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On the symmetry of the divisor function in almost all short intervals. (English) Zbl 1122.11062

Summary: We study the symmetry of the divisor function \(d(n)\) in almost all short intervals; by elementary methods (based on the large sieve) we give an exact asymptotic estimate for the mean-square \(I(N,h)\) (over \(N<x\leq 2N\)) of its “symmetry sum” \(\sum_{|n-x|\leq h} \text{sgn}(n-x)d(n)\) (here \(\text{sgn}(t):=t/|t|\), with \(\text{sgn}(0)=0\)). Let
\[ D(N,h):=8N\sum_{t\leq\sqrt N}\frac{\mu(t)}{t^2}\sum_{k\leq\sqrt N/t}\| \frac hk\|\log^2\frac{\sqrt N}{tk}. \]
Theorem 1: Let \(N\) and \(h=h(N)<\sqrt n/2\) be large enough natural numbers (with \(h\to\infty\) as \(N\to\infty\)) and \(I(N,h)\), \(D(N,h)\) as above. Then \[ I(N,h)=D(N,h)+O(NhL^{5/2}\sqrt{\log L}). \]
Then under the hypothesis of Theorem 1 one obtains Corollary 1:
\[ I(N,h)=\frac{16}{\pi^2}Nh\log^3\frac{\sqrt N}{h}+O(NhL^{5/2}\sqrt{\log L}). \]

MSC:

11N37 Asymptotic results on arithmetic functions
11N36 Applications of sieve methods
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