[seqfan] Asymptotics of sum of the first n primes with a remainder term

[Vladimir Shevelev]

I am writing about A007504. Bach&Shallit proved that a(n) ~ n^2 * log(n) / 2. It is not difficult also to prove a more precise formula (with a remainder term). Using the best known Rosser's result for n-th prime: p_n>=n*log(n), we have a(n)=sum[i=1,...,n}p_i>=sum{i=2,...,n}i*log(i)>=int_2^n (t-1)*log(t-1)dt=(n-1)^2log(n-1)/2-(n-1)^2/4+1/4.
On the other hand, it is known (cf. P.Dusart,arxiv:1002.0442, formulas (4.1),(4.2)) that for n>=4, p_n<=n*(log(n)+log(log(n))+1) and a(n)=sum[i=1,...,n}p_i<=c+sum{i=1,...,n}(i*(log(i)+log(log(i))+1)<=c+(log(n)+log(log(n))+1)n*(n+1)/2 (with constant c). Thus a(n)=n^2 * log(n) / 2 + O(n^2*log(log(n))). 

Regards,
Vladimir

 Shevelev Vladimir‎