Let d(n) be the divisor function. The Dirichlet divisor problem concerns the error term
Δ(x) := n≤x∑d(n) − x(logx+2γ−1),
where γ is Euler's constant.
[Tsa2010-def-Delta]
Define the divisor-problem exponent
α := inf{a≥0: Δ(x)=O(xa+ε) for all ε>0}.
[Tsa2010-def-alpha]
We define
C63 := α,
the Dirichlet divisor problem exponent.
It is conjectured that α=1/4.
[Tsa2010-conj-1-4]
The best known upper bound is due to Huxley,
α ≤ 416131.
[Tsa2010-ub-131-416]
Hardy's omega result implies that Δ(x)=Ω(x1/4) (up to logarithmic factors), hence α≥1/4.
[Tsa2010-omega-1-4]
The best established range currently is
41 ≤ C63 ≤ 416131.