Let $b:\mathbb N\to \{0,1\}$ be the indicator function of integers that are a sum of two non-zero integer squares. Let $f(t)\in \mathbb Z[t]$ be an irreducible polynomial of degree $2$ with positive leading coefficient and not of the form $(at+b)^2+c^2$ for some rationals $a,b,c$. Then standard sieve heuristics would suggest that $$ \frac{x}{\sqrt{\log x} }\ll \sum_{n\leq x } b(|f(t)|) \ll \frac{x}{\sqrt{\log x}}.$$ My question is: has this problem been studied before and are there any results confirming the lower bound, even for at least one polynomial $f$? The upper bounds are directly proved by the Brun or the Selberg sieve.
Number Theory – Quadratic Polynomial Values as Sum of Two Integer Squares
analytic-number-theorynt.number-theorysums-of-squares
Related Solutions
The proof given in Iwaniec-Kowalski is, as it stands, wrong. It can be easily fixed, as I explain below.
In general, one can think of $\nu(n)^2$ as the characteristic function of integers with $P^-(n):=\min(p|n)\ge y$. So $$ \sum_{n\le x} \frac{\nu(n)^2 f(n)}{n} \approx \sum_{ n\le x,\ P^-(n)\ge y } \frac{f(n)}{n} \asymp \prod_{y\le p\le x} \left(1+\frac{f(p)}{p}\right) $$ for any reasonable multiplicative function $f$ that is bounded on primes. However, an important restriction is that $f(p)<\kappa$ on average, where $\kappa$ is the sifting dimension. In this case the dimension is 1, whereas $f(p)=8$. So the sum in question, say $S$, does not satisfy a priori the claimed bound. In fact, in this case $S\gg(\log x/\log y)^8$:
If $P^+(n)=\max(p|n)$, then we have that $$ S= \sum_{n\le x}\frac{\nu(n)^2\tau(n)^3}{n} \asymp \sum_{P^+(n)\le x}\frac{\nu(n)^2\tau(n)^3}{n} $$ (this step is heuristic and employed for simplicity). If we let $\sigma_m=\sum_{[d_1,d_2]=m}\theta_{d_1}\theta_{d_2}$ and we open $\nu(m)^2$, we have that $$ S \approx \sum_{P^+(m)\le x} \sigma_m \sum_{P^+(n)\le x,\ m|n} \frac{\tau(n)^3}{n} = P(x) \sum_{P^+(m)\le x} \frac{\sigma_m g(m)}{m}, $$ where $g(m)$ is a multiplicative function with $g(p)=8+O(1/p)$ and $P(x)=\prod_{p\le x}(1+\tau(p)^3/p+\tau(p^2)^3/p^2+\cdots)\asymp(\log x)^8$. Hence $$ S/P(x)\approx \sum_{P^+(d_1d_2)\le x} \frac{\theta_{d_1}\theta_{d_2}g([d_1,d_2])}{[d_1,d_2]} = \sum_{P^+(d_1d_2)\le x} \frac{\theta_{d_1}\theta_{d_2}g(d_1)g(d_2)}{d_1d_2} \frac{(d_1,d_2)}{g((d_1,d_2))}. $$ Writing $f(m)=\prod_{p|m}(1-g(p)/p)$ so that $m/g(m)=\sum_{n|m}f(n)n/g(n)$, we deduce that $$ S/P(x)\approx \sum_{P^+(n)\le x}\frac{f(n)n}{g(n)} \left( \sum_{P^+(d)\le x,\ n|d} \frac{\theta_{d}g(d)}{d}\right)^2. $$ When $y/2 \lt n\le y$, we have that $$ \sum_{P^+(d)\le x,\ n|d} \frac{\theta_{d}g(d)}{d} = \frac{\theta_n g(n)}{n} = \frac{\mu(n)g(n)}{G} \sum_{k\le y,\ (k,q)=1,\ n|k} \frac{\mu^2(k)}{\phi(k)} = \frac{\mu(n)g(n)}{G} \cdot \frac{\textbf{1}_{(n,q)=1}}{\phi(n)} $$ (note that there is an error in the definition of $\theta_b$ in Iwaniec-Kowalski, as one has to restrict them on those $b$ which are coprime to $q$. In fact, $\theta_b=(\mu(b)b/G) \sum_{k\le y,\ (k,q)=1,\ b|k}\mu^2(k)/\phi(k)$). So $$ S \gtrsim \frac{P(x)}{G^2} \sum_{y/2 \lt n\le y,\ (n,q)=1} \frac{\mu^2(n) f(n)g(n)n}{\phi(n)^2} \gg_q (\log x)^8(\log y)^5, $$ by the Selberg-Delange method. This is certainly bigger than what we would need.
In order to remedy this deficiency, one would have to choose $\nu(n)$ in another way, as weights from a sieve of dimension 8 or higher. The easiest choice to work with is, as GH also points out, is to define $\nu$ via the relation $\nu(n)^2=(1*\lambda)(n)$, where $(\lambda(d):d\le D)$ is a $\beta$ upper bound sieve of level $y$ and dimension 8, so that $$ S = \sum_{n\le x} \frac{(1*\lambda)(n)\tau(n)^3}{n}. $$ The point is that the sequence $(\lambda(d))_{d\le D}$ satisfies the Fundamental Lemma (Lemma 6.3 in Iwaniec-Kowalski) in the following strong sense:
(1) $\lambda(d)$ is supported on square-free integers with $P^+(d)\le y$
(2) whenever $\{a(d)\}_{d\ge1}$ is a sequence such that $$ \bigg|\sum_{P^+(d)\le z}\frac{\mu(d)a(dm)}{d}\bigg| \le Ag(m)\prod_{y_0\le p\le z}(1-g(p)/p) \quad\text{when}\ P^-(m)>z, $$ where $A\ge0$ and $y_0\ge1$ are some parameters and $g\ge0$ is multiplicative with $$ \prod_{w\le p\le w'} (1-g(p)/p)^{-1} \le \left(\frac{\log w'}{\log w}\right)^8\left(1+\frac{K}{\log w}\right)\qquad(w'\ge w\ge y_0), $$ we have $$ \sum_{d\le D} \frac{\lambda(d)a(d)}{d} = \sum_{P^+(d)\le y} \frac{\mu(d)a(d)}{d} + O_K\left( Ae^{-u}\prod_{y_0\le p\le y}\left(1-\frac{g(p)}{p}\right) \right), $$ where $u=\log D/\log y$.
