We establish sharp lower bounds for the $2k$-th moment in the range $k \geq 1/2$ of the family of quadratic twists of modular $L$-functions at the central point.

In this note, we give a summary of the article ``The distribution of prime ideals of imaginary quadratic fields'' by G. Harman, A. Kumchev and P.A. Lewis and establish analogous results for real quadratic fields based on the same method.

We prove that for any positive integers $k$, $q$, $n$ with $n>N(k)$, integer $c$, and polynomials $f_i(x)$ of degree $k$ whose leading coefficients are relatively prime to $q$, there exists a solution $\underline{x}$ to the congruence $$\sum_{i=1}^n f_i(x_i) \equiv c \pmod q$$ that lies in a cube of side length at least $\max\{q^{1/k},k\}$. Moreover, the result is best possible up to the determination of $N(k)$.

Let $p, m$ be positive integers and $p\geq 2, F_n$ is the Fibonacci number. In this paper, we consider the Diophantineequation $F_{n-d}^m+F_n^m+F_{n+d}^m=y^p$, where $q$ is a prime andwe prove that there are no integer solutions when $3\nmid d, 2\mid m$. Moreover, we study this equation in a particular case $d=2$, $m=1$ or $m=q$, where $q$ is a prime of the type $4k+3$.

As is well known, we can average the eigenfunction $y^s$ of the hyperbolic Laplacian on the hyperbolic plane by $\Gamma$ a lattice in $\mathbf{SL}(2,\mathbb{R})$ to obtain an automorphic form: the non-holomorphic Eisenstein series $E_\mathfrak{a}(z,s)$. In this note, we choose a particular eigenfunction $y^s dx$ of the Hodge-Laplace operator for 1-forms on the hyperbolic plane $\mathbb{H}$. We can average $y^s dx$ by $\Gamma$ to define a 1-form $E_\mathfrak{a}^1 \big( (z,v), s \big)$. We shall prove that $E_\mathfrak{a}^1$ admits a Fourier expansion and calculates the corresponding coefficients. Also, we evaluate the integral $\int_{\gamma} E_\mathfrak{a}^1$ for when $\gamma$ is a lifting of horocycles and closed geodesics in the unit tangent bundle. Finally, we will obtain an analog to the Rankin-Selberg method for $E_\mathfrak{a}^1$ via horocycles.

In [B2], we discussed the multiple base discrete logarithm problem involving two primes in the $n$-th power of $\mathbb{G}_m$ or two points in the $n$-th power of the CM elliptic curve $E_d:y^2=x^3-d^2x$. In this paper, we conduct our research for three primes and three points. These computations lead to new constructions of families of wild $1$-motives (see, [BB2]).

We show that if $h\in\mathbb{Z}[x]$ is a polynomial of degree $k$ such that the congruence $h(x)\equiv 0\pmod{q}$ has a solution for every positive integer $q$, then any subset of $\{1,2,\ldots,N\}$ with no two distinct elements with difference of the form $h(n)$, with $n$ positive integer, has density at most $(\log N)^{-c\log\log\log N}$, for some constant $c$ that depends only on $k$. This improves on the best bound in the literature, due to Rice, and generalizes a recent result of Bloom and Maynard.

The main work of this paper is to study the computational problem of the fourth power mean of one kind generalized Kloosterman sums, and using the analytic methods, the properties of Gauss sums and trigonometric sums; and algebraic geometric techniques to give an asymptotic formula for the fourth power mean. The results in this paper give advancement to the known methods and techniques, which can enlighten the related problems.