It can be proven that the following forms all sum to the same constant:
where σ0(n) = d(n) is the
divisor function, a
multiplicative function that equals the number of positive
divisors of the number n. To prove the equivalence of these sums, note that they all take the form of
Lambert series and can thus be resummed as such.[2]
Despite its irrationality, the
binary representation of the Erdős–Borwein constant may be calculated efficiently.[5][6]
Applications
The Erdős–Borwein constant comes up in the
average case analysis of the
heapsort algorithm, where it controls the constant factor in the running time for converting an unsorted array of items into a heap.[7]
^Knuth (1998) observes that calculations of the constant may be performed using Clausen's series, which converges very rapidly, and credits this idea to
John Wrench.