素数上的短多项式指数和的均值估计
Mean Estimates for Short Polynomial Exponential Sums over Primes
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中文总结 AI 辅助
本文针对素数上的短多项式指数和，建立了关于平均量 V_n 的上界，利用 Fejér 核、差分和除数函数矩，并给出对数节省的充分条件。
中文摘要 AI 辅助
设 $n\ge2$ 为固定整数，并设 $K,x,y$ 为正整数，满足 $2\le K\le y<x$。考虑具有实系数的多项式 $f(u)=\alpha u^n+\alpha_{n-1}u^{n-1}+\cdots+\alpha_1u+\alpha_0$。我们研究量 $V_n(K,x,y) = \sum_{k=1}^{K} \left| \sum_{x-y<p\le x}e(kf(p)) \right|$，其中 $e(t)=e^{2\pi it}$，内和遍历素数。对于 $\alpha=\frac{a}{q}+\frac{\theta}{q^2}$，其中 $a\in\mathbb Z$，$q\in\mathbb N$，$(a,q)=1$，$|\theta|\le1$，我们建立了界 $V_n(K,x,y) \ll Ky\left[ \frac{1}{\sqrt{K\log(2y)}} + \min\left\{ \Delta^{\frac{1}{2^n}} (\log(2y))^{\frac{n^2-1}{2^n}}, \Delta^{\frac{1}{3\cdot 2^{n-2}}} (\log(2y))^{\frac{n^3-1}{3\cdot 2^{n-1}}} \right\} \right]$，其中 $\Delta= \frac{1}{q}+\frac{1}{y}+\frac{q}{Ky^n}$。该界对区间的位置是一致的，对数因子依赖于区间的长度。证明使用了非负 Fejér 核、逐次差分以及广义除数函数的二阶和三阶矩。Brun-Titchmarsh 不等式用于计算区间内素数的个数，并在平均项中提供了额外的对数节省。我们获得了对区间长度的对数任意固定幂次节省的充分条件，以及一个对所有整数求和的类似均值估计。
英文摘要
Let $n\ge2$ be a fixed integer, and let $K,x,y$ be positive integers satisfying $2\le K\le y<x$. Consider a polynomial \[ f(u)=αu^n+α_{n-1}u^{n-1}+\cdots+α_1u+α_0 \] with real coefficients. We study the quantity \[ V_n(K,x,y) = \sum_{k=1}^{K} \left| \sum_{x-y<p\le x}e(kf(p)) \right|, \qquad e(t)=e^{2πit}, \] where the inner sum is over primes. For $α=\frac{a}{q}+\fracθ{q^2}$, $a\in\mathbb Z$, $q\in\mathbb N$, $(a,q)=1$, $|θ|\le1$, we establish the bound \[ \begin{aligned} V_n(K,x,y) &\ll Ky\Biggl[ \frac{1}{\sqrt{K\log(2y)}}+ \min\left\{ Δ^{\frac{1}{2^n}} \bigl(\log(2y)\bigr)^{\frac{n^2-1}{2^n}}, Δ^{\frac{1}{3\cdot 2^{n-2}}} \bigl(\log(2y)\bigr)^{\frac{n^3-1}{3\cdot 2^{n-1}}} \right\} \Biggr], \end{aligned} \] where \[ Δ= \frac{1}{q}+\frac{1}{y}+\frac{q}{Ky^n}. \] The bound is uniform in the position of the interval, and the logarithmic factors depend on its length. The proof uses the nonnegative Fejér kernel, successive differencing, and the second and third moments of the generalized divisor function. The Brun-Titchmarsh inequality accounts for the number of primes in the interval and provides an additional logarithmic saving in the term arising from averaging. We obtain sufficient conditions for a saving of any fixed power of the logarithm of the interval length, as well as an analogous mean estimate for sums over all integers.
发表机构
- Tajik State University of Finance and Economics（塔吉克国立财经大学）
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