= Solution
In the level-$D$ convention used here, the <Selberg upper-bound sieve> states that a nonnegative sequence with <sieve distribution> $A_d=Xg(d)+r(d)$ satisfies
$$
S\leq\frac XJ+\sum_{d\leq D}3^{\omega(d)}|r(d)|,\qquad
J=\sum_{\substack{d\leq\sqrt D\\d\text{ squarefree}\\p\mid d\Rightarrow p\in\mathcal P}}\prod_{p\mid d}\frac{g(p)}{1-g(p)}.
$$
It follows by minimizing the quadratic form with the <Selberg sieve weights> and using the <Selberg least-common-multiple weights> bound $|\lambda_d|\leq3^{\omega(d)}$.
Sieve the interval $x<n\leq x+z$ by two and by the <primes> congruent to three modulo four, up to $L=\sqrt D$. The interval <divisor> counts give
$$
A_d=\left\lfloor\frac{x+z}{d}\right\rfloor-\left\lfloor\frac xd\right\rfloor
=\frac zd+O(1).
$$
Thus $X=z$, $g(p)=1/p$, and $r(d)=O(1)$ uniformly in the location of the interval. Take $D=z^{1/2}$ and $L=z^{1/4}$.
To bound $J$, let $y=L^{1/4}$ and form the <Euler product> over the forbidden <primes> at most $y$. The given <reciprocal-prime sum in residue class one modulo four>, subtracted from the <Mertens second theorem>, yields
$$
\sum_{\substack{p\leq y\\p=2\text{ or }p\equiv3\pmod4}}\frac1p
=\tfrac12\log\log y+O(1),\qquad
W=\prod_{\substack{p\leq y\\p=2\text{ or }p\equiv3\pmod4}}(1-1/p)^{-1}\asymp\sqrt{\log y}.
$$
Moreover, the <Mertens first theorem> bounds the logarithmic mean by $\sum_{p\leq y}(\log p)/p=\log y+O(1)\leq\tfrac12\log L$ for large $L$. The <truncated Euler-product lower bound> gives $J\geq W/2\gg\sqrt{\log z}$.
The full-range <summatory bound for three to the prime omega> bounds the error by $O(D\log^2D)=O(\sqrt z\log^2z)$, which is $O(z/\sqrt{\log z})$. Every integer whose <prime factors> are all congruent to one modulo four survives this finite sieve. Therefore the <half-dimensional interval sieve> gives
$$
\boxed{\#\{x<n\leq x+z:\ p\mid n\Rightarrow p\equiv1\pmod4\}\ll\frac z{\sqrt{\log z}}.}
$$
The constants are independent of $x$. Enlarging them covers the bounded range of $z$ before the asymptotic product estimates apply, so the result holds throughout $1000\leq z\leq x$.
Back to article page
