关于k重除数函数的Montgomery-Hooley定理
A Montgomery-Hooley theorem for the k-fold divisor function
摘要：
令$d_k(n)$表示$k$倍除数函数.对于广泛的大$q$的预期绑定$$\sum_{n\leq x\atop {n\equiv a(q)}}d_k(n)-\text { main term }\approx \sqrt {x/q}$$被证明是真实的，在平均意义上-所有$k$。这概括了Pongsriiam和Vaughan [15]研究$k=2$的工作，并回答了Rodgers和Soundarajan [17]的工作，他们使用渐近大筛来研究问题的平滑版本。我们使用Goldston和Vaughan [7]开发的圆方法来研究非光滑问题。
英文摘要：
Let $d_k(n)$ denote the $k$-fold divisor function. For a wide range of large $q$ the expected bound $$\sum_{n\leq x\atop {n\equiv a(q)}}d_k(n)-\text { main term }\approx \sqrt {x/q}$$ is shown to be true in an average sense -- for all $k$. This generalises the work of Pongsriiam and Vaughan [15] who studied $k=2$, and answers the work of Rodgers and Soundararajan [17], who used the asymptotic large sieve to study a smoothed version of the problem. We use a circle method approach as developed by Goldston and Vaughan [7] to study the unsmoothed problem.
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