On the asymptotic behavior of the Dickman-de Bruijn function (Q1318143)

The Dickman-de Bruijn function \(\rho(u)\) for \(u\geq 0\) is defined as the continuous solution of the delay equation \(u\rho'(u)+ \rho(u-1)=0\) with the initial condition \(\rho(u)=1\) for \(0\leq u\leq 1\). In this paper the following result is proved: for \(u\geq u_ 0\) \[ \rho(u)=e^{\gamma-u \xi(u)+I(\xi(u))} {1\over {\sqrt{2\pi w_ 2(u)}}} \left\{1+ \sum_{n=1}^ \infty f_ n(u)+ O(e^{-u/12})\right\}, \] where \(\xi(u)\) is the unique positive solution of \(e^ \xi= u\xi+1\), \(I(z)= \int_ 0^ Z (e^ v-1) v^{-1} dv\), \(w_ 2(u)= e^{\xi(u)}(\xi(u)- 1)\xi(u)^{-2}\) and \(f_ n(u)\) are complicated functions, the expression being too long to quote here, the series is uniformly convergent. From this result we deduce the following corollary: for \(u\geq u_ 0\) \[ \rho(u)= e^{\gamma-u\xi(u)+I(\xi(u))} {1\over {\sqrt{2\pi w_ 2(u)}}} \left\{1+ \sum_{n=1}^ N {1\over {u^ m}} \sum_{r=0}^ \infty {{a_{nr}} \over {\xi(u)^ r}}+ O_ N\left( {1\over {u^{N+1}}}\right)\right\}, \] where \(N\) is fixed, \(a_{nr}\) are constants and the series is uniformly convergent. For the functions \(\rho(u-v)\) \((u\geq u_ 0\), \(0\leq v\leq u-u_ 0)\) and \(\rho^{(k)}(u)\) (\(u\geq u_ 0\), \(k\) is fixed) similar results are given.