Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA note on Linniks Theorem on quadratic non-residues
267
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $N^varepsilon$.
rate research
Read More
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 $N_p(a,b)$ is the number of positive integers $x<p/2$ with ${x^2+b}_p>{ax^2+b}_p$, and ${m}_p$ with $minmathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$.
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 $a_{E_i}(p) := p+1-#E_i(mathbb{F}_p)$ the trace of Frobenius. Under the assumption of the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power $L$-functions $L(s, mathrm{Sym}^i E_1otimesmathrm{Sym}^j E_2)$ where $i,jin{0,1,2}$, we prove an explicit result that can be stated succinctly as follows: there exists a prime $p mid N_{E_1}N_{E_2}$ such that $a_{E_1}(p)a_{E_2}(p)<0$ and [ p < big( (32+o(1))cdot log N_{E_1} N_{E_2}big)^2. ] This improves and makes explicit a result of Bucur and Kedlaya. Now, if $Isubset[-1,1]$ is a subinterval with Sato-Tate measure $mu$ and if the symmetric power $L$-functions $L(s, mathrm{Sym}^k E_1)$ are functorial and satisfy GRH for all $k le 8/mu$, we employ similar techniques to prove an explicit result that can be stated succinctly as follows: there exists a prime $p mid N_{E_1}$ such that $a_{E_1}(p)/(2sqrt{p})in I$ and [ p < left((21+o(1)) cdot mu^{-2}log (N_{E_1}/mu)right)^2. ]
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 $$ pequiv 1pmod{q}, quad loglog p > frac{log 6.83}{frac{1}{2}-epsilon} mbox{ and } frac{phi(p-1)}{p-1} leq frac{1}{2} - epsilon, $$ there exists a quadratic non-residue $g$ which is not a primitive root modulo $p$ such that $gcdleft(g, frac{p-1}{q}right) = 1$.
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 be $C_0^infty$-smooth and to vanish near the singularity of the quadric. In out work we allow the weight function to be finitely smooth and not vanish near the singularity, and we give also an explicit dependence on the weight function.
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 projected integral vectors from a large sphere into the unit sphere. Using the above result we generalize the work of Aka, Einsiedler and Shapira in various directions. For example, we compute the joint distribution of the residue classes modulo $q$ and the properly normalized orthogonal lattices of primitive integral vectors lying on the level set $-(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})+x_{4}^{2}=N$ as $Ntoinfty$, where the normalized orthogonal lattices sit in a submanifold of the moduli space of rank-$3$ discrete subgroups of $mathbb{R}^{4}$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