Remark: if $a=g$, then the condition on $a$ holds trivially with $A=y_0=1$. Hence, the above statement is indeed a generalization of the usual Fundamental Lemma.
We have that $$ S = \sum_{d\le D} \frac{\lambda(d)a(d)}{d}, $$ where $$ a(d) = \sum_{m\le x/d} \frac{\tau(dm)^3}{m}. $$ If $P^-(m)>z$, then \begin{align*} \sum_{P^+(d)\le z} \frac{\mu(d)a(dm)}{d} &= \sum_{P^+(d)\le z} \frac{\mu(d)}{d}\sum_{k\le x/(dm)}\frac{\tau(dkm)^3}{k} \\ &= \sum_{n\le x/m} \frac{\tau(nm)^3}{n} \sum_{dk=n,\,P^+(d)\le z}\mu(d). \end{align*} By M\"obius inversion, we then conclude that \begin{align*} \sum_{P^+(d)\le z} \frac{\mu(d)\tau(d)^3a(dm)}{d} &= \sum_{n\le x/m,\,P^-(n)>z} \frac{\tau(nm)^3}{n} \\ &\ll \tau(m)^3 (\log x/\log z)^8 \\ &\asymp (\log x)^8\tau(m)^3\prod_{11\le p\le z}(1-\tau(p)^3/p), \end{align*} so that the second axiom holds for $a$ with $A\asymp(\log x)^8$, $y_0=11$ and $g=\tau^3$. We thus conclude that \begin{align*} S = \sum_{d\le D} \frac{\lambda(d)a(d)}{d} &= \sum_{P^+(d)\le y} \frac{\mu(d)a(d)}{d} + O\left( e^{-\log x/\log y} \left(\frac{\log x}{\log y}\right)^8\right) \newline &= \sum_{n\le x,\ P^-(n)>y} \frac{\tau(n)^3 }{n} + O\left( e^{-\log x/\log y} \left(\frac{\log x}{\log y}\right)^8\right) \newline &\ll \left(\frac{\log x}{\log y}\right)^8. \end{align*}
A more general remark: Selberg's sieve is not as flexible as the $\beta$-sieve as far as ``preliminary sieving'' is concerned because it carries inside it the sieve problem it is applied to, in contrast to the $\beta$-sieve weights that only depend on the sifting dimension via the $\beta$ parameter.
EDIT: the monotonicity principle II described in page 49 of "Opera de Cribro" by Friedlander-Iwaniec is a way to use Selberg's sieve weights for preliminary sieving. See also Proposition 7.2 in the same book.
There is some useful information in the paper P. Erdôs and I. Kátai. On the growth of some additive functions of small intervals. Acta Math. Hungar. 33 (1979), 345-359.
Let $$O_{k}(n)=\max _{j=1, \ldots, k} \omega(n+j), \quad o_{k}(n)=\min _{j=1, \ldots, k} \omega(n+j).$$ Then (see I. Kátai. Local growth of the number of the divisors of consecutive integers. Publ. Math. Debrecen 18 (1971), 171-175.) for every $\varepsilon>0$, apart from a set of $n$'s having zero density, the inequalities $$ O_{k}(n) \leqq(1+\varepsilon) \varrho\left(\frac{\log k}{\log \log n}\right) \log \log n, \quad o_{k}(n) \geqq(1-\varepsilon) \bar{\varrho}\left(\frac{\log k}{\log \log n}\right) \log \log n $$ hold for every $k=1,2, \ldots$ Here $\varrho(u)$ $(u \ge 0)$ is defined as the inverse function of $\psi(z)=z \log \frac{z}{e}+1$ defined in $z \ge 1$, and $\bar{\varrho}(n)$ $(n \ge 0)$ is the inverse function of the same $\psi(z)$ defined in $0<z \le 1$. In the same paper it was conjectured that $$ O_{k}(n) \geqq(1-\varepsilon) \varrho\left(\frac{\log k}{\log \log n}\right) \log \log n $$ and $$ o_{k}(n) \leqq(1+\varepsilon) \bar{\varrho}\left(\frac{\log k}{\log \log n}\right) \log \log n $$
They prove that for every $\varepsilon>0$ these irequalities hold for every $k \ge 1$, apart from a set of $n$'s having zero density.
Best Answer
This problem has been addressed in a paper of Friedlander and Iwaniec in Acta Mathematica 1978, called Quadratic polynomials and quadratic forms. Under general conditions they count the number of integers $n\le x$ for which the values $g(n) = an^2+ bn+c$ (with $a>0$, and $a$, $b$, $c$ integers) may be represented by a given quadratic form $\phi(u,v) = Au^2+Buv+Cv^2$. The main result gives a lower bound of order $x/\sqrt{\log x}$ as may be expected, provided a (necessary) local condition is met.