Other Gamma Applications

Number theory is an area of mathematics that studies the general properties of numbers, particularly the integers. It is here that we find a couple interesting applications of gamma.

Numbers Dividing Numbers

Consider all the integers from 1 to n, where n is any number that we please. Each of these integers has a certain number of divisors.

For example the number 30 is divisible by 1, 2, 3, 5, 6, 10, 15, and 30, so it has 8 divisors. The number 1 has only itself as a divisor. Every number greater than 1 has at least 2 divisors, namely 1 and itself.

Averaging Divisors

If we wish to know the average number of divisors of all the numbers from 1 to n, there is actually a really simple answer! The average number of divisors is approximately:

ln n +2γ -1.

This result was proved by Lejeune Dirichlet in 1838.

Testing It Out!

Although this is only an approximation it gets better and better as n gets larger. For n=1000 our approximation gives 7.06219…, while the true answer is 7.069. That’s pretty accurate! This wouldn’t be possible without gamma.

Another Application

Another great number theory application of gamma occurs when looking at the remainders from division.

If you divide an integer n by all the integers less than it, sometimes you are going to get an even division and sometimes you are going to be left with an extra fractional bit.

The Distance of Division

Consider the distances from all these divisions to the next integer up. If you take the average of them, you are going to get a number close to gamma!

This value gets closer to gamma the higher the integer n that you consider. This result was proved by Charles de la Vallèe Poussin in 1898.

Interestingly enough, the average is still gamma if, instead of dividing by all the integers less than n, you consider only the prime integers less than n.