Blomer 2006 estimates representation numbers quadratic forms

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corollary_1: Blomer and Granville's two-term asymptotic for the sum over n up to x of r_f(n) to an integer power beta >= 1: the leading term a_K (log x)^(K-1) x plus the term pi (1 + (2^(beta-1) - 1)/u) x / sqrt(D), with relative error a negative power of log x, uniformly in x >= D (log D)^(2 rho) / a.

corollary_2: Blomer and Granville's bounds for the count V(x) of integers up to x that are sums of two powerful numbers: V(x) lies between x (log log x)^A / (log x)^(1 - 2^(-1/3)) for some real A and x (log log x)^(2^(2/3) - 1) / (log x)^(1 - 2^(-1/3)), up to constants.

theorem_1: Blomer and Granville's asymptotic for the sum over n up to x of r_f(n) to an integer power beta >= 1, as x over log x times a polynomial of degree K = 2^(beta-1) in log x plus a power-saving error, uniformly in D <= x^(1/2^(beta-2) - epsilon), with the leading and lowest coefficients given explicitly and a pole of order K for the Dirichlet series.

theorem_2: Blomer and Granville's elementary estimate, for every real beta >= 0, of the sum over n up to x of r_f(n)^beta as pi (1 + (2^(beta-1) - 1)/u) x / sqrt(D) plus an explicit error term, where u is the least positive integer represented by a form in the coset of f by the ambiguous classes.

theorem_3: Blomer and Granville's order of magnitude, for a fixed discriminant -D and every real beta >= 0, of the sum over n up to x of r_f(n)^beta as x (log x)^(2^(beta-1) - 1), with implied constants depending on beta and D.

theorem_4: Blomer and Granville's comparison of two primitive binary quadratic forms of the same discriminant -D: for every real beta >= 0 and D = o(x) their beta-th moments of r(n) up to x agree within a factor 2^(|1-beta| + o(1)), the o(1) tending to 0 as x/D tends to infinity.

theorem_5: Blomer and Granville's order of the sum over n up to x of r_f(n)^beta for every real beta >= 0 in the range where h/g = (log x)^(kappa log 2) with kappa <= L: it is x (log x)^(E(kappa,beta) + o(1)) for an explicit piecewise exponent E, and, if there are no Siegel zeros, it is x (log x)^E(kappa,beta) / g up to a factor (log log x)^O(1).

theorem_6: Blomer and Granville's bounds for the number N_f(x) of integers up to x represented by a form f: under the assumptions of Theorem 5, with kappa defined by h/g = (l log x)^(kappa log 2), the upper bounds of (1.1) to (1.3) hold in their ranges, and the lower bound of (1.1) holds for 0 <= kappa <= 1/2 - epsilon once D is large in terms of epsilon.

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V. Blomer and A. Granville, Estimates for representation numbers of quadratic forms, Duke Math. J. 135 (2006), no. 2, 261--302; DOI 10.1215/S0012-7094-06-13522-6. Received 18 April 2005, revision received 3 May 2006. 2000 MSC primary 11E16, secondary 11N56.

The copy read for this card is not the final journal PDF but the journal's typeset proof of the article: every page after the first carries the running head "xxx dmj5134 June 27, 2006 18:0", every page carries crop marks, the first page names the journal, volume and issue, and the 42 pages are numbered 1--42 rather than 261--302. The count matches the printed extent, but the page correspondence was not checked, so the page numbers below are the proof's. It has a text layer. Its margin on p. 34 carries an editorial query on the conclusion of the proof of Theorem 5. Provenance: downloaded in the repository's survey download set of September 2026; the download URL was not recorded; 284,127 bytes. The typeset proof's only notice is "Vol. 135, No. 2, © 2006" under "DUKE MATHEMATICAL JOURNAL" on p. 1, naming no holder or license; the article's Project Euclid page (DOI 10.1215/S0012-7094-06-13522-6) could not be read on 2026-10-02 (it served a bot challenge); the Duke University Press journal page (dukeupress.edu/duke-mathematical-journal, read 2026-10-02) prints "© 2024 Duke University Press. All Rights Reserved.", states that authors "make standard copyright assignments once the paper has been accepted", and names no Creative Commons license, every other right reserved.

Read status: claims checked for Corollary 2 and for the statements of Theorems 1--6 and Corollary 1, each read clause by clause on the printed pages; no proof was verified. The claim page Blomer and Granville 2006 rests on Corollary 2, and the claim page Odoni's disproof cites it as a later refinement, in agreement with p. 8; the problem page itself consumes no statement from this source.

