Bounding the summatory function of the inverses of the terms of an arithmetic progression
Let \(a,b,x\) be positive numbers and let \(v\) be the integer part of \((x-a)/b\). Define \[ A_{a,b} (x)= \sum^v_{n=0} (a+bn)^{-1} \text{ if } x\geq a\quad \text{and} \quad A_{a,b} (x)= -\sum^{-1}_{n=v+1} (a+bn)^{-1} \text{ if } x<a. \] In particular, the \(A_{1,1} (x)\), \(x\geq 1\), are the partial sums of the harmonic series. Define \(q_{a,b} (x)=x^{-1} (\{(x-a)/b\} -{1\over 2})\), where \(\{c\}\) denotes the fractional part of a number \(c\), \[ J_{a,b}(x) =\int^\infty_x u^{-1} q_{a,b} (u)du \quad\text{and} \quad \gamma_{a,b}= -b^{-1} \ln a-J_{a,b} (a). \] The main result states that \(A_{a,b} (x)= b^{-1} \ln x+ \gamma_{a,b} -q_{a,b} (x)+J_{a,b}(x)\). Approximate values of the integral \(J_{a,b} (x)\) are needed to employ this result. To this end, it is shown that \(-b/12 x^2\leq J_{a,b} (x)\leq b/24x^2\) and that the coefficients of \(bx^{-2}\) cannot be improved. Some bounds for \(h(x)= (24x^2)^{-1} -J_{a,1} (x)\) are also given.
scientific article; zbMATH DE number 1013903
scientific article