Correlations of the divisor function

Abstract

Let τ(n)= ∑d 1d|n denote the divisor function. Employing the methods Green and Tao developed in their work on prime numbers, we give an asymptotic for the following correlation E n ∈ [ − N , N ] d ∩ K ∏ i = 1 t τ ( ψ i ( n ) ) , where Ψ=(ψ1,…, ψt) is a non-degenerated system of affine-linear forms no two of which are affinely related, and where K is a convex body. These correlations include, for example, the following averages along k-term arithmetic progressions for fixed integers k⩾ 2 E ( τ ( n ) τ ( n + d ) … τ ( n + ( k − 1 ) d ) | n , d ∈ N , n + ( k − 1 ) d ⩽ N ) . In the course of the proof, we construct a pointwise majorant function for a slightly smoothed version of τ that behaves quasi-random in certain precise sense. This construction is based on Erdős's fundamental work on sums of multiplicative functions.

Locations

• Proceedings of the London Mathematical Society - View
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