In this paper we establish a general asymptotic formula for the sum of the first n prime numbers, which leads to a generalization of the most accurate asymptotic formula given by Massias and Robin in 1996.

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In this paper we establish several results concerning the generalized Ramanujan primes. For $n\in\mathbb{N}$ and $k \in \mathbb{R}_{> 1}$ we give estimates for the $n$th $k$-Ramanujan prime which lead both to generalizations and to improvements of the results presently in the literature. Moreover, we obtain results about the distribution of $k$-Ramanujan primes. In addition, we find explicit formulae for certain $n$th $k$-Ramanujan primes. As an application, we prove that a conjecture of Mitra, Paul and Sarkar concerning the number of primes in certain intervals holds for every sufficiently large positive integer.Comment: The final publication is available at link.springer.com under http://link.springer.com/article/10.1007/s11139-015-9693-

Robin’s criterion states that the Riemann hypothesis is true if and only if $\unicode[STIX]{x1D70E}(n)<e^{\unicode[STIX]{x1D6FE}}n\log \log n$ for every positive integer $n\geq 5041$. In this paper we establish a new unconditional upper bound for the sum of divisors function, which improves the current best unconditional estimate given by Robin. For this purpose, we use a precise approximation for Chebyshev’s $\unicode[STIX]{x1D717}$-function.

Let $$\sigma (n)$$ σ ( n ) denote the sum of divisors function of a positive integer n. Robin proved that the Riemann hypothesis is true if and only if the inequality $$\sigma (n) < \textrm{e}^{\gamma }n \log \log n$$ σ ( n ) < e γ n log log n holds for every integer $$n > 5040$$ n > 5040 , where $$\gamma $$ γ is the Euler–Mascheroni constant. In this paper we establish a new family of integers for which Robin’s inequality $$\sigma (n) < \textrm{e}^{\gamma }n \log \log n$$ σ ( n ) < e γ n log log n hold. Further, we establish a new unconditional upper bound for the sum of divisors function. For this purpose, we use an approximation for Chebyshev’s $$\vartheta $$ ϑ -function and for some product defined over prime numbers.