[SeqFan] sum of primes, product of primes

[Jeffrey Shallit]

> Sequence A2110 is the sum of the first n primes.  Let S(x) = the sum of the
> primes < x.  Is the growth rate of S(x) known?  A quick check of the sum of
> primes < 10^11 suggests that the growth rate is at least x^1.846, and of
> course it is <= x^2.  Is there a more accurate growth rate known?

For this, see Section 2.7 of my book with Eric Bach, _Algorithmic
Number Theory_, MIT Press, 1996.  There it is proved that the sum
over primes p <= x is asymptotically x^2/(2 log x).  Furthermore, a
general technique for estimating such sums is provided.

> Sequence A7504 is the product of the first n primes.  If P(x) = log of
> product of prime p = sum of log(p) for all p <= x, then empirically
> log(P(x))/log(x) -> 1 as x -> infinity.  Is this correct?

The sum of the logs of the primes <= x is the well-known number
theoretic function theta(x), discussed in nearly
any book on number theory (esp. analytic number theory).  One of 
the statements of the prime number theorem is that this function
is asymptotically equal to x (not log (x), as you wrote, which I
presume is a typo).  See, for example, p. 206 and p. 233 of my
book with Eric Bach.

Jeffrey Shallit, Computer Science, University of Waterloo,
Waterloo, Ontario  N2L 3G1 Canada shallit at graceland.uwaterloo.ca
URL = http://math.uwaterloo.ca/~shallit/