For any abelian group $A$, we prove an asymptotic formula for the number of $A$-extensions $K/\mathbb {Q}$ of bounded discriminant such that the associated norm one torus $R_{K/\mathbb {Q}}^1 \mathbb {G}_m$ satisfies weak approximation. We are also able to produce new results on the Hasse norm principle and to provide new explicit values for the leading constant in some instances of Malle's conjecture.

We prove a lower bound for the large sieve with square moduli.

Let $\lambda $ be a fixed integer exceeding 1 and ${{s}_{n}}$ any strictly increasing sequence of positive integers satisfying ${{s}_{n}}\le {{n}^{15/14+o(1)}}$. In this paper we give a version of the large sieve inequality for the sequence ${{\lambda }^{{{s}_{n}}}}$. In particular, we obtain nontrivial estimates of the associated trigonometric sums “on average” and establish equidistribution properties of the numbers ${{\lambda }^{{{s}_{n}}}},n\le p{{(\log p)}^{2+\varepsilon }}$, modulo $p$ for most primes $p$.

In this paper, we study the orthogonalities of Hecke eigenvalues of holomorphic cusp forms. An asymptotic large sieve with an unusually large main term for cusp forms is obtained. A family of special vectors formed by products of Kloosterman sums and Bessel functions is constructed for which the main term is exceptionally large. This surprising phenomenon reveals an interesting fact: that Fourier coefficients of cusp forms favor the direction of products of Kloosterman sums and Bessel functions of compatible type.

For an integer $n$, let $d\left( n \right)$ denote the ordinary divisor function. This paper studies the asymptotic behavior of the sum

$$S\left( x \right)\,:=\sum\limits_{m\le x,n\le x}{d\left( {{m}^{2}}+{{n}^{2}} \right)}$$

It is proved in the paper that, as $x\,\to \,\infty $,

$$S(x):={{A}_{1}}{{x}^{2}}\log x+{{A}_{2}}{{x}^{2}}+{{O}_{\in }}({{x}^{\frac{3}{2}+\in }}),$$

where ${{A}_{1}}$ and ${{A}_{2}}$ are certain constants and $\in $ is any fixed positive real number.

The result corrects a false formula given in a paper of Gafurov concerning the same problem, and improves the error $O({{x}^{\frac{5}{3}}}\,{{(\log \,x)}^{9}})$ claimed by Gafurov.