On the distribution of modular square roots of primes

Let $q$ be a prime, $P \geq 1$ and let $N_q(P)$ denote the number of rational primes $p \leq P$ that split in the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$. The first part of this paper establishes various unconditional and conditional (under existence of a Siegel zero) lower bounds for $N_q(P)$ in the range $q^{1/4+\varepsilon} \leq P \leq q$, for any fixed $\varepsilon>0$. This improves upon what is implied by work of Pollack and Benli-Pollack. The second part of ...

In this paper we study the distribution of squares modulo a square-free number $q$. We also look at inverse questions for the large sieve in the distribution aspect and we make improvements on existing results on the distribution of $s$-tuples of reduced residues.

The equidistribution of roots of quadratic congruences with prime moduli depends crucially upon effective bounds for a special Weyl linear form. Duke, Friedlander and Iwaniec discovered a strong estimate for this Weyl linear form when the quadratic polynomial has negative discriminant. T\'oth established an analogous but weaker bound when the quadratic polynomial has positive discriminant. We obtain a stronger estimate for the Weyl linear form for quadratics of positive discr...

In our paper, we apply additive-combinatorial methods to study the distribution of the set of squares $\mathcal{R}$ in the prime field. We obtain the best upper bound on the number of gaps in $\mathcal{R}$ at the moment and generalize this result for sets with small doubling.

We study the distribution of spacings between squares modulo q as the number of prime divisors of q tends to infinity. In an earlier paper Kurlberg and Rudnick proved that the spacing distribution for square free q is Poissonian, this paper extends the result to arbitrary q.

Let $p$ be a prime number, $p=2^nq+1$, where $q$ is odd. D. Shanks described an algorithm to compute square roots $\pmod{p}$ which needs $O(\log q + n^2)$ modular multiplications. In this note we describe two modifications of this algorithm. The first needs only $O(\log q + n^{3/2})$ modular multiplications, while the second is a parallel algorithm which needs $n$ processors and takes $O(\log q+n)$ time.

We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory and the geometry of numbers. We give applications of these results to new bounds on correlations between {\it Sali{\'e}} sums and to a new equidistribution estimate for the set of modular roots of primes.

We use the spectral theory of Hilbert-Maass forms for real quadratic fields to obtain the asymptotics of some sums involving the number of representations as a sum of two squares in the ring of integers.

The equation $x^2 + 1 = 0\mod p$ has solutions whenever $p = 2$ or $4n + 1$. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. That the roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec from 1995. We show that a subsequence of the roots of the equation remains equidistributed even when one adds a restriction on the primes which has to do with the angle in...

We investigate the error term of the asymptotic formula for the number of squarefree integers up to some bound, and lying in some arithmetic progression a (mod q). In particular, we prove an upper bound for its variance as a varies over $(\mathbb{Z}/q\mathbb{Z})^{\times}$ which considerably improves upon earlier work of Blomer.