What is Gauss's logarithmic integral Li(N)? Using the model of a prime number dice Gauss guessed that the number of primes less than N might be estimated by the number

1/log(2)+1/log(3) +…+1/log(N).

We can draw a picture of this number. The sum of all these terms represents the area under a descending staircase where the nth step has height 1/log(n). Although this area very accurately estimates the number of primes, Gauss found that the best fit of all came by drawing a smooth curve through this staircase where the height at each value x is 1/log(x).

[The area under the staircase is 1/log(2)+1/log(3)+…+1/log(9). The area under the smooth graph is what Gauss called Li(9), the logarithmic integral of 9.]

By calculating the area under this smooth graph, rather than the descending staircase, Gauss achieved an even better guess for the number of primes less than N. The area under the graph to N is given the name Li(N) which stands for the logarithmic integral of N. Integration is the way mathematicians calculate the area under a curve.