Half integer approximations for the partial sums of the harmonic series (Q1178474)

Let \(S_ n=1+1/2+\cdots+1/n\) be the \(n\)-th partial sum of the harmonic series. For a real number \(A\) let \(n_ A\) denotes the largest integer for which \(S_{n_ A}>A\). If \(e^{A-\gamma}=m+\delta\) where \(\gamma\) is Euler's constant, \(m\) is integral and \(0\leq\delta<1\) then Comtet has shown that \(n_ A\) is either \(m\) or \(m+1\). We call \(\omega(A)\) as Comtet function for which Boas has shown that when \(\omega(A)\geq\delta\) \(n_ A=m\) and when \(\omega(A)<\delta\) \(n_ A=m+1\). In the present paper the authors have proved three theorems. In Theorem 1 an asymptotic approximation of \(S_ n\) is obtained in terms of the half integer variable \(n+1/2\). In two other theorems improved bounds for Comtet functions are obtained and computation for its asymptotic expansion has been done which sharpens the previous result of Boas.