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We present a short, self-contained, and purely combinatorial proof of Linniks theorem: for any $varepsilon > 0$ there exists a constant $C_varepsilon$ such that for any $N$, there are at most $C_varepsilon$ primes $p leqslant N$ such that the least positive quadratic non-residue modulo $p$ exceeds $N^varepsilon$.
Let $p>3$ be a prime, and let $(frac{cdot}p)$ be the Legendre symbol. Let $binmathbb Z$ and $varepsilonin{pm 1}$. We mainly prove that $$left|left{N_p(a,b): 1<a<p text{and} left(frac apright)=varepsilonright}right|=frac{3-(frac{-1}p)}2,$$ where $N_p(
Let $E_1$ and $E_2$ be $overline{mathbb{Q}}$-nonisogenous, semistable elliptic curves over $mathbb{Q}$, having respective conductors $N_{E_1}$ and $N_{E_2}$ and both without complex multiplication. For each prime $p$, denote by $a_{E_i}(p) := p+1-#E_
Let $qgeq 1$ be any integer and let $ epsilon in [frac{1}{11}, frac{1}{2})$ be a given real number. In this short note, we prove that for all primes $p$ satisfying $$ pequiv 1pmod{q}, quad loglog p > frac{log 6.83}{frac{1}{2}-epsilon} mbox{ and } fra
In his paper from 1996 on quadratic forms Heath-Brown developed a version of circle method to count points in the intersection of an unbounded quadric with a lattice of short period, if each point is given a weight. The weight function is assumed to
We compute the statistics of $SL_{d}(mathbb{Z})$ matrices lying on level sets of an integral polynomial defined on $SL_{d}(mathbb{R})$, a result that is a variant of the well known theorem proved by Linnik about the equidistribution of radially proje