_n e(n\alpha)|\,d\alpha \gg N^{-3/8}(\log N)^{-1/2}(\sum_{n=1}^N |b_n|^2)^{1/2}$, a square-root-of-logarithm improvement over the earlier bound. In the special case $b_n=\mu(n)$, this yields $\int_0^1 |\sum_{n\le N}\mu(n)e(n\alpha)|\,d\alpha \gg N^{1/8}/(\log N)^{1/2}$. The paper also proves analogous lower bounds for prime-supported sums and offers a new proof, via Ramanujan sums and Linnik's large sieve, of the known $N^{1/2}$ lower bound for the exponential sum with the von Mangoldt function.

What carries the argument

The load-bearing device is a pointwise approximation of the Fejér kernel $T_N(\alpha)=N^{-1}|F_N(\alpha)|^2$, where $F_N(\alpha)=\sum_{n\le N}e(n\alpha)$, by a modified kernel $G_N^*$ in the square-free case or $H_{N,P}$ in the prime case, whose nonzero Fourier coefficients are supported on the complementary set—non-square-free integers or non-primes. The uniform approximation error is bounded by Linnik's large sieve, the inequality $\sum_r |\sum_n a_n e(n\alpha_r)|^2 \le (N+\delta^{-1}-1)\sum_n |a_n|^2$ for separated points $\alpha_r$. Because the target coefficients and the approximating kernel have disjoint support, their convolution vanishes, leaving only the controlled approximation error to be integrated. In Part III, Ramanujan's sum $c_q(n)=\sum_{(a,q)=1}e(an/q)$ supplies the Fourier coefficients of the kernel $K_{N,Q}$, allowing the central integral $V$ to be evaluated asymptotically.

What would settle it

Take $n=9$ and square-free $q=15$: the two numbers share the factor 3, yet $q$ is not the prime 3, so the proof's assertion that $(q,n)>1$ implies $q=p$ is not true. Computing the contribution of such $q$, a square-free number containing the shared prime together with an extra prime, to the error term would determine whether the asymptotic for $V$, and hence the claimed new proof of the $N^{1/2}$ bound, can be repaired.

Watch

Extended reading notes

Core claim

On the paper's own terms, the main new quantitative result is Theorem 1: for arbitrary complex coefficients $b_n$ supported on square-free integers, the $L^1$ norm of the exponential sum is at least a constant multiple of $N^{-3/8}(\log N)^{-1/2}$ times the $\ell^2$ norm of the coefficients. The proof transfers the earlier construction in which the Fejér kernel $T_N(\alpha)=N^{-1}|\sum_{n\le N}e(n\alpha)|^2$ is approximated by a modified kernel whose nonzero coefficients live only on non-square-free integers, integrates against the target sum, and uses support-disjointness plus Linnik's large sieve to control the error. The same template with primes in place of square-free integers gives lower bounds of size $N^{-1/4}(\log N)^{-1/2}$ for prime-supported coefficients, $N^{1/2}/(\log N)^2$ for the unweighted prime sum, and $N^{1/4}/\log N$ for the prime sum weighted by the non-principal character modulo 3. For the von Mangoldt function, the paper claims a new proof of the existing lower bound $\int_0^1|\sum_{n\le N}\Lambda(n)e(n\alpha)|\,d\alpha \gg N^{1/2}$, evaluating a related integral asymptotically as $3Q/\pi^2 N^2$ with the help of Ramanujan's sum and the large sieve.

Load-bearing premise

The new proof of the prime-sum lower bound rests on the assumption that a square-free number sharing a factor with a prime power must be exactly that prime itself; this is false when the square-free number also contains other primes, and the claimed bound on the error term depends on it.

Editorial extensions

If this is right

Square-free-supported exponential sums cannot be smaller on average than $N^{-3/8}(\log N)^{-1/2}$ times their coefficient norm, so cancellation among square-free coefficients is limited.
For the Möbius function the paper's bound $\gg N^{1/8}/(\log N)^{1/2}$ removes the final logarithmic factor from the earlier $N^{1/8}/\log N$ lower bound.
For prime-supported coefficients the paper proves $L^1 \gg N^{-1/4}(\log N)^{-1/2}(\sum_{p\le N}|a_p|^2)^{1/2}$; the unweighted prime exponential sum consequently satisfies $L^1 \gg N^{1/2}/(\log N)^2$.
The character-weighted prime sum $\sum_{p\le N}\chi_3(p)e(p\alpha)$ has $L^1$ norm $\gg N^{1/4}/\log N$, so even though its value at $\alpha=0$ oscillates, its average size grows as a power of $N$.
The paper also claims a new proof that $\int_0^1|\sum_{n\le N}\Lambda(n)e(n\alpha)|\,d\alpha \gg N^{1/2}$, matching the known lower bound.

