Dirichlet's divisor problem

Dirichlet's divisor problem involves estimating how many pairs of positive integers multiply to a number less than or equal to a given value x. Mathematically, it looks at the sum of the divisor function up to x, which counts how many divisors each number has. The challenge is to accurately approximate this sum and understand the fluctuations around the main term, which is roughly proportional to x times the logarithm of x. Despite significant progress, the problem remains open in precisely determining the size of these error terms, reflecting deep complexities in number theory.