Tao's shifted convolution conjecture for generalised divisor functions

Let dk(n)d_k(n) denote the number of ordered factorizations of nn into kk positive integers, and let dl(n)d_l(n) be defined similarly. Let ϵ>0\boldsymbol{\epsilon>0} and k,l2k,l\ge 2. For 1hx1ϵ1\le h\le x^{1-\epsilon}, Tao's shifted convolution conjecture.

nxdk(n+h)dl(n)ck,l(h)(k1)!(l1)!x(logx)k+l2\sum_{n\le x}d_k(n+h)d_l(n) \sim \frac{c_{k,l}(h)}{(k-1)!(l-1)!}x(\log x)^{k+l-2}

as xx\to\infty, for an explicit constant ck,l(h)c_{k,l}(h). This extends the conjectural asymptotic of Conrey and Gonek for equal indices and remains open in the stated range; Ng and Thom proved a lower bound of the correct order.

Sources & referencesView supporting material

Primary source

Bikram Misra and Biswajyoti Saha, “Triple convolution sums of the generalised divisor functions and related sums over primes”, arXiv:2509.04610 (2026).

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