Reading between the lines

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

The same convolution-with-complementary-kernel strategy should adapt to other sparse coefficient sets, such as $r$-free integers or integers with a prescribed number of prime factors, with the exponent governed by the sieve density of the forbidden set.
Because the large-sieve approximation errors in the paper are uniform in $\alpha$, the method may yield pointwise information about these exponential sums, not just their $L^1$ averages.
For any fixed non-principal Dirichlet character, the proof behind Theorem 9 should go through unchanged, since the only feature used is the absence of a large contribution at $\alpha=0$.
The same Ramanujan-sum evaluation could be reused to estimate higher moments or correlations of the von Mangoldt kernel, since the relevant sums are controlled at rational points by the same large-sieve inequality.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing you should know first: equation (17) is false as written. The integral of g_N(beta) g_N(alpha+beta) expands to sum_{m,n} b_m b_n e(m alpha) times a beta-integral that forces n+m=0, impossible for positive indices. So the left side is identically zero, and the inequality that drives both Theorem 1 and Theorem 8 is not established. The intended step is almost certainly the autocorrelation identity with a complex conjugate on one factor, and with that one-character repair the argument seems to go through. But the text as submitted is wrong at the central step.

What is genuinely good: the paper gives a complete proof attempt for the sqrt(log N) improvement that Balog and Ruzsa had stated for squarefree-supported coefficients, and it extends the framework to primes with arbitrary or oscillating coefficients. Theorems 8 and 9, especially the chi_3 case, look new and useful. The large sieve setup with the Fejer kernel and the constructed approximations G*_N and H_{N,P} is clean and instructive, and the citation pattern is honest — no circularity, standard external tools only.

Now the soft spots, in proportion. The stress-test note is correct and is the more serious issue: (17) is not a minor typo in an aside, it is the tool that converts the pointwise upper bound (16) into the L1 lower bound. Without the conjugate, the proof of Theorem 1 collapses. The fix is one character, but it must be stated and the computation redone. The same applies to Theorem 8, which explicitly leans on (17).

The second gap is in Lemma 10 (Part III). The claim that for squarefree q with (q,n)>1 one must have q=p is false; q can share a prime factor with n without equaling p (e.g., q=6, n=12). The bound in (33) therefore does not follow from the given reasoning. This invalidates the new proof of Vaughan's lower bound as written, though the intended bound is plausible and likely obtainable with a more careful summation over divisors.

Net assessment: the paper is not publishable as is, but the core ideas are sound enough that I would not throw it away. It deserves a serious referee only after the authors repair (17) and rework Lemma 10. For a reading group it is a useful case study on how a single missing conjugate can sink a proof, but I would not cite it in its current form. Recommend: major revision, with the burden on the authors to fix both gaps.

Referee Report

2 major / 5 minor

Summary. The paper studies the L1 norm of exponential sums with coefficients supported on squarefree integers, on primes with arbitrary coefficients, and with the von Mangoldt function. In Part I, the authors prove a lower bound (Theorem 1) that improves the Balog-Ruzsa bound by a factor of sqrt(log N), and derive a corresponding bound for the Mobius function (Corollary 2). In Part II, they construct an exponential sum supported off the primes, prove a lower bound for sums over primes (Theorem 8), and apply it to the character chi_3 to obtain a lower bound of size N^{1/4}/log N (Theorem 9). In Part III, they attempt a new proof of Vaughan's lower bound for the L1 norm of the von Mangoldt exponential sum, using Ramanujan sums and Linnik's large sieve. The paper is written in the style of the Balog-Ruzsa method, with the central novelty being the use of the sharp large sieve to obtain the stated lower bounds.

Significance. If the proofs are correct, the results are of interest to analytic number theorists. The squarefree improvement over Balog-Ruzsa is explicitly noted as a consequence of the method, and the new proof of Vaughan's lower bound, even if not a new result, offers a different technique via Ramanujan sums. The paper gives explicit constants and uses standard tools (large sieve, prime number theorem, Rosser-Schoenfeld bounds), and the statements are falsifiable and concrete. However, the claimed novelty is moderate: Theorem 1 improves a known bound by a logarithmic factor, and the Vaughan bound is already known. The significance depends on the validity of the proofs, which currently contain load-bearing gaps.

major comments (2)

