On Generalized Landau Identities

Pentti Haukkanen

Abstract: The identity $$ \sum_{d|r}\frac{\mu^{2}(d)}{\phi(d)}=\frac{r}{\phi(r)}, $$ where $\mu$ is the Möbius function and $\phi$ is the Euler function, is known in the literature as the Landau identity. The present paper collects several extensive generalizations of that identity given by the author and P.J. McCarthy. Some of the extensive generalizations are further generalized. Also a large number of known special cases of the identities here are listed.

Keywords: Number-theoretic identities; even arithmetical functions; regular convolutions; Möbius function; Euler totient; Ramanujan sums.

Classification (MSC2000): 11A25

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