The problem
Littlewood (1914): π(x) − li(x) changes sign infinitely often. Open effective companion: give a concrete x₀ with π(x₀) > li(x₀). Best bracket: somewhere between \(10^{19}\) and \(e^{727}\).95.
π(x) versus li(x): The Skewes Number
historicReferences
History & significance
Every computation into the billions shows li ahead. Littlewood's proof used the explicit-formula machinery — sign changes forced by ζ behaviour, with no construction. Skewes (1933 under RH, 1955 unconditionally) produced bounds so large they required logarithms of logarithms to write down.
The resolution (human proof)
Existence proven by John E. Littlewood, 1914. The effective chase became its own literature: Sherman Lehman's method, te Riele's 1986 computation finding negative regions near x ≈ \(10^{316}\) (Bayes–Hudson–Zeindler refined), Kotnik's verification none occurs before \(10^{14}\), Platt–Trudgian improvements. Canonical entry for the gap between knowing THAT and knowing WHERE.