Further Considerations

I mentioned in  a recent blog entry how the definition of the "factors" of a number is somewhat arbitrary, where the consideration as whether both 1 and the number itself should be considered as factors is a matter of convention.

The convention with respect to the "proper factors" of a number is to include 1, while excluding the number in question.

So once again the proper factors of 6 (the first perfect number) are therefore 1, 2 and 3. And the sum of these proper factors = 6 (which is why it qualifies as a perfect number).

However if we exclude 1 and include the number in question as a factor (which is the approach I have adopted with respect to estimating the frequency of Riemann zeros) an interesting alternative way for defining a perfect number arises.

So now we the factors of 6 as 2, 3 and 6.

Certainly this leads to the inelegant result of 11 i.e. (2 * 6  – 1) when we sum the factors.

However if we take the sum of reciprocals of these factors and then obtain the answer = 1.

So 1/2 + 1/3 + 1/6 = 1.

So this would then provide an interesting alternative way of defining a perfect number i.e. as the number where the sum of reciprocals of factors = 1.

Of course if we include both 1 and the number itself, another interesting scenario arises.

Here the sum of all factors (in our example 1 + 2 + 3 + 6) is double the perfect number in question (6), while the sum of the reciprocals of its factors (i.e. 1 + 1/2 + 1/3 + 1/6) = 2.

It also struck me that if all natural numbers were prime then  the Riemann zeros would thereby not exist.

This would follow by definition from my simple estimation method where - by definition - the accumulated sum of the factors of each number = 0. Therefore no non-trivial zeros would arise.

This points once again to the complementary nature of the primes and non-trivial zeros, which properly must be understood in a dynamic interactive manner.

Thus whereas the primes represent the independent aspect (where numbers have no factors) the non-trivial zeros represent the corresponding interdependent aspect (where composite numbers do contain such factors).

There is another interesting complementary aspect of behaviour worth mentioning.
As we know we can manually test for primes by establishing whether each successive number can be broken down into constituent factors.

Now with respect to the trivial zeros we are doing something similar i.e. testing each natural number to see how many constituent factors it contains.

However whereas in the case of the primes we get an individual estimate (i.e. whether each particular number is prime) in the case of the trivial zeros we get a collective measurement (i.e. an estimate of how many trivial zeros exist up to a given number).