Primes in higher-order progressions on average
In this paper, we establish some theorems on the distribution of primes in higher-order progressions on average.
math.NT↗
arXiv subjects
Nianhong Zhou
Publications and source records attributed to Nianhong Zhou.
The explicit asymptotic formula of divisor function on average over values of quadratic polynomial
Let $F({\bf x})={\bf x}^tQ_{\bf x}+\mathbf{b}^t{\bf x}+c\in\mathbb{Z}[{\bf x}]$ be a quadratic polynomial in $\ell (\ge 3 )$ variables ${\bf x} =(x_{1},...,x_{\ell})$, where $F({\bf x})$ is positive when ${\bf x}\in\mathbb{R}_{\ge 1}^{\ell}$, $Q\in {\rm M}_{\ell}(\mathbb{Z})$ is an $\ell\times\ell$ matrix and its discriminant $\det\left(Q^t+Q\right)\neq 0$. It gives an explicit asymptotic formula for the following sum \[\sum_{{\bf x}\in [1,X]^{\ell}\cap\mathbb{Z}^{\ell}}τ\left(F({\bf x})\right), \] where $τ$ is the divisor function.
math.NT↗
The sum of multidimensional divisor function over values of quadratic polynomial
Let $F({\bf x})={\bf x}^tQ_m{\bf x}+\mathbf{b}^t{\bf x}+c\in\mathbb{Z}[{\bf x}]$ be a quadratic polynomial in $\ell (\ge 3 )$ variables ${\bf x} =(x_{1},...,x_{\ell})$, where $F({\bf x})$ is positive when ${\bf x}\in\mathbb{R}_{\ge 1}^{\ell}$, $Q_m\in {\rm M}_{\ell}(\mathbb{Z})$ is an $\ell\times\ell$ matrix and its discriminant $\det\left(Q_m^t+Q_m\right)\neq 0$. It gives explicit asymptotic formulas for the following sum \[ T_{k,F}(X)=\sum_{{\bf x}\in [1,X]^{\ell}\cap\mathbb{Z}^{\ell}}τ_{k}\left(F({\bf x})\right) \] with the help of the circle method. Here $τ_{k}(n)=\#\{(x_1,x_2,...,x_{k})\in\mathbb{N}^{k}: n=x_1x_2...x_{k}\}$ with $k\in\mathbb{Z}_{\ge 2}$ is the multidimensional divisor function.
math.NT↗