bibkey: baier2016squaresieve authors: Stephan Baier year: 2016 title: The square sieve and the large sieve with square moduli doi: null url: https://arxiv.org/abs/1506.06148v2 claim: “Lemma 2 recalls Heath-Brown’s weighted square sieve, retaining the Jacobi correlations and the support restriction |d|<exp(L) for L sieve primes.” strata_touched: [] license: citation-only triage: anchor

The weighted square-sieve input

The inspected primary text is arXiv:1506.06148v2, updated 2016-06-07; the first version was submitted 2015-06-19. The locator used here is Lemma 2, printed p.2, rather than the paper’s main large-sieve theorem. This is an existing theorem, not a new estimate or a statement of the latest large-sieve bound.

Baier credits D. R. Heath-Brown, The square sieve and consecutive square-free numbers, Mathematische Annalen 266 (1984), 251–259, Theorem 1. The lemma below was checked in Baier’s primary text; this note does not claim an independent full-text inspection of Heath-Brown’s article.

Let $\mathcal{S}$ contain $L\ge 1$ distinct primes. For a nonnegative weight $w$ supported on positive integers, impose the lemma’s additional restriction

$$w(d)=0\text{if }d=0\text{ or }|d|\ge \exp (L).$$

The weighted number of positive squares then satisfies

$$\sum _{t\ge 1}w(t^2)\ll \frac{1}{L}\sum _{d\in \mathbb{Z}}w(d)+\frac{1}{L^2}\sum _{\begin{array}{c}p,q\in \mathcal{S}\\ p\ne q\end{array}}\left|\sum _{d\in \mathbb{Z}}w(d)\left(\frac{d}{pq}\right)\right|.\text{\hspace{1em}(B1)}$$

The pair sum is ordered. In applications below all sieve primes are odd, so the Jacobi symbol is unambiguous. Its value at a nonunit is zero; replacing that value by one changes the correlation term. The support cap is also part of the quoted lemma. A proposed use with larger discriminants needs another justified sieve formulation or a new choice of primes before (B1) can be applied.

No Lean verification or originality claim is supplied. The primary input alone does not bound the correlations of a population selected by its own prime divisibility conditions.

Conditioning on the actual FIB prime support

For each actual positive integer $n$, let $h\in \{0,1\}$ be its canonical Zeckendorf unit bit and $(A,B)$ its complete remaining five-window composition. Use the same source throughout:

$$n=h+2A+3B,Q=A^2+AB-B^2,c=4A+7B,D=5h^2-4Q.$$

The existing FIB theory, §202.1 gives

$$c^2-D=5n(n-2h).\text{\hspace{1em}(B2)}$$

Take a finite population $\mathcal{N}$ with $D(n)>0$, nonnegative weights $a_n$, and a common set $\mathcal{S}$ of $L$ odd primes dividing every $n\in \mathcal{N}$. Define

$$w(d)=\sum _{n\in \mathcal{N}}{a}_n{1}_{D(n)=d},k(n)=\#\{p\in \mathcal{S}:p\nmid D(n)\}.$$

For each $p\in \mathcal{S}$, (B2) forces $D(n)\equiv c(n{)}^2(\mathrm{mod}p)$. Thus every nonzero symbol is one, while a zero remains zero. For distinct $p,q\in \mathcal{S}$,

$$\sum _{d}w(d)\left(\frac{d}{pq}\right)=\sum _{n\in \mathcal{N}}{a}_n{1}_{\gcd (D(n),pq)=1}\ge 0.$$

Consequently the entire normalized correlation term is exactly

$$\frac{1}{L^2}\sum _{\begin{array}{c}p,q\in \mathcal{S}\\ p\ne q\end{array}}\left|\sum _{d}w(d)\left(\frac{d}{pq}\right)\right|=\sum _{n\in \mathcal{N}}{a}_n\frac{k(n)(k(n)-1)}{L^2}.\text{\hspace{1em}(B3)}$$

If every discriminant is coprime to ${\prod }_{p\in \mathcal{S}}p$, (B3) equals $(1-1/L){\sum }_n{a}_n$. There is no cancellation saving from these support primes, even when the discriminants are nonsquares. As a concrete canonical example, $n=360$ has $h=1$, $(A,B)=(52,85)$, $c=803$, and $D=409$. Both $(409/3)$ and $(409/5)$ equal one. This example verifies the symbol mechanism; with just those two sieve primes, $409>\exp (2)$ prevents applying (B1).

