Let $[·]$ be the floor function. In this paper, we prove by asymptotic formula that when $1<c<\frac{3441}{2539}$, then every sufficiently large positive integer $N$ can be represented in the form $$N=\left[p_1^c\right]+\left[p_2^c\right]+\left[p_3^c\right]+\left[p_4^c\right]+\left[p_5^c\right],$$ where $p_1$, $p_2$, $p_3$, $p_4$, $p_5$ are primes such that $p_1=x^2+y^2+1$.

Consider the linear congruence equation $x_1+...+x_k\equiv b(mod{n}^s)$ for $b\in \mathbb{Z}$, $n,s\in \mathbb{N}$. Let ${(a,b)}_s$ denote the generalized gcd of $a$ and $b$ which is the largest $l^s$ with $l\in \mathbb{N}$ dividing $a$ and $b$ simultaneously. Let $d_1,...,d_{\tau \left(n\right)}$ be all positive divisors of $n$. For each $d_j\mid n$, define ${𝒞}_{j,s}\left(n\right)=\{1\le x\le n^s:{(x,n^s)}_s=d_j^s\}$. K. Bibak et al. (2016) gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on $x_i$. We generalize their result with generalized gcd restrictions on $x_i$ and prove that for the above linear congruence, the number of solutions...

Let $q\ge 3$ be a positive integer. For any integers $m$ and $n$, the two-term exponential sum $C(m,n,k;q)$ is defined by $C(m,n,k;q)={\sum }_{a=1}^{q}e((m{a}^k+na)/q)$, where $e\left(y\right)={\mathrm{e}}^{2\pi \mathrm{i}y}$. In this paper, we use the properties of Gauss sums and the estimate for Dirichlet character of polynomials to study the mean value problem involving two-term exponential sums and Dirichlet character of polynomials, and give an interesting asymptotic formula for it.

The main purpose of this paper is to use the M. Toyoizumi's important work, the properties of the Dedekind sums and the estimates for character sums to study a hybrid mean value of the Dedekind sums, and give a sharper asymptotic formula for it.

The main purpose of this paper is using the mean value formula of Dirichlet L-functions and the analytic methods to study a hybrid mean value problem related to certain Hardy sums and Kloosterman sums, and give some interesting mean value formulae and identities for it.

Soit $f\left(x\right)$ une fraction rationnelle à coefficients entiers, vérifiant des hypothèses assez générales. On prouve l’existence d’une infinité d’entiers $n$, ayant exactement deux facteurs premiers, tels que la somme d’exponentielles ${\sum }_{x=1}^{n}exp(2\pi if\left(x\right)/n)$ soit en $O\left(n^{\frac{1}{2}-{\beta }_f}\right)$, où ${\beta }_f\&gt;0$ est une constante ne dépendant que de la géométrie de $f$. On donne aussi des résultats de répartition du type Sato-Tate, pour certaines sommes de Salié, modulo $n$, avec $n$ entier comme ci- dessus.