Logarithmic integral function

In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function.

It is relevant in problems of physics and has number theoretic significance.

Equivalently, The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 83381 05028 39684 85892 02744 94930... OEIS: A070769; this number is known as the Ramanujan–Soldner constant.

It must be understood as the Cauchy principal value of the function.

The function li(x) is related to the exponential integral Ei(x) via the equation which is valid for x > 0.

This identity provides a series representation of li(x) as where γ ≈ 0.57721 56649 01532 ... OEIS: A001620 is the Euler–Mascheroni constant.

A more rapidly convergent series by Ramanujan [1] is The asymptotic behavior for x → ∞ is where

The full asymptotic expansion is or This gives the following more accurate asymptotic behaviour: As an asymptotic expansion, this series is not convergent: it is a reasonable approximation only if the series is truncated at a finite number of terms, and only large values of x are employed.

denotes the number of primes smaller than or equal to

Assuming the Riemann hypothesis, we get the even stronger:[2] In fact, the Riemann hypothesis is equivalent to the statement that: For small

but the difference changes sign an infinite number of times as