The Titchmarsh divisor problem asks for the asymptotic behavior of the sum of the divisor function at shifted primes.
Titchmarsh (1930) studied the sum of the numbers of divisors
of 
over primes 
up to a given bound. The higher-order problem uses the 
-fold
divisor function 
, which counts ordered 
-tuples of positive integers
with product 
.
Hu et al. (2026, Theorem 1.1) report that for every fixed integer 
and nonzero integer 
,
 |
as 
, where 
is the Mangoldt function
and 
is a positive constant given by an Euler product.
The argument adapts dilation-graph estimates from the reported Poisson-Dirichlet
law for prime predecessors (OpenAI 2026).
