From: sfinch@math28.skiles.gatech.edu (Steven Finch -Rs. Calkin) Newsgroups: sci.math.research Subject: Re: A problem in number theory Date: 26 Jun 1996 13:50:19 -0400 Hello! If B(n) is the number of positive integers not exceeding n which can be expressed as a sum of two squares, then K = lim_{n\to\infty} (sqrt(ln(n))/n) * B(n) exists and K = .764223653... No simple expression for K is known. K is called the Landau-Ramanujan constant and much more information about it can be found at http://www.mathsoft.com/asolve/constant/lr/lr.html K is now known to 10000 decimal places (a recent result). Also, a second-order constant lim_{n\to\infty} (ln(n)^(3/2))/(K*n)) * * (B(n) - K*n/sqrt(ln(n))) = 0.581948659... has recently been computed to 5000 places. Steve Finch sfinch@mathsoft.com ============================================================================== From: Peter-Lawrence.Montgomery@cwi.nl (Peter L. Montgomery) Newsgroups: sci.math.research Subject: Re: A problem in number theory Date: Wed, 26 Jun 1996 18:26:50 GMT In article <31D14FAD.41C6@math.uni-hamburg.de> Torsten Kemps writes: > >Let \phi(n) be the number of positive integers k<=n that can be written >k=x^2+y^2 with x,y also integers. Does the limit > lim_{n\to\infty} \phi(n)/n >exist and what is it's value? Numerical calculations suggest that it >exists. > The limit is zero. A positive integer k is expressible in the form x^2 + y^2 if and only if the exponents of 3, 7, 11, 19, 23, ... in the prime factorization of k are all even. If p == 3 (mod 4) is prime, then the density of integers whose exponent of p is even is 1 - 1/p + 1/p^2 - 1/p^3 + ... = p/(p + 1). The infinite product (3/4)(7/8)(11/12)(19/20) ... diverges to zero, by comparison with 1/3 + 1/7 + 1/11 + 1/19 + ... [It is well-known that the sum of the reciprocals of the primes diverges. Is there a simple demonstration that the sum of the reciprocals of the primes == 3 mod 4 also diverges?] -- Peter L. Montgomery pmontgom@cwi.nl San Rafael, California My sex is male. I am ineligible for an abortion. ============================================================================== From: Paul.Zimmermann@loria.fr (Paul Zimmermann) Newsgroups: sci.math.research Subject: Re: A problem in number theory Date: 27 Jun 1996 13:23:55 GMT In article <4qrr6h$3kt@gannett.math.niu.edu> rusin@vesuvius.math.niu.edu (Dave Rusin) writes: In article <31D14FAD.41C6@math.uni-hamburg.de>, Torsten Kemps wrote: >Let \phi(n) be the number of integers k\leq n that can be expressed as >k=a^2+b^2, a,b integers. Does the limit > \lim_{n\to\infty} \phi(n)/n >exist and what is it's value? Numerical computations suggest that it >exists. The notation "phi(n)" is sort of taken, but we'll let you borrow it. Some old posts on the topic are at 93_back/2squares [I updated the URL -- djr] but the long and short of it is that your phi(n) is approximately (3/4)* n / sqrt(log(n)), so that your limit exists and equals zero. dave More precisely, the number of integers less than n that are sums of two squares behaves like K*n/sqrt(log(n)) where K is the Landau-Ramanujan constant. Dave Hare from the Maple group just computed 10000 digits of K. The first digits are K=0.764223653... See http://www.mathsoft.com/asolve/constant/lr/lr.html for more information. Therefore the sums of two squares have density 0. As far as I know it is not known whether the sums of *three cubes* have density 0 or not. (I have found 100735175 sums of 3 cubes less than 10^9.) Paul -- Paul Zimmermann Projet Eureca INRIA Lorraine Technopole de Nancy-Brabois 615 rue du Jardin Botanique BP 101 F-54600 Villers-les-Nancy Paul.Zimmermann@loria.fr http://www.loria.fr/~zimmerma