Papers › On correlation of the 3-fold divisor function with itself
On correlation of the 3-fold divisor function with itself
David T. Nguyen
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Let ζᵏ(s) = ∑ₙ₌₁^∞ τₖ(n) n⁻ˢ, s > 1. We present three conditional results on the ternary additive correlation sum ∑_(n≤X) τ₃(n) τ₃(n+h), (h≥1), and give numerical verifications of our method. The first is a conditional proof for the full main term of the above correlation sum for any composite shift 1 ≤h ≤X^(2/3), on assuming an averaged level of distribution for the three-fold divisor function τ₃(n) in arithmetic progressions to level two-thirds. The second is a conditional derivation for the leading order main term asymptotics of this correlation sum, also valid for any composite shift 1 ≤h ≤X^(2/3). The third result gives a complete expansion of the polynomial for the full main term for the special case h=1 from both our method and from the delta-method, showing that our answers match. Our method is essentially elementary, especially for the h=1 case, uses congruences, and, as alluded to earlier, gives the same answer as in prior prediction of Conrey and Gonek [Duke Math. J. 107 (3) 2002], previously computed by Ng and Thom [Funct. Approx. Comment. Math. 60(1) 2019], and unpublished heuristic probabilistic arguments of Tao. Our procedure is general and works to give the full main term with a power-saving error term for any correlations of the form ∑_(n≤X) τₖ(n) f(n+h), to any composite shift h, and for a wide class of arithmetic function f(n).
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