On the set F of complex-valued arithmetic functions we construct an infinite family of convolutions, that is, binary operations ψ of the form
![](//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190726102149601-0971:S0008439500008894:S0008439500008894_inline01.gif?pub-status=live)
so that (F, +, ψ) is a commutative ring, for which the unity is unbounded. Here + denotes pointwise addition.