Mellin transforms and Fourier-Ramanujan expansions
For Re \(\alpha\geq 0\) let \(K_{\alpha}\) be the class of all arithmetic functions \(f_{\alpha}(n)=\sum_{d | n}d^{1-\alpha} w(d),\) where \(w=M\{F\}\) is an inverse Mellin transform with \(F\in L(-\infty,+\infty)\). In this paper it is proved that each \(f_{\alpha}\in K_{\alpha}\) has an absolutely convergent Fourier-Ramanujan expansion \(f_{\alpha}(n)=\sum_{q\geq 1}a_ q(f_{\alpha}) c_ q(n),\) where \(c_ q(n)\) is Ramanujan's trigonometric sum and the Fourier coefficients \(a_ q(f_{\alpha})\) are integrals of Mellin type represented conversely by a modified Ramanujan series. The method of proof is based on the theory of Fourier transforms and on Ramanujan's classical Fourier expansion \(\sigma_{1-s}(n)=\zeta (s)\sum_{q\geq 1}q^{-s} c_ q(n)\) (Re s\(>1\), \(\sigma_ s(n)=\sum_{d | n}d^ s,\) \(\zeta\) (s) the Riemann zeta-function). In the applications some examples from the classes \(K_ 0\) and \(K_ 1\) are investigated including e.g. the logarithms of Jacobi's elliptic theta-functions. It here turns out that Ramanujan's outstanding identities \(0=\sum_{q\geq 1}q^{-1} c_ q(n)\) and \(-\sigma_ 0(n)=\sum_{q\geq 1}q^{-1} \log q\quad c_ q(n),\) which do not fit into the popular theory of Ramanujan expansions, are useful tools in the author's scheme of Fourier analysis for the class \(K_{\alpha}\).
logarithms of Jacobi elliptic theta-functions
scientific article; zbMATH DE number 3957233