On the distribution of modular square roots of primes

We propose a method for determining which integers can be written as a sum of two integral squares for quadratic fields $\Q(\sqrt{\pm p})$, where $p$ is a prime.

Uniformly for small $q$ and $(a,q)=1$, we obtain an estimate for the weighted number of ways a sufficiently large integer can be represented as the sum of a prime congruent to $a$ modulo $q$ and a square-free integer. Our method is based on the notion of local model developed by Ramar\'e and may be viewed as an abstract circle method.

The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0\pmod\ell$ for the primes $\ell=5, 7, 11$, and it is known that there are no others of this form. On the other hand, for every prime $\ell\geq 5$ there are infinitely many examples of congruences of the form $p(\ell Q^m n+\beta)\equiv 0\pmod\ell$ where $Q\geq 5$ i...

We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\sum_g(\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\ldots,p-1$, and $(\frac{\cdot}p)$ denotes the Legendre symbol. On the basis of our numerical computations, we formulate 35 conjectures involving primitive roots modulo primes. For example, we conjecture that for any prime $p$ there is a primitive root $...

We show that the exponent of distribution of the sequence of squarefree numbers in arithmetic progressions of prime modulus is $\geq 2/3 + 1/57$, improving a result of Prachar from 1958. Our main tool is an upper bound for certain bilinear sums of exponential sums which resemble Kloosterman sums, going beyond what can be obtained by the Polya-Vinogradov completion method.

We find an asymptotic formula for the number of primitive vectors $(z_1,\ldots,z_4)\in (\mathbb{Z}_{\neq 0})^4$ such that $z_1,\ldots, z_4$ are all squareful and bounded by $B$, and $z_1+\cdots + z_4 = 0$. Our result agrees in the power of $B$ and $\log B$ with the Campana-Manin conjecture of Pieropan, Smeets, Tanimoto and V\'{a}rilly-Alvarado.

Given a negative $D>-(\log X)^{\log 2-\delta}$, we give a new upper bound on the number of square free integers $<X$ which are represented by some but not all forms of the genus of a primitive positive definite binary quadratic form $f$ of discriminant $D$. We also give an analogous upper bound for square free integers of the form $q+a<X$ where $q$ is prime and $a\in\mathbb Z$ is fixed. Combined with the 1/2-dimensional sieve of Iwaniec, this yields a lower bound on the numbe...

We investigate the average distribution of primes represented by positive definite integral binary quadratic forms, the average being taken over negative fundamental discriminants in long ranges. In particular, we prove corresponding results of Bombieri-Vinogradov type and of Barban-Davenport-Halberstam type, although with shorter ranges than in the original theorems for primes in arithmetic progressions: The results imply that, for all $a>0$, the least prime that can be repr...

Artin's Conjecture on Primitive Roots states that a non-square nonunit integer $a$ is a primitive root modulo $p$ for the positive proportion of $p$. This conjecture remains open, but on average, there are many results due to P. J. Stephens. There is a natural generalization of the conjecture for composite moduli. We can consider $a$ as the primitive root modulo, $(\mathbb{Z}/n\mathbb{Z})^{*}$ if $a$ is an element of the maximal exponent in the group. The behavior is more com...

In this paper we extend and improve all the previous results known in literature about weighted average, with Ces\`aro weight, of representations of an integer as sum of a positive arbitrary number of prime powers and a non-negative arbitrary number of squares. Our result includes all cases dealt with so far and allows us to obtain the best possible outcome using the chosen technique.