Logarithmic integral function: Difference between revisions

Revision as of 21:23, 19 November 2002

Revision as of 07:22, 20 November 2002

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The logarithmic integral is mainly important because it occurs in estimates of [[prime number]] densities, especially in the [[prime number theorem]]:

:π(''x'') ~ liLi(''x'')

where π(''x'') denotes the number of primes smaller than or equal to ''x'', and Li(x) is the [[offset logarithmic integral]] function. This is related to the logarithmic integral described above since Li(x) = li(x) - li(2). It gives a better estimate to the π function than li(x).

In order to avoid the principal value calculation, the prime number theorem is sometimes presented in terms of the integral ₂<font size="+1">&int;</font>^x 1/ln ''t'' d''t'', which differs from li(''x'') by the value li(2) &asymp; 1.04516. This is the [[offset logarithmic integral]] function.

The function li(''x'') is related to the ''[[exponential integral]]'' Ei(''x'') via the equation