cs.CC — Computational Complexity
Non-computable derivatives of the fractional-part autocorrelation
We describe a natural new example of non-computability in analysis. Namely, the multiplicative autocorrelation of the fractional-part function, \[ A(x)=\int_0^\infty \{t\}\{xt\}\,\frac{dt}{t^2}, \] is computable and differentiable almost everywhere, but its finite derivative can be non-computable even for a computable . We derive these results from an adaptation of our earlier work on computability of Brjuno-like functions.
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