When one considers the sequence of values of some reasonable arithmetic function, like the divisor function , or the function the number of prime factors of , one may notice that excluding a few abnormal values, the sequence looks very much like a neat, recognizable sequence. Or at least, smoothing the sequence by averaging it produces a sequence which looks neat.

Thus has normal order : almost all numbers have about ; this is the theorem of Hardy-Ramanujan. It’s even better than this : for almost all numbers, is within of its model . In fact the values of follow a Gaussian distribution, as the Erdös-Kac theorem reveals.

In contrary the divisor function doesn’t have any normal order, but it does have an average order which is quite good-looking, namely (while has a normal order which, surprisingly isn’t but a bit a less, namely , showing that a few exceptionally highly divisble numbers are able to make the average deviate substantially).

Now of course on the primes and collapse violently, being stuck at respectively 2 and 1. The question is whether just after such a shocking value, at , one recovers more normal arithmetic indices, whether there are traces of the past catastrophe (or signs of the coming castrophe just before it).

Of course isn’t absolutely any number, for instance it’s surely even; but then is just any number? It must also be said that it has higher chances to be mutiple of 3 than a generic number, in fact one in two chance instead of one in three, because whith equal probability.

**From the point of view of **

There the answer is: yes, perfectly. Indeed, Heini Halberstam established that the Erdös-Kac theorem holds when is constrained to range in the shifted primes . That is, lies within of for a positive proportion of primes, the density being again gaussian.

**From the point of view of **

Not quite. For this consider the Titchmarsh divisor problem, consisting in estimating . If was replaceable by as we suppose, then this sum would be asymptotic to by Mertens formula. It turns out that it is in fact asymptotic to , the constant prefactor being well over 1,9, so that it can be said that has almost twice as many divisors as banal numbers of his size. Now remember it’s always even; now there are good heuristic reasons to believe that , so that in average is close to , but 1,9 is still much larger sensibly larger than 4/3. Here the higher chances of to be divisible by 3, 5 etc. in comparison to a banal number weigh enough to make the average deviate.

It would be interesting to determine whether has a normal order, in the same vein as Halberstam’s result.

**Another criterion of banality: the probability of being squarefree**

As we know, the density of squarefree numbers is . It is then reasonable to wonder if among numbers of the forme , the proportion of square free is the same. It’s clear to see that it can’t quite be so: indeed, has one on two chance of being divisible by the square 4 (while a generic number has of course one on four chance), one on six chances of being divisible by 9, one on chance of being divisibly by the prime . So one guesses that is a little less likely to be squarefree than any number; indeed, it’s been proven by Mirsky that the proportion of squarefree numbers among numbers of the form is , to compare with .

The property of being squarefree can be replaced by the property of being -free for instance, the density among shifted primes being then , to compare with .

**Appendix: average of the divisor function on odd and even numbers**

We know that

And let us suppose that and .

Now the sum on even numbers can be decomposed into a sum on numbers of 2-adic valuation 1,2,3… But numbers of 2-adic valuation k are numbers which are of the form 2^km with odd, so

for fixed . Pretending this asymptotic also holds when is allowed to grow with , we infer from that . Hence and .

It might be that this intuition has been proven more rigorously, I’d like to see where!

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