Effective bounds for certain functions concerning prime numbers (Q1919163)

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.