{"id":"7db9e311-1290-4ef7-bb94-09421f41c486","arxiv_id":"1908.06946","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves a square-root-of-logarithm improvement over Balog and Ruzsa's lower bounds for L1 norms of exponential sums on squarefree integers, plus new prime-supported bounds, and a flawed new proof of Vaughan's theorem.","lead":"This paper improves lower bounds on the average size of exponential sums supported on squarefree integers and primes. It uses Linnik's large sieve, and it also attempts a new proof of Vaughan's classical lower bound for the von Mangoldt function.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (17) is false as written: the un-conjugated product integrates to zero, so Theorem 1's proof lacks its key step; inserting a complex conjugate repairs it.","rationale":"The reader's VERDICT of CONDITIONAL is appropriate, but the weakest point is not Lemma 10 in Part III; it is the false identity (17) used in the proof of the headline Theorem 1. Expanding both sides shows the un-conjugated integral vanishes, so the proof as written has no valid derivation of the key L1-lower bound. This is nevertheless an easily repaired typo: replacing the second factor by its complex conjugate gives the standard autocorrelation identity, and the desired inequality follows. Thus the theorem is probably correct but the manuscript requires a correction. The reader's Lemma 10 concern is also real: the assertion that (q,n)>1 forces q=p for squarefree q is false, since q may be any squarefree divisor of rad(n) sharing a prime with n. However, because Lambda(n) is nonzero only for prime powers, the error term is still o(Q N^2) for Q = N^{1/2}, so Part III is also repairable. Since both gaps are repairable, the conditional verdict should stand and no new verdict movement is recommended.","tokens_in":12063,"tokens_out":15618,"duration_ms":149658,"concrete_test":"Expansion check for N=2 with b_1=b_2=1: the left side of Eq. (17) is the integral of (e(beta)+e(2 beta))(e(alpha+beta)+e(2 alpha+2 beta)) d beta, which is 0, while the right side is e(alpha)+e(2 alpha). Replacing the second factor by its complex conjugate gives e(-alpha)+e(-2 alpha), whose modulus integral agrees with the claimed upper bound. This confirms that the proof is repaired by inserting the conjugate, and that without it the derivation fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Part I's proof of Theorem 1 rests on Eq. (17), which states that the integral of g_N(beta) g_N(alpha+beta) over beta equals the sum of |b_n|^2 e(n alpha). With g_N(beta) = sum_{n=1}^N b_n e(n beta) and e(x) = e^{2 pi i x}, the integrand expands as sum_{m,n} b_n b_m e(m alpha) e((n+m) beta). The beta-integral is nonzero only if n+m=0, which is impossible for n,m >= 1, so the left side is identically zero. Thus the displayed identity, and the inequality that follows from it, are not established by the text. The intended formula is the autocorrelation identity with a complex conjugate: the integral of g_N(beta) overline{g_N(alpha+beta)} is sum |b_n|^2 e(-n alpha), and after alpha -> -alpha it yields the same upper bound. With that one-character correction, the Part I argument goes through. As written, however, the central proof of Theorem 1 contains a false displayed identity; the same step is also used in the proof of Theorem 8. This is a more load-bearing gap than the Lemma 10 squarefree issue, which concerns only the Part III Vaughan proof.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12253,"tokens_out":6485,"duration_ms":60509,"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":[{"comment":"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.","section":"Part I, Proof of Theorem 1, Eq. (17)"},{"comment":"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.","section":"Part III, Proof of Lemma 10, Eq. (33)"}],"minor_comments":[{"comment":"The abstract contains a typo: \"prim es\" should be \"primes\".","section":"Abstract"},{"comment":"The name \"Rusza\" in \"Balog and Rusza [3]\" should be \"Ruzsa\".","section":"Introduction, page 2"},{"comment":"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).","section":"Part II, Eq. (24)"},{"comment":"The Ramanujan sum c_q(n) is defined in Eq. (31) after being used in Eq. (30); reorder for clarity.","section":"Part III, Eq. (30)-(31)"},{"comment":"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.","section":"Part I, Proof of Theorem 1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has two load-bearing technical errors, one in Eq. (17) affecting Parts I and II, and one in Lemma 10 affecting Part III. Both appear repairable, and the stated results are plausible. However, as written the proofs are not valid, so the paper requires a major revision before it can be considered for publication. The contribution is modest in scope but suitable for a specialized journal if corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nNow 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).