On a problem of Rényi
Denoting by \(\nu_ q (x)\) the number of integers \(n\leq x\), for which \(\Omega(n)- \omega(n)= q\), \textit{A. Rényi} [Publ. Inst. Math. Acad. Serbe Sci. 8, 157--162 (1955; Zbl 0066.03203)] showed \[ \nu_ q (x)= d_ q x+ o(x), \qquad x\to \infty, \] where \[ \sum_{q=0}^ \infty d_ q z^ q= {6\over {\pi^ 2}} \prod_ p {{1- {z\over {p+1}}} \over {1- {z\over p}}}= F(z) \quad \text{in } | z|<2. \] H. Delange, the reviewer (for \(q=1\)), B. Saffari, I. Kátai, A. S. Fainleib, and finally, \textit{D. Wolke} [Monatsh. Math. 111, 323--330 (1991; Zbl 0742.11045)] gave asymptotic expansions for \(\nu_ q (x)\) with rather good error terms. The author improves on Wolke's result and shows the asymptotic formula \[ \nu_ q (x)= d_ q x+ x^{1/2} \int_ 0^{1/7} V_ q (u) x^{-u} du+ O(x^{1/2} \exp\{- C_ 6 N(x)\}) \tag \(*\) \] with some function \(V_ q(u)\), given as a power series in \(u\) and \(\log {1\over u}\); \(N(x)\) denotes the function \((\log x)^{3/5} (\log \log x)^{-1/5}\), known from sharp versions of the prime number theorem. From \((*)\) the author deduces an asymptotic expansion \[ \nu_ q (x)= d_ q x+ x^{1/2} \sum_{j=1}^ J {1\over {(\log x)^{j+1}}} P_{j,q-1} (\log \log x)+ R(x), \] where \(1\leq J\leq {1\over 7} \log x\), with polynomials \(P_{j,q-1}\) of degree \(\leq (q-1)\), and an error term \[ \begin{multlined} R(x)\ll x^{1/2} {{(\log \log x)^{q-1}} \over {(\log x)^{J+ 2}}} \Biggl( x^{-1/14} (\log x)^{J+2}+ (q-1)! 7^ J {\textstyle {1\over J}} +(J+1)! 7^ J+ \exp (-C_ 8 N(x))\Biggr) \\ \ll x^{1/2} \exp (-C_ 9 N(x)); \end{multlined} \] the last estimate is uniform in \(1\leq q\leq C_ 9 N(x)\), \(J= N(x)\). An important tool for the proof is the representation \[ \sum_{n\leq x} z^{\Omega(n)- \omega(n)}= x F(z)+ x^{1/2} \Phi^* (x,z)+ O_ \varepsilon \Bigl( x^{1/2} \exp (-C_ 5 N(x)) \Bigr) \] in \(x\geq 10\), \(| z|\leq 2-\varepsilon\), where \(\Phi^* (x,z)\) is defined by an integral of the shape \[ -{{\sin\pi_ z} \over\pi} \int_ 0^{1/7} {{\zeta({1\over 2}-u)} \over {{1\over2}-u}} \exp\{(z-1) {\mathcal Q} ({\textstyle {1\over2}}- u)\times {\mathcal G}({\textstyle {1\over 2}}- u,z)\} u^{1-z} x^{-u} du. \]
local behaviour of \(\Omega (n)- \omega(n)\)
scientific article; zbMATH DE number 617402