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Effective bounds for certain functions concerning prime numbers. (Bornes effectives pour certaines fonctions concernant les nombres premiers.) (French) Zbl 0856.11043

Let \(p_k\) be the \(k\)-th prime number and let \(\theta(p_k)= \sum^k_{i=1}\log p_i\). The authors obtain new estimates and improvements of the bounds given by Rosser and Schoenfeld, Schoenfeld, and Robin for the functions \(p_k\), \(\theta(p_k)\), \(S_k= \sum^k_{i=1}p_i\) and \(S(\chi)=\sum_{p\leq x}p\). The estimates are obtained by using methods based on the Stieltjes integral and by direct calculation for small values.

MSC:

11N25 Distribution of integers with specified multiplicative constraints
11N56 Rate of growth of arithmetic functions
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References:

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