On correlation of the 3-fold divisor function with itself

arXiv2206.05877MaRDI QIDQ6401863

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Publication date: 12 June 2022

Abstract: Let $zet{a}^k(s)=su{m}_{n=1}^{i}nftya{u}_k(n)n^{-s},Res>1$. We present three conditional results on the additive correlation sum ${\displaystyle }$sum_{nle X} au_3(n) au_3(n+h)${\displaystyle }$ and give numerical verifications of our method. The first is a conditional proof for the full main term of this correlation sum for the case $h=1$, on assuming an averaged level of distribution for the three-fold divisor function $a{u}_3(n)$ in arithmetic progressions to level two-thirds. The second is a derivation for the leading term asymptotics of this correlation sum, valid for any composite shift $h$. The third result gives a complete expansion of the polynomial for the full main term also for the case $h=1$ but from the delta-method, showing that our answers match. We also refine an unconditional result of Heath-Brown on the classical correlation of the usual divisor function and numerically study the error terms, showing square-root cancellation supporting our approach. 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, previously computed by Ng and Thom, and 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 $su{m}_{nleX}a{u}_k(n)f(n+h)$, to any composite shift $h$, and for any arithmetic function $f(n)$, such as $f(n)=a{u}_{e}ll(n),Lambda(n),$ et cetera.

Has companion code repository: https://github.com/nguyen-d-8/correlations

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