[Part I, Proof of Theorem 1, Eq. (17)] The displayed identity ∫_0^1 g_N(β)g_N(α+β)dβ = Σ_{n=1}^N |b_n|^2 e(nα) is false as written. Because g_N(β) = Σ_{n=1}^N b_n e(nβ) and g_N(α+β) = Σ_{m=1}^N b_m e(m(α+β)), the product expands to Σ_{n,m} b_n b_m e(mα) e((n+m)β). The β-integral vanishes unless n+m=0, which is impossible for n,m ≥ 1, so the left-hand side is identically zero. The intended identity is the autocorrelation formula with a complex conjugate: ∫_0^1 g_N(β) overline{g_N(α+β)} dβ = Σ |b_n|^2 e(-nα). With this one-character correction, the inequality below (17), and consequently the proof of Theorem 1 and its reuse in Theorem 8, become valid. As written, however, the central step is not established.
[Part III, Proof of Lemma 10, Eq. (33)] In bounding the error term in Eq. (32), the proof states that for squarefree q with (q,n)>1 and n=pm, the condition (q,n)>1 implies q=p. This is false: q may be any squarefree integer sharing at least one prime factor with n, not necessarily the specific prime p, and q could be composite or involve primes dividing m. Consequently, the summation over q in the error term is not reduced to ∑_{p≤Q} p log p, and the bound O(N log N ∑_{p≤Q} p) in Eq. (33) is not justified. A correct argument would estimate ∑_{q≤Q} μ(q)^2 q ω(q) N log N, which gives O(N Q^2 log N log log Q) and is still o(QN^2) for Q=N^{1/2}, but the proof as written needs to be repaired.

minor comments (5)

[Abstract] The abstract contains a typo: "prim es" should be "primes".
[Introduction, page 2] The name "Rusza" in "Balog and Rusza [3]" should be "Ruzsa".
[Part II, Eq. (24)] The notation 1p(n) for the indicator of the primes is potentially confusing because p is used both as a summation index and as the prime in the definition; consider using a different symbol such as 1_P or P(n).
[Part III, Eq. (30)-(31)] The Ramanujan sum c_q(n) is defined in Eq. (31) after being used in Eq. (30); reorder for clarity.
[Part I, Proof of Theorem 1] The variable M is introduced for N/2 while M is also used in the large sieve inequality (12) as the starting index; this notational collision is harmless but could be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derived lower bounds are not assumed as inputs; the argument relies on external standard results and contains no fitted-parameter or self-citation loops that carry the proof.

full rationale

The paper's central claims are lower bounds for L1 norms of exponential sums over squarefree integers (Theorem 1, Corollary 2), primes (Theorems 6-7), and the von Mangoldt function (Part III), plus a character-weighted prime sum (Theorem 9). Each proof starts from external inputs: Selberg's sharp large sieve, the prime number theorem, Rosser-Schoenfeld bounds, the asymptotic density of squarefree integers from Montgomery-Vaughan, and standard Ramanujan-sum identities. None of these inputs contains the target lower bound, and none is a parameter fitted to the quantity being predicted. The Balog-Ruzsa construction is cited as prior technique, not as an unverified premise, and no uniqueness theorem from the present authors is invoked to forbid alternatives. The displayed identity in Eq. (17) is algebraically incorrect as written (the un-conjugated product integrates to zero; a complex conjugate is needed), and Lemma 10's claim that a squarefree q with (q,n)>1 must equal the prime p dividing n is false (e.g., q=6 and n=2). These are correctness gaps, not circularity: the derivation does not assume its conclusion, and correcting them would not make the argument depend on its own output. Because no load-bearing step reduces by construction to an input, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The results rest on standard analytic number theory inputs rather than on new postulates. The main axioms are the sharp large sieve inequality, the prime number theorem, and the Rosser-Schoenfeld lower bound for π(x). The proof of Lemma 10 also uses the asymptotic density of squarefree integers, but the application assumes a uniformity in the coprimality condition that is not stated.

assumptions (4)

standard math Sharp large sieve inequality (Eq. 12) with constant N + 1/δ - 1
Used to bound sums of |F_N|^2 over well-separated points in Lemmas 3, 4, and 10; cited to Davenport [5] and attributed to Selberg.
standard math Prime number theorem
Used to evaluate the main term in Theorem 7 and Eq. (34) in Lemma 10.
standard math Rosser-Schoenfeld bound π(N) > N/log N for N ≥ 17
Used in Lemma 3, Lemma 4, and the alternative proof of Lemma 3 to lower-bound π(P) in terms of P/log P; cited to [13].
standard math Asymptotic density of squarefree integers up to Q is (6/π^2)Q + O(Q^{1/2})
Used in Lemma 10, Eq. (32). The application to the sum over n with the coprimality condition (q,n)=1 requires a uniformity in n that is not justified in the paper, a gap.

