Math Conjectures

Dirichlet Divisor-Problem Exponent Conjecture

OPENMajorExact constant problemProposed c. 1916 · Standard version

Canonical statement

Let d(n)=#{a∈Z>0:a∣n}d(n)=\#\{a\in\mathbb Z_{>0}:a\mid n\} and γ=lim⁡N→∞(∑k=1N1/k−log⁡N)\gamma=\lim_{N\to\infty}(\sum_{k=1}^{N}1/k-\log N), and put
Δ(x)=∑n≤xd(n)−xlog⁡x−(2γ−1)x. \Delta(x)=\sum_{n\le x}d(n)-x\log x-(2\gamma-1)x .
For every ε>0\varepsilon>0,
Δ(x)=Oε(x1/4+ε)(x→∞). \Delta(x)=O_\varepsilon(x^{1/4+\varepsilon}) \quad(x\to\infty).
View source LaTeX
Let \(d(n)=\#\{a\in\mathbb Z_{>0}:a\mid n\}\) and \(\gamma=\lim_{N\to\infty}(\sum_{k=1}^{N}1/k-\log N)\), and put
\[
  \Delta(x)=\sum_{n\le x}d(n)-x\log x-(2\gamma-1)x .
\] For every \(\varepsilon>0\),
\[
  \Delta(x)=O_\varepsilon(x^{1/4+\varepsilon})
  \quad(x\to\infty).
\]
NT-AN-028:[Dirichlet1849Divisor][Ivic1985Zeta][Huxley2003Lattice][BourgainWatt2017]
Number theoryAnalysis and functional analysis

Notes

Let d(n)d(n) count the divisors of nn and let Δ(x)=∑n≤xd(n)−xlog⁡x−(2γ−1)x\Delta(x)=\sum_{n\le x}d(n)-x\log x-(2\gamma-1)x be the error term in the classical asymptotic for the average of the divisor function, which Dirichlet established in 1849 with Δ(x)=O(x)\Delta(x)=O(\sqrt x) by his hyperbola method [Dirichlet1849Divisor]. The divisor-problem exponent conjecture asserts that Δ(x)=Oε(x1/4+ε)\Delta(x)=O_\varepsilon(x^{1/4+\varepsilon}) for every ε>0\varepsilon>0; this sharp form emerged in the Voronoï era, around 1916, rather than in a single dated source.

Omega results show that Δ(x)\Delta(x) is infinitely often of size comparable to x1/4x^{1/4}, so the conjectured exponent is the natural threshold and cannot be improved. Upper bounds have advanced from the classical Voronoï-era estimates through the exponential-sum results surveyed in Ivić's book [Ivic1985Zeta] to the strongest current exponents of Huxley [Huxley2003Lattice] and of Bourgain and Watt [BourgainWatt2017], the latter exploiting the close analogy with the mean square of the Riemann zeta function and with the circle problem. All proved exponents remain strictly above 1/41/4, so the conjecture stays open; a resolution demands closing that final gap.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.