\n\nThe 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.\n\nNet 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.","headline":"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.","tokens_in":12858,"tokens_out":1401,"would_cite":false,"duration_ms":17554,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11L03","11L07","11L20","11N36","42A05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Linnik's large sieve gives a sqrt-log improvement for square-free sums and new prime-sum bounds.","keywords":["Möbius function","von Mangoldt function","prime number","square-free integer","Linnik's large sieve","bound for L1 norm of exponential sum","Ramanujan sum"],"falsifier":"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.","tokens_in":11805,"feed_emoji":"🧮","tokens_out":20072,"duration_ms":178941,"temperature":0.7,"pith_summary":"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.","feed_headline":"Square-free exponential sums get a sharper lower bound","feed_subtitle":"The same sieve method boosts prime-supported sums and gives a new proof of a known prime bound.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the approximating-sum construction and the square-free-support method that Theorem 1 improves by a logarithmic factor.","marker":"[2]"},{"why":"Provides the sharp large-sieve inequality used to bound every pointwise approximation error in the paper.","marker":"[5]"},{"why":"Gives the lower bound $\\pi(N)>N/\\log N$ used to convert the large-sieve estimates into the stated powers of $\\log N$.","marker":"[13]"},{"why":"States the known $N^{1/2}$ lower bound for the von Mangoldt exponential sum that Part III attempts to reprove and that Part II's prime results are measured against.","marker":"[15]"},{"why":"Defines Ramanujan's sum and its basic evaluation, which supplies the Fourier coefficients in the kernel $K_{N,Q}$.","marker":"[11]"},{"why":"Provides the asymptotic density of square-free integers, used to evaluate the main term in Lemma 10.","marker":"[10]"}],"fun_headline_variants":["Square-free sums: sharper L1 lower bound via Linnik sieve","Prime exponential sums get new L1 lower bounds","New proof of von Mangoldt sum bound using Ramanujan's sum","Tighter L1 norms for exponential sums over square-free integers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Square-free sums: sharper L1 lower bound via Linnik sieve","Prime exponential sums get new L1 lower bounds","New proof of von Mangoldt sum bound using Ramanujan's sum","Tighter L1 norms for exponential sums over square-free integers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000255,"raw_usage":{"total_tokens":1563,"prompt_tokens":930,"completion_tokens":633,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":563}},"tokens_in":546,"tokens_out":633,"duration_ms":6836,"temperature":1.0,"reasoning_tokens":563,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:31:18.397799+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Balog and I","cited_arxiv_id":null,"evidence_quote":"Supplies the approximating-sum construction and the square-free-support method that Theorem 1 improves by a logarithmic factor."},{"cited_title":"Davenport, Multiplicative number theory , 3rd","cited_arxiv_id":null,"evidence_quote":"Provides the sharp large-sieve inequality used to bound every pointwise approximation error in the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the lower bound $\\pi(N)>N/\\log N$ used to convert the large-sieve estimates into the stated powers of $\\log N$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the known $N^{1/2}$ lower bound for the von Mangoldt exponential sum that Part III attempts to reprove and that Part II's prime results are measured against."},{"cited_title":"Ramanujan, On certain trigonometrical sums and their applications in the theory of numbers, Trans","cited_arxiv_id":null,"evidence_quote":"Defines Ramanujan's sum and its basic evaluation, which supplies the Fourier coefficients in the kernel $K_{N,Q}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the asymptotic density of square-free integers, used to evaluate the main term in Lemma 10."}],"review_version":1}