The lcm{1,2,...,n} as a product of values of the sine function sampled over half of the points in a Farey sequence.

Recently Greg Martin (arXiv:0907.4384) showed a cute little formula for the lcm{1,2,3,..,n} = LCM(n).

Jonathan Vos Post reported this on OEIS in A003418 as follows:

"The product of the gamma-function sampled over the set of all rational numbers in the open interval (0, 1) whose denominator in lowest terms is at most n equals ((2*pi)^(1/2))*LCM(n)^(-1/2)."

Thus we look at a Farey sequence. If a/b is a member of this sequence 1−a/b is also a member. We have to compute a product and so we can trade in two Gamma evaluations for one sine evaluation by the reflection formula.

Number theorists like walking the log ...

Therefore let's look at two more items: the number of terms in a Farey sequence of order n, which is Sum{i=1..n} φ(i) − 1 (see A015614). And a nice cup of tea.

Here φ(n) denotes Euler's totient function, which is the number of positive integers not exceeding n and relatively prime to n. Further let Λ(n) denote von Mangoldt's function, then the second equation can be rewritten as

The last term, Sum lbs(k), can be approximated by

prox(x) = x( 3x / Pi^2 − 1) − sqrt(x) log(x).

The plot below shows the difference between the true value and the approximation for 3 ≤ n ≤ 3000.

See the Wikipedia article for more on Farey sequences.

Jutta Gut's theorem

Concerned with sequences:

 OEIS seq Number of fractions in a Farey sequence of order n A005728 1, 2, 3, 5, 7, 11, 13, 19 Sum of phi A002088 0, 1, 2, 4, 6, 10, 12, 18 Lcm of {1,2,3,...n} A003418 1, 1, 2, 6, 12, 60, 60 Lcm of the proper divisors of n A048671 1,1,1, 2,1, 6, 1, 4,3, 10 Exponential von Mangoldt A014963 1, 2, 3, 2, 5, 1, 7, 2, 3 Not a prime power A024619 6, 10, 12, 14, 15, 18, 20 2^phi A066781 2, 2, 4, 4, 16, 4, 64, 16 Prime powers A000961 1, 2, 3, 4, 5, 7, 8, 9 Twice the prime powers A138929 2, 4, 6, 8, 10, 14, 16,18 Cyclotomic polynomials at x = 1 A020500 0, 2, 3, 2, 5, 1, 7, 2, 3 Cyclotomic polynomials at x =-1 A020513 -1, -2,0,1,2,1,3,1,2,1, 5

Here is a Maple worksheet.