Riemann in his original article on the zeta function in 1859, was enabled through appreciation of the important nature of the zeta zeros to provide a formula – given the incorporation of the influence of a sufficient number of zeros – could exactly predict the magnitude of primes up to a given number.
However the other side of the coin as it were is the corresponding accumulated total of all the divisors (or factors) of the natural numbers up to a given magnitude.
Just as we can give the simple formula n/ln n that will approximate the primes up to a given number n, we can give the corresponding simple formula n(log n) to calculate the accumulated number of factors.
So – counting all divisors of a number (including 1 and the number in question) – the total number of factors to 100 = 482. And the corresponding answer given by our simple formula is 461 (when rounded to the nearest integer) which is already surprisingly accurate (over 95%).
Now Dirichlet proposed a slight modification on this formula as n(log n + 2γ – 1) + Δ(n), where γ (the Euler-Mascheroni constant) = .5772156649… and Δ(n) represents a remaining variable which can have a positive or negative value
Using this modified formula, the total number of factors to 100 = 476 (to the nearest integer). Now the reason why we still get a slight underestimate here of 6 is mainly due to the fact that the cut-off point we are using for n, i.e. 100 is in itself a highly composite number with 9 divisors. For example if we had taken the estimate with respect to 101 (which is prime) the total number of factors would increase by 2 to 478 while the estimated formula would give 476 (when rounded) that reduces the underestimate to 2.
However, because of the random nature in which the factor composition of numbers varies, there will still always be a certain absolute discrepancy (positive or negative) using the Dirichlet formula in the estimate of the total number of factors.
Now various attempts have been made to put an upper bound on the value of this last variable term. Dirichlet himself suggested n1/2. Later estimates have reduced the value down close to .3, though it is widely believed that .25 would effectively be the lowest power that can be assigned. However this would only strictly apply as n approaches ∞. This would entail for example that when n = 100, the difference between our estimate and the actual number of divisors would be less than 10 for Dirichlet’s power of ½. In fact it is 6! However if the power is taken as .25 this would means the limit on our bound would be just over 3, whereas in fact the actual discrepancy is 6. So it would not work in this case where n is a relatively small finite value.
In one way however it is surprising that an exact formula cannot be provided – at least in principle to measure the sum of divisors up to a given number – when a formula exists i.e. the Van Mangoldt formula, to do this for the primes.
There are however two distinct distributions with respect to the primes.
The commonly understood distribution relates to the frequency of individual primes up to a given natural number magnitude. So for example we have 25 such primes up to 100 and the primes in this regard are randomly distributed.
However the alternative distribution relates to the collective distribution of prime factors within a given natural number. In this sense 100 for example has just 2 (distinct) prime factors i.e. 2 and 5.
The question then arises as to the nature of the distribution of all the divisors of a natural number. Again the presumption would be that this distribution is likewise random in nature but its precise relationship to the corresponding distribution of prime factors may be much more difficult to define.
Going back however to Dirichlet’s formula for the (accumulated) sum of divisors, I have generally used the slightly modified version where one divisor for each number is treated as redundant.
This in fact is what happens for example in the case of perfect numbers.
So 6 for example has 4 divisors i.e. 1, 2, 3 and 6. However the last of these is excluded giving a measurement therefore of what are sometimes referred to as the proper factors of the number. (There is however considerable confusion evident in the definition of proper factors with some definitions excluding both 1 and the number in question). And 1 + 2 + 3 = 6. Therefore 6 is a perfect number.
So the formula for the accumulated sum of proper factors can be given as
n(log n + 2γ – 2) + Δ(n).
Therefore for estimates we can leave out the variable term and use
n(log n + 2γ – 2) = n(log n – .8456...)
This is then quite close to the simpler approximation n(ln n – 1).
Therefore using the former more precise measurement we have 376 accumulated (proper) factors up to 100, whereas the latter estimate gives 361. The actual number of factors is 382. Then in relative terms as n becomes progressively larger, the latter simpler estimate will approach closer and closer to 100% accuracy.
The advantage of all of this is that the corresponding formula for frequency of zeta zeros up to t is t/2π(ln t/2π – 1) where n = t/2π. And as this formula is remarkably accurate in terms of predicting the actual frequency of zeta zeros to t, we can then use it in turn to calculate the (accumulated) frequency of proper factors to n.
For example as shown in the above table. the actual number of divisors up to n =1000, is 6061. The corresponding estimate of the frequency of the Riemann zeta zeros to t, where n = t/2π, is 5908.
And this is already 97.46% accurate.
And as we shall see in future contribution all factor distributions for Dirichlet L-functions can in turn be linked to the distribution of the Riemann zeta zeros.
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