Values of the Euler phi function not divisible by a given odd prime
Let \(\varphi(n)\) be the Euler function of the positive integer \(n\). This is always even if \(n\geq 3\). Given an odd prime \(q\), the authors determine an asymptotic formula for the number of positive integers \(n\leq x\) such that \(q\) does not divide \(\varphi(n)\). Denoting this number by \(E_q(x)\), the theorem on page 170 shows that \[ E_q(x)=e_0(q){{x}\over {(\log x)^{1/(q-1)}}}+O\Biggl({{x}\over {(\log x)^{q/(q-1)+\varepsilon}}}\Biggr) \] where \(\varepsilon\in (0,1)\) and the constant implied by the \(O\)-symbol depends on \(q\) and \(\varepsilon\). The authors gives an explicit formula for the positive constant \(e_0(q)\) involving invariants of the cyclotomic number field \({\mathbb K}(q)={\mathbb Q}(e^{2\pi i/q})\). The proof involves a Mertens type theorem for the product \[ \prod_{p\leq x,~p\equiv 1\pmod{q}}\Biggl(1-{1\over p}\Biggr). \] \{Reviewer's remark: The latter product can be derived in an easier way by invoking the Mertens' theorem for the algebraic number field \({\mathbb K}(q)\). Furthermore, a standard application of the Selberg-Delange method gives an asymptotic expansion \[ E_q(x)= {e_0(q)x\over (\log x)^{1/(q-1)}} \Biggl(1 + {e_1(q)\over \log x}+\cdots+{e_k(q)\over \log^k x}+O_k\Bigl({1\over \log^{k+1} x} \Bigr)\Biggr), \] with \(e_j(q)\) being certain constants depending on \(q\), and thus an extension of the main result of the paper. The above remarks (in more detail) along with a study of the numbers \(e_1(q)\), which are closely related to the Euler-Kronecker constant for \({\mathbb K}(q)\), can be found in a recent preprint [``Values of the Euler \(\varphi\)-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields'', Preprint (2011)] of the reviewer with \textit{K. Ford} and \textit{P. Moree}\}.
scientific article; zbMATH DE number 5526902