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Cite this review

Pith. "Pith review of Linnik's large sieve and the $L^{1}$ norm of exponential sums." pith.science (2026). https://pith.science/paper/W27GHCLA

@misc{pith2026190806946,
  author       = {Pith},
  title        = {Pith review of: Linnik's large sieve and the $L^1$ norm of exponential sums},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W27GHCLA}},
  note         = {Machine review of arXiv:1908.06946}
}

abstract

The method of proof of Balog and Ruzsa and the large sieve of Linnik are used to investigate the behaviour of the $L^{1}$ norm of a wide class of exponential sums over the square-free integers and the primes. Further, a new proof of the lower bound due to Vaughan for the $L^{1}$ norm of an exponential sum with the von Mangoldt $\Lambda$ function over the primes is furnished. Ramanujan's sum arises naturally in the proof, which also employs Linnik's large sieve.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Balog and A

A. Balog and A. Perelli, On the L1 mean of the exponential sum formed with the M¨ obius function, J. London Math. Soc. (2) , 57 (1998), 275–288

work page 1998
[2]

Balog and I

A. Balog and I. Z. Ruzsa, A new lower bound for the L1 mean of the exponential sum with the M¨ obius function,Bull. London Math. Soc. , 31 (1999), 415–418

work page 1999
[3]

Balog and I

A. Balog and I. Z. Ruzsa, On the exponential sum over r-free integers, Acta Math. Hungar. , 90 (2001), no. 3, 219–230

work page 2001
[4]

Bombieri, On the large sieve, Mathematika, 12 (1965), 201–225

E. Bombieri, On the large sieve, Mathematika, 12 (1965), 201–225

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[5]

Davenport, Multiplicative number theory , 3rd

H. Davenport, Multiplicative number theory , 3rd. ed., Graduate Studies in Mathematics, 74, Springer-Verlag, New York, 2000

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[6]

D. A. Goldston, The major arcs approximation for an expone ntial sum over primes, Acta Arith., 92 (2000), no. 2, 169–179

work page 2000
[7]

G. H. Hardy and J. E. Littlewood, A new proof of a theorem on r earrangements, J. London Math. Soc. , 23 (1948), 163–168

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[8]

Yu. V. Linnik, The large sieve, Dokl. Akad. Nauk SSSR , 30 (1941), 292–294

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[9]

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Ramanujan, On certain trigonometrical sums and their applications in the theory of numbers, Trans

S. Ramanujan, On certain trigonometrical sums and their applications in the theory of numbers, Trans. Cambridge Philos. Soc. , 22, (1918), 259–276

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[12]

R´ enyi, On the large sieve of Ju

A. R´ enyi, On the large sieve of Ju. V. Linnik, Compositio Math. , 8 (1950), 68–75

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J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. of Math. , 6 (1962), 64–94

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R. C. Vaughan, The L1 mean of exponential sums over primes, Bull. London Math. Soc. , 20 (1988), 121–123. Department of Mathematics, Emory University, 201 Dowman Dri ve, Atlanta, Georgia 30322, USA E-mail address : eneckel@emory.edu Department of Mathematics, University of Mar...

1988

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Balog and A

A. Balog and A. Perelli, On the L1 mean of the exponential sum formed with the M¨ obius function, J. London Math. Soc. (2) , 57 (1998), 275–288

work page 1998

[2] [2]

Balog and I

A. Balog and I. Z. Ruzsa, A new lower bound for the L1 mean of the exponential sum with the M¨ obius function,Bull. London Math. Soc. , 31 (1999), 415–418

work page 1999

[3] [3]

Balog and I

A. Balog and I. Z. Ruzsa, On the exponential sum over r-free integers, Acta Math. Hungar. , 90 (2001), no. 3, 219–230

work page 2001

[4] [4]

Bombieri, On the large sieve, Mathematika, 12 (1965), 201–225

E. Bombieri, On the large sieve, Mathematika, 12 (1965), 201–225

work page 1965

[5] [5]

Davenport, Multiplicative number theory , 3rd

H. Davenport, Multiplicative number theory , 3rd. ed., Graduate Studies in Mathematics, 74, Springer-Verlag, New York, 2000

work page 2000

[6] [6]

D. A. Goldston, The major arcs approximation for an expone ntial sum over primes, Acta Arith., 92 (2000), no. 2, 169–179

work page 2000

[7] [7]

G. H. Hardy and J. E. Littlewood, A new proof of a theorem on r earrangements, J. London Math. Soc. , 23 (1948), 163–168

work page 1948

[8] [8]

Yu. V. Linnik, The large sieve, Dokl. Akad. Nauk SSSR , 30 (1941), 292–294

work page 1941