Re: On the partial sums of reciprocals of primes

From: Olivier Bordellès (borde43_at_wanadoo.fr)
Date: 08/30/04 

Date: Mon, 30 Aug 2004 13:30:02 +0000 (UTC)

Explicit inequalities on some functions of prime numbers can be found
in the (very good) paper entitled "Approximate formulas for some
functions of prime numbers" by J.B. Rosser & L. Schoenfeld (Illinois
J. Maths 6, 1962, 64 - 94). For example, they showed that :

1. |Sum ( log(p)/p , p , 2 , x ) - log(x) + E | < 1/(2*log(x))

Where E := Gamma + Sum( lop(p)/(p^2-p) , p ) = 1.33258227573322087,

2. Pi(x) < 1.25506 x/log(x) (x>1) where Pi(x) is the number of prime
numbers up to x (see the prime number theorem),

3. p_n > n.log(n) (n>1) where p_n is the nth prime,

4. theta(x) < 1.000081x (x>1) where theta is one of the Tchebytchev
function (ie theta(x) = Sum ( log(p) ,p ,2 , x)). This result is more
recent and may be found in the paper "Sharper bounds for the functions
theta(x) and psi(x) II" (Maths. Comp. 30 (1976), 337-360).

More recent results can be found in Dusart's PHD thesis 1998 (see
online). For example, he has proved that :

p_n >= n.(log(n) + loglog(n) - 1) (n>1)

and that

|theta(x)-x| < 515.x/log(x)^3 (x>1).

The proofs are based on the computation of zeros of the Riemann zeta
function on the critical line (sigma = 1/2), and hence are not
elementary.

Olivier Bordellès.