Contents

• Setting (pp. 2--4): $f$ is a primitive positive integral binary quadratic form of fundamental discriminant $-D$; $r_f(n)$ counts representations of $n$ by $f$ up to automorphisms; $N_f(x)$ counts the $n\le x$ represented by $f$; $h$ is the class number and $g$ the number of genera. With $\mathcal L_{-D}=L(1,\chi_{-D})\varphi(D)/D$ and $\kappa=\log(h/g)/((\log2)\log(\mathcal L_{-D}\log x))$, the paper splits $N_f(x)$ into three ranges: $0\le\kappa\le1/2$, which extends the range of Bernays's result, where $N_f(x)\asymp L(1,\chi_D)x/(\tau(D)\sqrt{\mathcal L_{-D}\log x})$ (1.1; the character is printed $\chi_D$ there and $\chi_{-D}$ elsewhere), the intermediate range $1/2<\kappa<1$ (1.2), and the elementary range $1\le\kappa\ll\log D/\log\log D$ where $N_f(x)\asymp x/\sqrt D$ (1.3). The paper offers (1.1)--(1.3) as the estimates it believes hold (p. 2) and proves them except for the lower bounds when $\kappa\in[1/2-\varepsilon,1/(\log2)+\varepsilon]$ (p. 3).
• Theorems 1--6 and Corollary 1 (pp. 3--8): asymptotics for $\sum_{n\le x}r_f(n)^\beta$ for integer $\beta\ge1$ uniformly in $D\le x^{1/2^{\beta-2}-\varepsilon}$, with a pole of order $2^{\beta-1}$ at $s=1$ for the Dirichlet series (Theorem 1; Corollary 1 states the two main terms); elementary estimates for small $x$ with main term $\pi(1+(2^{\beta-1}-1)/u)x/\sqrt D$ (Theorem 2); for fixed $D$, $\sum_{n\le x}r_f(n)^\beta\asymp x(\log x)^{2^{\beta-1}-1}$ (Theorem 3); two forms of the same discriminant have moments within a factor $2^{|1-\beta|+o(1)}$ when $D=o(x)$ (Theorem 4); for fixed $L>0$, $\sum_{n\le x}r_f(n)^\beta=x(\log x)^{E(\kappa,\beta)+o(1)}$ once $x$ is so large that $\kappa\le L$ and $E(\kappa,\beta)\ge-1-L\log2$, with $\kappa$ here defined by $h/g=(\log x)^{\kappa\log2}$ (the paper calls this the range $D\le(\log x)^L$); in the same range, if there are no Siegel zeros in the sense of (1.15), $\sum_{n\le x}r_f(n)^\beta=x(\log x)^{E(\kappa,\beta)}(\log\log x)^{O(1)}/g$ (Theorem 5, p. 7); the upper bounds in (1.1)--(1.3) in their stated ranges and the lower bound in (1.1) for $0\le\kappa\le1/2-\varepsilon$ once $D$ is large in terms of $\varepsilon$, under the assumptions of Theorem 5 with $\kappa$ as in the setting above (Theorem 6, p. 8).
• Corollary 2 (p. 8; proof in section 9.3, pp. 38--41): with $V(x)$ the count of $n\le x$ expressible as a sum of two powerful numbers,

$$\frac{x(\log\log x)^A}{(\log x)^{1-2^{-1/3}}}\ll V(x)\ll \frac{x(\log\log x)^{2^{2/3}-1}}{(\log x)^{1-2^{-1/3}}}$$

for some $A\in\mathbb R$. Statement read clause by clause on the printed page. The authors conjecture $V(x)\asymp x(\log x)^{-1+2^{-1/3}}(\log\log x)^{2^{2/3}-1}$ (p. 8; p. 3 says the upper bound is proved and the lower bound misses it by a power of $\log\log x$). The proof writes a powerful number uniquely as $a^3b^2$ with $a$ squarefree and counts the $n\le x$ represented by the forms $a_1^3x_1^2+a_2^3x_2^2$ with $(a_1,a_2)=1$; the lower bound uses the forms $x_1^2+p^3x_2^2$ for primes $p\equiv3\pmod4$ with $(\log x)^{(2^{2/3}/3)\log2}\le p\le2(\log x)^{(2^{2/3}/3)\log2}$ such that $L(s,\chi_{-4p})$ has no Siegel zero, through (1.16), and the upper bound uses (1.2) in the intermediate range.

• Background (p. 3): the first author's earlier work, the print's [3], which combines J. Reine Angew. Math. 569 (2004), 213--234 (the problem's [Bl04]) and its part II, J. London Math. Soc. (2) 71 (2005), 69--84, gave $V(x)=x/(\log x)^{(1-2^{-1/3})+o(1)}$. The zbMATH review of part I (Zbl 1051.11050) and summary of part II (Zbl 1166.11312), place this asymptotic in part II; part I proves $x(\log x)^{-0.253}\ll V(x)\ll x(\log x)^{-1/6}\log\log x$.

Compiled scope

The introduction (pp. 1--9) was read on the printed pages, and the statements of Theorems 1--6 and Corollaries 1--2 were checked clause by clause; section 6 (pp. 22--24), the end of section 7 (p. 27) and section 9 (pp. 34--41) were read in outline. No proof was verified, and sections 2--5, the rest of 7, and 8 were not read beyond their headings and openings. The journal pagination was not compared. Nothing here is independently reviewed.

Results. Theorem 1 (p. 5), integer moments of $r_f(n)$; Corollary 1 (pp. 3--4), their two main terms; Theorem 2 (p. 5), the elementary estimate for every $\beta\ge0$; Theorem 3 (p. 6), fixed $D$; Theorem 4 (p. 6), comparison of forms of one discriminant; Theorem 5 (p. 7), the range $\kappa\le L$; Theorem 6 (p. 8), bounds for $N_f(x)$; Corollary 2 (p. 8), sums of two powerful numbers.

Bears on. #1081: the problem's $A(x)$ is the paper's $V(x)$, and Corollary 2 bounds it above and below by $x/(\log x)^{1-2^{-1/3}}$ times powers of $\log\log x$, the lower power unspecified. Since $1-2^{-1/3}\approx0.2063<1/2$, the lower bound alone gives $A(x)\sqrt{\log x}/x\to\infty$, so $A(x)$ is not asymptotic to $cx/\sqrt{\log x}$ for any $c>0$. The paper also conjectures that the upper bound gives the true order. Theorems 5 and 6 bear on the problem only as the inputs of Corollary 2's lower and upper bounds.

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