We prove some distribution results for the k-fold divisor function in arithmetic progressions to moduli that exceed the square-root of length X of the sum, with appropriate constrains and averaging on the moduli, saving a power of X from the trivial bound. On assuming the Generalized Riemann Hypothesis, we obtain uniform power saving error terms that are independent of k.

We follow and specialize Y.T. Zhang's method on bounded gaps between primes to our setting. Our arguments are essentially self-contained, with the exception on the use of Deligne's work on the Riemann Hypothesis for varieties over finite fields. In particular, we avoid the reliance on Siegel's theorem, leading to some effective estimates.

我们证明了k倍除数函数的一些分布结果，其算术级数超过了总和的长度X的平方根，具有适当的约束并取平均值，从而节省了X的幂。假设广义黎曼假设，我们获得独立于k的统一节能误差项。

我们遵循并专门研究YY Zhang的方法在素数与我们的环境之间的有限间隙上。我们的论点本质上是独立的，除了将Deligne的黎曼假说工作用于有限域上的变种之外。特别是，我们避免了依赖Siegel定理，从而得出了一些有效的估计。