## Equidistribution of roots of a quadratic congruence to prime moduli.(English)Zbl 0840.11003

It is shown that for $$f(X)= aX^2+ 2bX+ c\in \mathbb{Z}[X]$$ with $$D= ac- b^2> 0$$, the number of pairs $$(p, \nu)$$ of primes $$p\leq x$$ and integers $$\nu\in [\alpha p, \beta p)$$ satisfying $$f(\nu)\equiv 0\bmod p$$, is asymptotic to $$(\beta- \alpha) \pi(x)$$ as $$x\to \infty$$, for any $$0\leq \alpha< \beta\leq 1$$. This theorem implies a uniform distribution result for angles of Salié sums. With the Weyl principle, the proof consists of showing that $\sum_{p\leq x} \rho_h(p)= \sum_{p\leq x} \sum_{\nu\bmod p, f(\nu)\equiv 0\bmod p} e^{2\pi ih\nu/p}= o(\pi(x))\quad (x\to \infty)$ for all $$h\in \mathbb{Z}\backslash \{0\}$$. Here $$\pi(x)$$ is the prime counting function.
The proof starts from a similar equidistribution result where the modulus runs through all natural numbers [see C. Hooley, Mathematika 11, 39-49 (1964; Zbl 0123.25802)]. The reduction to a sum over the primes is obtained by a complicated sieving procedure. Estimates are needed for the sums $$\sum_{M< m\leq 2M} \rho_h(dm)$$ for a given $$d\geq 1$$, and $$\sum_{n, m} \alpha_m \beta_n \rho_h(nm)$$, where $$M< m\leq 2M$$, $$N< n\leq 2N$$ and $$(n, m)= 1$$. These estimates are obtained by studying the spectral decomposition of certain Poincaré series for congruence groups of $$\text{SL}(2, \mathbb{Z})$$.

### MSC:

 11A07 Congruences; primitive roots; residue systems 11L20 Sums over primes 11N36 Applications of sieve methods

Zbl 0123.25802
Full Text: