From: Christian.Radoux@skynet.be Newsgroups: sci.math.research,sci.math Subject: Re: New(?) prime conjectures Date: Wed, 25 Mar 1998 17:41:55 -0600 In article <6fbhcr$kcs$1@broadway.interport.net>, "Bill Daly" wrote: > > Does anyone know if the following conjectures are new? > > Let p[j] be the j-th prime. Then > > Conjecture 1. Let s(n) = sum(j=1..n, 1/p[j]) - ln(ln(n)). Then s(n) > converges as n goes to infinity. > > The hand-waving argument is that p[j] is approximately j*ln(j), and the > similar series t(n) = sum(j=2..n, 1/(j*ln(j))) - ln(ln(n)) is convergent. > > Conjecture 2. sum(j=1..oo, 1/(p[j]*ln(p[j]))) is convergent. > > The hand-waving argument is that p[j]*ln(p[j]) is approximately j*ln(j)^2, > and sum(j=2..oo, 1/(j*ln(j)^2)) is convergent. > > Regards, > > Bill > > These problems are well known : Conjecture 1. Yes Actually, we have the theorem : s(n) = ln(ln p[n]) + alpha + O(1/(ln(p[n])), where alpha is 0.261 497 212 8... Conjecture 2. Yes Generalization Let f be a continuous function on [2,OO[ Let us call S(n) = f(2) + f(3) + f(5) + f(7) + f(11) +... + f(p[n]) Then, to get an asymptotic approximation of S(n), integrate f(x)/ln(x) from 2 to p[n]. If this integral converges when n -> OO, the S(n) converges also. And this is indeed the case for your particular function f(x)/x*ln(x),since the integral, from 2 to OO of 1/(x*(ln x)^y) is convergent for any y > 1 (here y = 2) With best regards e-mail : Christian.Radoux@skynet.be URL : http://users/skynet.be/radoux (case sensitive server !...) -----== Posted via Deja News, The Leader in Internet Discussion ==----- http://www.dejanews.com/ Now offering spam-free web-based newsreading ============================================================================== [In this "generalization", f must be decreasing (to zero) -- djr] ============================================================================== From: Jan Kristian Haugland Newsgroups: sci.math Subject: Re: New(?) prime conjectures Date: Wed, 25 Mar 1998 19:31:58 +0000 Your first conjecture is a theorem of Mertens from 1874. He proved that sum (p prime < x) (1/p) = log log x + A + O(1 / log x) for some constant A. See H. Davenport: Multiplicative Number Theory. Your second conjecture is a direct consequence of the Prime Number Theorem. On Wed, 25 Mar 1998, Bill Daly wrote: > Does anyone know if the following conjectures are new? > > Let p[j] be the j-th prime. Then > > Conjecture 1. Let s(n) = sum(j=1..n, 1/p[j]) - ln(ln(n)). Then s(n) > converges as n goes to infinity. > > The hand-waving argument is that p[j] is approximately j*ln(j), and the > similar series t(n) = sum(j=2..n, 1/(j*ln(j))) - ln(ln(n)) is convergent. > > Conjecture 2. sum(j=1..oo, 1/(p[j]*ln(p[j]))) is convergent. > > The hand-waving argument is that p[j]*ln(p[j]) is approximately j*ln(j)^2, > and sum(j=2..oo, 1/(j*ln(j)^2)) is convergent.