On Peano and Riemann derivatives

For a given real function of one real variablef the conceptsn-th order Peano and (modified) Riemann derivatives are introduced. The conjecture is formulated that Peano derivative ofn-th order exists if and only if all Riemann derivatives of order less or equal ton exist and then then-th order Peano and Riemann derivative coincide. It is shown that this conjecture is equivalent to an assertion about the value of certain functional determinant of order 2n+1. This assertion is checked forn≤8. The general case remains an open question.