On Peano and Riemann derivatives (Q1864676)

For a positive integer \(n\) denote by \(C_n\) the following statement: Let \(f\) be a real function defined in a neighborhood of a \(t_0\in\mathbb R\). Then the \(n\)-th order Peano derivative \(D^{(n)}_Pf(t_0)\) exists if and only if the Riemann derivative \(D^{(k)}_Rf(t_0)\) exists for \(k=0,1,\dots ,n\). In this case, \(D^{(n)}_Rf(t_0)=D^{(n)}_Pf(t_0)\). The authors conjecture that \(C_n\) is true for any positive integer \(n\). Actually, the implication Peano \(\Rightarrow\) Riemann is easy, the opposite one is a problem. In the paper, the authors formulate an equivalent conjecture involving a functional determinant of order \(2n+1\) and, using it, they check \(C_n\) for \(n\leq 8\).