Going through Derbyshire's Prime Obsession & trying to take LogIntegral of 20^ZetaZero[1] & comes up with a value of 1.99797 - 3.91384 I instead of -0.105384 + 3.14749 I quoted in the book. Am I doing something wrong?

Is it something to do with discrepancy between American definition of LogIntegral & European?


1 Answer 1


Please note the chapter on the zeta function in Wagon's Mathematica In Action, where he refers to a book by H.M. Edwards called Riemann's Zeta Function. Wagon cautions thatExpIntegralEi[r Log[x]]must be used instead ofLogIntegral[x^r]when x is real and r is complex. This is because the complex log function returns r Log[x]-2 Pi n i, withnchosen to make the imaginary part lie between -Pi and Pi, which leads to incorrect sums. For example,

N[ExpIntegralEi[ZetaZero[1] Log[20]]]

returns -0.105384 + 3.14749 I

  • $\begingroup$ Thank you for that reference & the explanation :) $\endgroup$
    – martin
    Mar 2, 2014 at 20:35
  • $\begingroup$ ... Just looked at it on Amazon - think I need to buy a copy $\endgroup$
    – martin
    Mar 2, 2014 at 20:37

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