论文标题
关于3倍除数函数的相关性
On correlation of the 3-fold divisor function with itself
论文作者
论文摘要
令$ζ^k（s）= \ sum_ {n = 1}^\inftyτ_k（n）n^{ - s}，\ re s> 1 $。 我们在三元添加剂总和$ $ $ $ \ sum_ {n \ le x}τ_3（n）τ_3（n+h），\ quad（h \ ge 1），$$上介绍三个条件结果。首先是任何复合移位$ 1 \ le H \ le h \ le x^{2/3} $的有条件证明，假设在Arithmetic进度中达到三分之二的三分之二的分隔函数$τ_3（n）$的平均分布水平。第二个是此相关总和的领先顺序主术语渐近学的条件推导，对于任何复合移位$ 1 \ le H \ le h \ le x^{2/3} $也有效。第三个结果给出了从我们的方法和Delta-method中的特殊情况$ h = 1 $的整个主术语的多项式的完整扩展，这表明我们的答案匹配了。 我们的方法本质上是基本的，尤其是对于$ h = 1 $的情况，使用一致性，正如早期所述，给出了与Conrey和Gonek先前预测的答案[Duke Math。 J. 107（3）2002]，以前由Ng和Thom [Funct。大约评论。数学。 60（1）2019]，以及未发表的陶氏概率论点。我们的过程是一般的，可以将整个主术语带有节省功率的错误项，以$ \ sum_ {n \ le x}τ_k（n+h）f（n+h）$的任何相关性与任何复合偏移$ h $，以及与广泛的Arithmetic函数$ f（n）$相关。
Let $ζ^k(s) = \sum_{n=1}^\infty τ_k(n) n^{-s}, \Re s > 1$. We present three conditional results on the ternary additive correlation sum $$\sum_{n\le X} τ_3(n) τ_3(n+h),\quad (h\ge 1),$$ and give numerical verifications of our method. The first is a conditional proof for the full main term of the above correlation sum for any composite shift $1 \le h \le X^{2/3}$, on assuming an averaged level of distribution for the three-fold divisor function $τ_3(n)$ in arithmetic progressions to level two-thirds. The second is a conditional derivation for the leading order main term asymptotics of this correlation sum, also valid for any composite shift $1 \le h \le X^{2/3}$. The third result gives a complete expansion of the polynomial for the full main term for the special case $h=1$ from both our method and from the delta-method, showing that our answers match. Our method is essentially elementary, especially for the $h=1$ case, uses congruences, and, as alluded to earlier, gives the same answer as in prior prediction of Conrey and Gonek [Duke Math. J. 107 (3) 2002], previously computed by Ng and Thom [Funct. Approx. Comment. Math. 60(1) 2019], and unpublished heuristic probabilistic arguments of Tao. Our procedure is general and works to give the full main term with a power-saving error term for any correlations of the form $\sum_{n\le X} τ_k(n) f(n+h)$, to any composite shift $h$, and for a wide class of arithmetic function $f(n)$.
