Let $f$ be a cusp form of half-integral weight $k+1/2$, whose Fourier coefficients $a(n)$ are not necessarily real. We prove an extension of the Bruinier-Kohnen conjecture on the equidistribution of the signs of $a(n)$ for the families $\{a(tp^{2\nu})\}_{p,\text{prime}}$, where $\nu$ and $t$ be fixed odd positive integer and square-free integer respectively.

We establish an analogue of a conjecture of Balasubramanian, Conrey, and Heath-Brown for the family of all Dirichlet characters with conductor up to Q. This forms another application of our work in developing an asymptotic large sieve.

We study some ``density functions'' related to the value-distributions of $L$-functions. The first example of such a density function was given by Bohr and Jessen in 1930s for the Riemann zeta-function. In this paper, we construct construct density functions for a wide class of $L$-functions. We prove that certain mean values of $L$-functions in this class are represented as integrals involving the related density functions.

We prove that the titled equation has only finitely many solutions, generalizing a work by Li and Yuan and another work by Mao Hua Le. We apply our method to the specific Diophantine equation considered by Mao Hua Le and obtain all the solutions.

In this paper we prove the Jawerth--Franke embeddings of Herz-type Besov and Triebel-Lizorkin spaces. Moreover, we obtain the Jawerth-Franke embeddings of Besov and Triebel-Lizorkin spaces equipped with power weights, and we prove the necessity of some assumptions on the parameters. As an application we present new embeddings between Besov and Herz spaces.

In this paper, we study the derivatives of an integer-valued polynomial of a given degree. Denoting by $Int_n(\mathbb{Z})$ the set of the integer-valued polynomials with degree $\leq n$, we show that the smallest positive integer $c_n$ satisfying the property: $\forall P \in Int_n(\mathbb{Z}), c_n P' \in Int_n(\mathbb{Z})$ is $c_n = \lcm(1 , 2 , \dots , n)$. As an application, we deduce an easy proof of the well-known inequality $\lcm(1 , 2 , \dots , n) \geq 2^{n - 1}$ ($\forall n \geq 1$). In the second part of the paper, we generalize our result for the derivative of a given order. Leaning on this study, we conclude the paper by determining, for a given natural number $n$, the smallest positive integer $\lambda_n$ satisfying the property: $\forall P \in Int_n(\mathbb{Z})$, $\forall k \in \mathbb{N}$: $\lambda_n P^{(k)} \in Int_n(\mathbb{Z})$.

We extend a bound of Roche-Newton, Shparlinski and Winterhof which says any subset of a finite field can be decomposed into two disjoint subset $\mathcal{U}$ and $\mathcal{V}$ of which the additive energy of $\mathcal{U}$ and $f(\mathcal{V})$ are small, for suitably chosen rational functions $f$. We extend the result by proving equivalent results over multiplicative energy and the additive and multiplicative energy hybrids.

We extend to logarithmic class groups the results on abelian principalization of tame ray class groups of a number field obtained in a previous article. As a consequence, for any extension $K/k$ of number fields which satisfies the Gross-Kuz'min conjecture for the prime $\ell$ and where at least one of the infinite places completely splits, we prove that there exists infinitely many abelian $\ell$-extensions $F/k$ such that the relative subgroup $\mathcal{C}\ell_{K/k} = \Ker (\mathcal{C}\ell_K\to\mathcal{C}\ell_\k)$ of the $\ell$-group of logarithmic classes of $K$ capitulates in the compositum $FK$.