Agnes Scott College

Emma T. Lehmer

A Numerical Function Applied to Cyclotomy
Bulletin of the American Mathematical Society, Vol. 36, No. 4 (April 1930), 291-298


A function \(\phi_2(n)\) giving the number of pairs of consecutive integers each less than n and prime to n, was considered first by Schemmel. In applying this function to the enumeration of magic squares, D. N. Lehmer has shown that if one replaces consecutive pairs by pairs of integers having a fixed difference λ prime to \(n=\prod_{i=1}^t p_i^{\alpha_i},\) then the number of such pairs (mod n) whose elements are both prime to n is also given by \[\phi_2(n) = \prod_{i=1}^t p_i^{\alpha_1-1}(p_i-2)\] As is the case for Euler's totient function \(\phi(n)\), the function \(\phi_2(n)\) obviously enjoys the multiplicative property \(\phi_2(m)\phi_2(n) = \phi_2(mn), (m,n)=1, \phi_2(1)=1\). In what follows we call an integer simple if it contains no square factor >1. For a simple number n we have the following analog of Gauss' theorem: \[\sum_{\delta|n} \phi_2(\delta) = \phi(n),\] where n is simple and where the summation extends over all the divisors of n. Using Dedekind's inversion formula, we can write \[\sum_{\delta|n} \phi(n/\delta)\mu(\delta) = \phi_2(n),\] where n is simple and where \(\mu(n)\) is Merten's inversion function.

It is the purpose of this note to develop another property of \(\phi_2(n)\), true only for simple numbers, and apply it to the evaluation of the discriminants and resultants of cyclotomic equations.