Functional equations and characterizations of Riemann's function
The Riemann function is defined by \(r(x)= \sum^\infty_{\nu=1} {1\over \nu^2} \sin (\nu^2x)\) and the authors let \(R(x)= r(2\pi x)\) be a scaled Riemann function. It satisfies the functional equations \(R(x+1) =R(x)\), \(R(-x)= -R(x)\), and \(R({x \over 2}) +R({x+1 \over 2})= {1 \over 2} R(2x)\) for all real \(x\). It is shown that \(R\) satisfies \(\sum^{q-1}_{k=0} R({x+k \over q})= {1\over q} R(qx)\) for \(q\) a square free positive integer and any real \(x\) and that \(R({1\over 4}) =\pi^2/8\). Then in Theorem 2 the authors show that \(g=R\) is the only continuous solution of the functional equations \[ \sum^{p-1}_{k=0} g\left( {x+k \over p} \right)= {1\over p} g(px) \tag{*} \] where \(g:\mathbb{R} \to\mathbb{R}\) is continuous, 1-periodic, non-even, \(g({1\over 4}) =\pi^2/8\), \(p\) is any prime, and \(x\) is any real. The conditions of this theorem are somewhat weakened without losing the characterization of \(R\) in a Theorem 3, where \(g:\mathbb{R} \to\mathbb{R}\) is continuous on \((0,1)\), non-even, \(q\)-periodic for some prime \(q\), has the value \(\pi^2/8\) at 1/4, and satisfies (*) for all \(x\in \mathbb{R}\) and all primes \(p\).
Riemann's continuous non-differentiable function
scientific article; zbMATH DE number 938669