The zero-sensitive formula (B3) holds without the coprimality assumption. Writing $z(n)=L-k(n)$ gives

$$k(n)(k(n)-1)=L(L-1)-(2L-1)z(n)+z(n{)}^2.\text{\hspace{1em}(B4)}$$

Any saving attributed to zeros must control the actual weighted incidence sufficiently to bound (B3); (B4) records the exact first- and second-moment combination. Dropping zeros, or assigning independent quadratic signs after imposing $p\mid n$, does not estimate this quantity. The common-prime assumption is essential; changing the sieve set separately for each candidate needs a separately justified argument.

The two canonical unit branches have different zero loci

If $h=0$ and $g=\gcd (A,B)>0$, then for every odd $p\mid n$,

$$p\mid D⟺p\mid c⟺p\mid g.\text{\hspace{1em}(B5)}$$

Indeed, (B2) gives the first equivalence, and the two rows $(2,3)$ and $(4,7)$ have determinant two. Over an odd prime, their simultaneous zero therefore forces both composition coordinates to vanish. Conversely $p\mid g$ makes $n,c,D$ vanish modulo $p$. Here $z(n)$ counts the selected support primes dividing this same composition gcd.

For $h=1$, $n=1+g(2a+3b)$ implies $\gcd (n,g)=1$ after writing $(A,B)=g(a,b)$. Nevertheless $p\mid n,D$ can occur when $4a+7b\equiv 0(\mathrm{mod}p)$. For example $n=30$ has $h=1$, $(A,B)=(4,7)$, $g=1$, and $D=25$: the support prime five divides $D$ while it does not divide $g$. The equivalence $p\mid D⟺p\mid c$ persists, but (B5)’s last equivalence does not transfer to this unit branch.

The quoted sieve’s size restriction

Suppose a candidate sequence additionally has $D(n)\asymp n$, $P^{+}(n)\asymp \log n$, and all selected sieve primes at most $P^{+}(n)$. The prime-counting bound then gives

$$L\le \pi (P^{+}(n))=O\left(\frac{\log n}{\log \log n}\right),\exp (L)=n^{o(1)}<D(n)$$

eventually. Its positive-weight discriminants violate the support cap of (B1). The hypothesis $D(n)\asymp n$ has not been established for actual SA/CA candidates; the fixed-support density example in the complementary-divisor note cannot supply it there.

These are paper applications of the cited lemma and the existing finite identity. They diagnose this choice of support primes, preserving zeros and all source weights. They supply neither a square-discriminant exclusion nor the pointwise signed Euler deficit required for Robin; modified sieves and primes outside the candidate support remain separate possibilities.

A polynomial score that also retains zeros

T. D. Browning, The polynomial sieve and equal sums of like polynomials, arXiv:1306.6767v1, submitted 2013-06-28. The inspected locators are Theorem 1.1, printed p.2, and its proof in §2, printed p.5. The source permits nonnegative summable weights, a fixed integer polynomial $f(x;y)$ of degree $d$ in $x$, any nonzero auxiliary polynomial, and a positive integer parameter $\alpha$. The theorem retains a height cutoff $|y|<\exp (L)$ and permits constants depending on the fixed polynomials.

Take $f(x;y)=x^2-y$ and auxiliary polynomial one. The coefficient gcd is one, so this choice removes no primes dividing $y$. The proof’s local root count and score are

$${\nu }_p(y)=\#\{x\mathrm{mod}p:x^2=y\},F_{\alpha }(\nu )=\alpha +(\nu -1)(2-\nu ).$$

For the common odd support primes above, ${\nu }_p(D(n))$ is two when $p\nmid D(n)$ and one when $p\mid D(n)$. Both have score $\alpha$. The expression in the proof therefore specializes exactly to

$$\sum _n{a}_n{\left(\sum _{p\in \mathcal{S}}{F}_{\alpha }({\nu }_p(D(n)))\right)}^2={\alpha }^2{L}^2\sum _n{a}_n.$$

Each $p,q$ contribution after combining the source’s nine coefficient terms equals ${\alpha }^2{\sum }_n{a}_n$; the individual root-count moments need not be equal. This zero-sensitive quadratic score also provides no saving on the forced support configuration.

For this monic specialization one can directly prove, at every height,

$$1_{\{y\text{ an integer square}\}}\le \frac{{\left({\sum }_{p\in \mathcal{S}}{F}_{\alpha }({\nu }_p(y))\right)}^2}{{\alpha }^2{L}^2}.$$

An integer square has a root at every prime and hence score $\alpha$ at every selected odd prime; the right side is nonnegative for every other $y$. This direct majorant has no height condition, but equals one on the support-conditioned discriminants. It does not remove the height condition from the quoted general theorem. Prime-power conditions, different polynomials and primes outside the forced support are not excluded by this computation.