The hyperbola \(xy=N\) (Q5939697)

Let \(xy=N\), be a given hyperbola \(xy=N\), where \(N\) is a natural number. The authors ask for the distribution of at most \(k\) lattice points on the curve or, what is the same, for the distribution of at most \(k\) divisors of \(N\). Let \(k\) be a fixed natural number and \(0< \gamma< 1\). Define NEWLINE\[NEWLINE\varepsilon_k(\gamma)= \liminf \{\varepsilon\mid N^\gamma\ll a_1<\cdots< a_k\leq a_1+ N^\varepsilon\},NEWLINE\]NEWLINE where all \(a_i\) are divisors of \(N\). Then it is proved that there is a constant \(c>0\) with NEWLINE\[NEWLINE\varepsilon_k \biggl( \frac 12 \biggr)< \frac 12- \frac{c} {\log k}.NEWLINE\]NEWLINE Another result concerns the divisor function \(d_\alpha(n)= \#\{(a,b)\mid a,b\in (N,N+N^\alpha]\), \(ab=n\}\). It is proved that for fixed \(0< \alpha< 1\) NEWLINE\[NEWLINE\sum d_\alpha^2(n)= 2N^{2\alpha}+ O(N^{3\alpha-1} \log N)+ O(N^\alpha).NEWLINE\]