## 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

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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