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Let $x$ and $y$ be given real numbers. We may suppose that $2\leqslant x \leqslant y$ and that $u:= \log(y)/\log(x)$ remains bounded in a compact set away from $1$ as $x,y\to\infty$. An integer $n$ is ...

In Shiu's work ('A Brun-Titchmarsh theorem for multiplicative functions') he proved that if $r\le x$ is a natural number, we have $$\sum_{r<n\le x}d(n)d(n-r)\ll x\log^2x\sum_{d|r}\frac{1}{d}.$$ I ...

I am reading section 12 (Flipping Moduli) of the paper "The polynomial $X^2+Y^4$ captures its primes" by Friedlander and Iwaniec. At page 997, just below equation (12.7) we start estimating the ...