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Sum of the serie 1/nextprime(n)

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  • Jacques Tramu
    Let s take the sequence : u(n) = nextprime =n (n = 0 ) - Sloane A007918 2,2,2,3,5,5,7,7,11,11,11,11,13,13,17,17,17,17,19,19,23,23,
    Message 1 of 1 , Jul 14, 2005
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      Let's take the sequence :

      u(n) = nextprime >=n
      (n >= 0 ) - Sloane A007918

      2,2,2,3,5,5,7,7,11,11,11,11,13,13,17,17,17,17,19,19,23,23,
      23,23,29,29,29,29,29,29,31,31,37,37,37,37,37,37,41,41,41,41

      Now take a look at the serie : s(n) = 1/u(0) + 1/u(1) + .... + 1/u(n)

      s(n) is divergent. (includes all primes as denominators)

      I seems that s(n) = log(n) + K -- where K = 0.2934581.... when n goes to
      infinity.

      Ex : (tested from = n=10 to n = 10^10)
      n = 10^10
      s(n)= 23.3193090775 log(n) = 23.0258509299 s(n) - log(n) = 0.2934581....

      ** Question: does anybody have an idea about that ?

      (trying to look into Samaranche papers, and found nothing).

      thx.



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