A note on Jordan's totient function (Q1115468)

\textit{P. Kesava Menon} [J. Indian Math. Soc., New. Ser. 27, 57-65 (1964; Zbl 0123.045)] defined the norm M(f) of a multiplicative function f as the multiplicative function given by \(M(f)(n)=(f*\lambda f)(n^ 2)\), where * is the Dirichlet convolution and \(\lambda\) is Liouville's function. In the present paper the norms of the Jordan totient function \(J_ k\) and some other totient functions have been derived. Also some congruence properties of \(J_ k\) have been given. The authors prove their results on the norms using the formula \(M(f*g)=M(f)*M(g)\), where f, g are multiplicative functions, and considering prime powers. A more general result can easily be obtained using the above formula and noting that if f is a completely multiplicative function, then \(M(f)=f^ 2\), where \(f^ 2(n)=f(n)^ 2\) for all n. Namely denoting by \(f^{(-1)}\) the Dirichlet inverse of f we have \[ M(f_ 1*... *f_ r*g_ 1^{(-1)}*... *g_ s^{(-1)})=f^ 2_ 1*... *f^ 2_ r*(g^ 2_ 1)^{(-1)}*... *(g^ 2_ s)^{(- 1)}, \] where \(f_ 1,...,f_ r,g_ 1,...,g_ s\) are completely multiplicative functions. As special cases of the above formula we obtain the norms given in the present paper. For example, the norm of \(J_ k\) is obtained taking \(r=s=1\), \(f_ 1=N^ k\), \(g_ 1\equiv 1\), where \(N^ k(n)=n^ k\) for all n. \textit{R. Sivaramakrishnan} [J. Reine Angew. Math. 280, 157-162 (1976; Zbl 0317.10010)] has proved the general result with \(r=2\), \(s=0\).