arXiv 1609.014112016-09math.NT

可加divisor和的界与猜想

Bounds and Conjectures for additive divisor sums

Nathan Ng, Mark Thom

备注：: This is the second version of the article. The article now considers the more general sums $D_{k,\ell}(x,h) = \sum_{n \le x} τ_k(n) τ_{\ell}(n+h)$. We also show that our probabilistic argument and Terry Tao's probabilistic argument give the same conjecture for $D_{k,\ell}(x,h)$

摘要：

加法因子和在黎曼ζ函数的矩理论中起着重要的作用。确定常因子函数的移位卷积和的精确渐近公式由来已久。近年来，人们发现三重因子函数的移位卷积和的一个精确的渐近公式在计算黎曼ζ函数的六阶矩时是有用的。本文研究了$D_{k,\ell}(x) = \sum_{n \le x} τ_k(n) τ_{\ell}(n+h)$，其中$τ_k$和$τ_{\ell}$分别是$k$次和$\ell$次除数函数.主要结果是$D_{k,\ell}(x,h)$的正确数量级的下界，在$h$中是一致的。此外，还研究了$D_{k,\ell}(x,h)$的渐近公式.利用Ivić和Conrey-Gonek的一个论证，简化了渐近公式中的首项.此外，给出了一种概率方法，它给出了相同的领先项。最后，我们表明，这两种方法给出了相同的答案，在最近的概率论的特里陶。

英文摘要：

Additive divisor sums play a prominent role in the theory of the moments of the Riemann zeta function. There is a long history of determining sharp asymptotic formula for the shifted convolution sum of the ordinary divisor function. In recent years, it has emerged that a sharp asymptotic formula for the shifted convolution sum of the triple divisor function would be useful in evaluating the sixth moment of the Riemann zeta function. In this article, we study $D_{k,\ell}(x) = \sum_{n \le x} τ_k(n) τ_{\ell}(n+h)$ where $τ_k$ and $τ_{\ell}$ are the $k$-th and $\ell$-th divisor functions. The main result is a lower bound of the correct order of magnitude for $D_{k,\ell}(x,h)$, uniform in $h$. In addition, the conjectural asymptotic formula for $D_{k,\ell}(x,h)$ is studied. Using an argument of Ivić and Conrey-Gonek the leading term in the conjectural asymptotic formula is simplified. In addition, a probabilistic method is presented which gives the same leading term. Finally, we show that these two methods give the same answer as in a recent probabilistic argument of Terry Tao.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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