A generalization of regular convolutions and Ramanujan sums

Abstract

Regular convolutions of arithmetical functions were first defined by Narkiewicz (Colloq Math 10:81–94, 1963). Useful identities regarding generalizations of the totient-counting function and Ramanujan sums were later proven for regular convolutions by McCarthy (Port Math 27(1):1–13, 1968) and Rao (Studies in arithmetical functions, PhD thesis, 1967). We introduce semi-regular convolutions as a generalization of the regular convolutions and show that many of these identities still hold. In particular, special cases of the generalized Ramanujan sums correspond to the corresponding expected generalizations of the totient-counting and Möbius functions. Then we demonstrate that the class of semi-regular convolutions is the broadest generalization to multiplicative-preserving convolutions possible in which even the most basic of these identities still hold. Finally, we introduce a convolution related to Cohen’s infinitary convolution (Int J Math Math Sci 16(2):373–383, 1993) that is semi-regular. This convolution has never been studied to the best of the authors’ knowledge and possesses a property that distinguishes it from all of the other semi-regular convolutions.