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We show that there exists a natural extention of the sum of divisors function to all unique factorization domains F having a finite number of units such that if a perfect number in F is defined to be an integer η whose proper divisors sum to η, then the analogue of Euclid's theorem giving the sufficient condition that an integer be an even perfect number holds in F, and an analogue of the Euclid-Euler theorem giving the necessary and sufficient condition that an even integer be perfect holds in those domains having more than two units, i. e., in Q(−1) …

The main result of this paper is that weak type multiplier theorems for Jacobi expansions yield weak type multiplier theorems for Hankel transforms.

In recent papers the authors studied multiplier theorems for ultraspherical and Jacobi expansions. An interesting paper of Igari suggested a new approach to multiplier theorems that used asymptotics instead of the elaborate machinery used earlier.

This paper extends the method of Igari to give the first weak type multiplier theorem for Hankel transforms. This extension is important in itself and because this method together with the authors’ results for Jacobi multipliers will generalize to the “radial” functions …