Ramanujan's Letter (5)

I would like to move on to another fascinating result from Ramanujan's 1st letter to Hardy.

Ramanujan asks us to take all numbers with an odd number of prime divisors. This had me initially scratching my head as to what this might entail! He then gives a list of these numbers up to 50, i.e..

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47

It is noteworthy that - apart from 30 and 42 - these provide a list of the prime nos. to 50.

So in all but 2 cases up to 50 an odd number of prime divisors implies just one divisor, i.e. the prime number itself.

The 2 exceptions here, 30 and 42 both contain 3 dissimilar prime divisors i.e. 30 = 2 * 3 * 5 and 42 = 2 * 3 * 7.

He then gives a formula for the sum of the squares of the reciprocals of these numbers,

i.e. 1/2² + 1/3² + 1/5² + ... + 1/30² + 1/31² + ...  = 9/2π².

Now this bears direct comparison with the Riemann zeta function (for  s = 2), where, the sum of the squares of the reciprocals of the natural numbers, i.e.

1/1² + 1/2² + 1/3² + 1/4² +  ...  = π²/6.

What is remarkable here, is that if one attempts to sum up the reciprocals of the prime numbers that it will converge very close to the value 9/2π² as all the terms up to 1/29² relate to primes.

So one could be easily tempted therefore to draw the wrong conclusion that the sum of the squares of reciprocals of the prime numbers converges to a neat expression involving the corresponding square of π.

However what is very interesting is that - whereas with the sums of squares of the natural numbers - the square of π relates to the numerator of the result, in the case of the reciprocals of the squares of the primes the corresponding square of π relates to the denominator.

However it is still noteworthy that the sum of squares of the reciprocals of primes approximates so closely to a square of π expression.  


This pattern is even more in evidence in relation to the reciprocals of the same numbers (i.e. with an odd no. of prime divisors) when raised to the power of 4.

Ramanujan gives the following neat formula (showing definite similarities with the previous formula
for reciprocals raised to the power of 2).
 
1/2⁴ + 1/3⁴ + 1/5⁴ + ... + 1/30⁴ + 1/31⁴ + ... = 15/2π⁴.

The potential prime number trap is even more closely in evidence here.

For example if one adds up the reciprocals (raised to the power of 4) of all the primes to 100, one obtains the result 14.999649.../2π⁴. One might then be forgiven for assuming that with the addition of further prime terms that the numerator of the expression here would converge eventually to 15.

However the true result of adding the reciprocals (raised to the power of 4) of all numbers divisible by an odd number of prime factors up to 100 = 14.999976.../2π⁴.

So sometimes initial appearances may prove deceptive. However once again it is indeed striking that the sum of the reciprocals of the primes (raised to the power of 4) approximates so closely to 15/2π⁴.


Then there are further surprises arising. One might again be tempted to think that with nice expression in powers of π arising where the reciprocals of relevant terms are raised to the power of 2 and power of 4 respectively, that this pattern would be repeated for all powers (entailing positive even integers).

However this does not appear to be the case. Certainly from my own investigations, there was no evidence of any simple number pattern (relating to powers of π) emerging with other even integer dimensions.

Likewise I could find no simple pattern entailing reciprocals of numbers with an even number of prime factors.

So again it is so ingenious how Ramanujan made such discoveries.

Incidentally, he also gives a simple formula for the calculation of the frequency of numbers with an odd number of prime factors.

This is given as 3n/π².

This would imply that where n = 100, we should expect  30.396...  i.e. 30 (rounded to the nearest integer). And in fact this is exactly the number we obtain containing the 25 primes up to 100 and 30, 42, 66, 70 and 78.  

Of course most of the numbers here to 100 are prime. However as we move higher up the number scale, the relative percentage of prime to all divisors (comprising an odd number of prime factors) would steadily decrease.