Distribution of arithmetic functions on certain subsets of integers (Q2478038)

Let \(d(n)\) denote the number, and \(\sigma(n)\) the sum, of the positive divisors of \(n\), \(\varphi\) denote Euler's totient function, and \(d_3(n)\) be the number of solutions of \(a_1a_2a_3= n\). For \(n\geq 2\), let \(p(n)\) be the least prime factor of \(n\) and \(\omega(n)\) the number of distinct primes dividing \(n\). In [Int. J. Math. Anal., to appear] the authors proved that \[ \#\{n\leq x: (nd(n),\sigma(n) = 1\}= (C+ o(1))\sqrt{{x\over\log x}} \] for a certain constant \(C\), thus improving a result of H. J. Kanold. The first three results of the current paper concern the analogous problem for the properties \[ \text{(i) }(nd(n),\varphi(p(n))=\text{ a power of 2}; \;\;\text{(ii) }(nd_3 (n), (p(n))= 1; \;\;\text{(iii) }U(n):= (nd_3(n),\sigma(n))= 1; \] the size of the corresponding sets are, respectively, \[ \text{(i) }(c_1+ o(1)){x\over \log\log\log_x};\;\text{(ii) }(c_2+ o(1)){x\over \sqrt{\log x\log\log\log x}}; \;\text{(iii) }(c_3+ o(1)){x\over \sqrt{\log x\log\log\log x}} \] as \(x\to\infty\) for certain positive constants \(c_1\), \(c_2\), \(c_3\). Theorem 4 establishes that, given any integer \(a\neq 0\), there exists a constant \(c_4> 0\) such that as \(x\to\infty\) \[ \sum_{\substack{ p\leq x\\ U(p+ a)= 1}} d(p+ a)= (c_4+ o(1)){li(x)\over \log\log\log x}. \] The final theorem states that, uniformly for \(r= (1+ o(1))\log\log x\), \[ \#\{n\leq x: (n,(p (n))= 1,\;\omega(n)= r\}\sim\#\{n\leq x: p(n)> \log\log x,\;\omega(n)= r\} \] as \(x\to\infty\), and an asymptotic formula for the right side is obtained. P. Erdős had derived in 1948 a corresponding result without the condition \(\omega(n)= r\). The proofs employ some classical methods including sieve arguments, known results and, in establishing the error terms, some intricate manipulations of various sums over primes.