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Chebyshev function help

  1. Jul 21, 2009 #1
    I'm a bit confused as to how to calculate the 2nd chebyshev function.

    I know [tex]\psi (x) = \sum_{p^k \le x} \ln p[/tex]

    but can someone show me how to expand it? Like, do I use for 2^k, 2, 4, 8, 16, 32... for all the [tex]p^k \le x[/tex], same with 3, and 4 and so on?

    MW gives the example:

    [tex]\psi (10) = \ln (2520) = 3\ln2+2\ln3+\ln5+\ln7[/tex]

    how would say, [tex]\psi (30)[/tex] be written?
     
  2. jcsd
  3. Jul 21, 2009 #2
    [tex]\spi(30)[/tex]
    Let's see...
    [tex]2,4,8,16 < 30[/tex]
    [tex]3,9,27 < 30[/tex]
    [tex]5,25 < 30[/tex]
    [tex]7 < 30[/tex]
    [tex]11 < 30[/tex]
    [tex]13 < 30[/tex]
    [tex]17 < 30[/tex]
    [tex]19 < 30[/tex]
    [tex]23 < 30[/tex]
    [tex]29 < 30[/tex]
    so ... [tex]\psi(30) = 4\log 2 + 3\log 3 + 2\log 5 + \log 7 + \log 11 + \log 13 + \log 17 + \log 19 + \log 23 + \log 29[/tex]
     
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