Complementary Formulas

As we have seen in recent blog entries I have used the accumulated sum of the factors of successive natural numbers to estimate (to a high level of accuracy) the frequency of the Zeta 1 (Riemann) non-trivial zeros.

There are close connections as between this procedure and the accepted general formula for estimating the frequency of these zeros. This entails therefore that just as there is a simple general formula for the estimation of the primes (containing - by definition - no factors), in like manner an equally simple alternative formula exists for the estimation of the sum of factors  of the composite numbers.

Once again the formula for calculating the frequency of the non-trivial zeros

=  t/2π{ln (t/2π) – 1} where t represents numbers on the imaginary line.

Now as we have seen there is a direct relationship with numbers on the real line where

n (representing real number) = t/2π.

Therefore we obtain the corresponding formula for calculation of the accumulated sum of factors (up to the number n) by substituting t/2π with n.

Therefore the required formula = n(ln n – 1).

So once again if we wish to estimate the accumulated sum of factors of the (composite) numbers up to 100,
we obtain 100(ln 100 – 1) = 361 (to nearest integer).

As we have seen the correct answer is 357! So the estimate is therefore surprisingly accurate.

As I have stated there is a direct complementarity  here with the corresponding simple formula for the frequency of primes i.e. n/ln n.

So the estimate for the number of primes up to 100 = 100/ln 100 = 22 (to the nearest integer).

This compares with the true number of primes = 25. So the formula for estimation of the (accumulated) sum of factors is therefore relatively much more accurate.

Now the complementarity of the two formulas can be readily appreciated.

The probability of a number being prime = 1/ln n. So if for example we are the point on the number scale where this probability = 1/4, this entails that 1/ln n = 1/4.

Therefore ln n = 4.

However if on average 1 in 4 numbers is prime then this implies that the remaining 3 are composite (which is given therefore as ln n - 1).

And fascinatingly whereas we divide n by ln n to obtain the frequency of primes, in complementary fashion we multiply (ln n - 1) by n to obtain the frequency of factors.

Now because Conventional Mathematics does not look at number behaviour in a dynamic interactive manner, it fails to properly appreciate this important complementary behaviour as between the behaviour of primes on the one hand and the behaviour of factors of composite numbers on the other. This complementarity then extends to the behaviour of the non-trivial zeros with respect to the imaginary line, which is directly related to the behaviour  of common factors (on the real line).