## 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

### Keywords:

symmetry; divisor function; short intervals
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