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Bounding the summatory function of the inverses of the terms of an arithmetic progression. (Encadrement de la fonction sommatoire des inverses des termes d’une progression arithmétique.) (French) Zbl 0873.40002

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.

MSC:

40A05 Convergence and divergence of series and sequences
40A25 Approximation to limiting values (summation of series, etc.)
65B15 Euler-Maclaurin formula in numerical analysis

Keywords:

harmonic series
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References:

[1] Hardy, G.H.) .- Divergent Series, Clarendon Press, Oxford, 1962. · Zbl 0032.05801
[2] Lehmer, D.H.) .- Euler constants for arithmetical progressions, Acta ArithmeticaXXVII (1975), pp. 125-142. · Zbl 0302.12003
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