An approximate expression related with RSA fixed points
Let \(n=pq\), where \(p\) and \(q\) are prime numbers and \(e\) be an integer such that \((e,(p-1)(q-1))=1\). \(P\) is said to be an \(\alpha\)-order fixed point if \(P^{e^\alpha}= P\pmod n\), \((P,n)=1\) holds for a positive integer \(\alpha\). The number of solutions of this congruence is denoted by \(T=T(n,e,\alpha)\). Let \(A=\{p:p\) a prime, \((p-1,e)=1\), \(1<p\leq x\}\), \(B=\{q:q\) a prime, \((q-1,e)=1\), \(1<q\leq y\}\), where \(x\) and \(y\) are large integers, and \[ M(A,B,e,\alpha)= \frac{1}{|A||B|} \sum_{p\in A,q\in B,(p,q)=1} \log T(pq,e,\alpha) \] where \(|A|,|B|\) denotes the number of elements in \(A\) and \(B\) respectively. In this article the author gives an estimate for the mean value \(M(A,B,e,\alpha)\).
scientific article; zbMATH DE number 2067410