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 \(\text{SL}(2, \mathbb{Z})\).


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


Zbl 0123.25802
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