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Unless I'm mistaken, this is quite easy and only involves the multiplicative property of the quadratic residue. If $n_p=n_q$ then we are done, because this very number is a simultaneous quadratic non …

Forgive me for my ignorance, but I'm very surprised to learn that there are two Vinogradovs, both famous in the field of analytic number theory. Guessing from their names and the Russian naming conven …

Let $N_{\bf n}$ be the number of solutions to $\sum x_i^s=n_i$. Them you only need to show $\sum N_{\bf m}N_{\bf m+n}\le\sum N_{\bf n}^2$, which is just Cauchy-Schwarz.

Bombieri and Pila had a well known bound for the count of lattice points on an algebraic curve in the plane. Does it generalize to a bound for the count of lattice points near (say within a distance o …

Suppose $f$ is a cusp form of half integral weight $k$ w.r.t. the group $\Gamma_0(4)$ ($k$ is not very low, can assume $k \ge 11/2$), and $a_n$ is its Fourier coefficient. The Linnik bound says that i …

I'm going through the last steps of Bourgain and Demeter's proof of the $l^2$ decoupling conjecture, but I'm unable to see how the first inequality in (43) goes through. I'll water down the question a …

By the Pólya-Vinogradov inequality, $|S(\chi_N,x)|\ll (p_1\cdots p_N)^{1/2}$, whose order you can estimate using the prime number theorem.

Let $f$ be a homogeneous polynomial with integral coefficients of 4 variables $a$, $b$, $c$ and $d$. Suppose $f$ is invariant under the rotation that rotates $(a,b)\in\mathbb{R}^2$ and $(c,d)\in\mathb …

Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the …