When one considers the sequence of values f(n) of some reasonable arithmetic function, like the divisor function \tau, or the function \omega(n) the number of prime factors of n, 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 \omega(n) has normal order \log\log n: almost all numbers n have about \log\log n ; this is the theorem of Hardy-Ramanujan. It’s even better than this : for almost all numbers, \omega(n) is within (\log\log n)^{1/2+\epsilon} of its model \log\log n. In fact the values of \frac{\omega(n)-\log\log n}{\sqrt{\log\log n}} follow a Gaussian distribution, as the Erdös-Kac theorem reveals.

In contrary the divisor function \tau doesn’t have any normal order, but it does have an average order which is quite good-looking, namely \log n (while \log\tau has a normal order which, surprisingly isn’t \log\log n but a bit a less, namely \log 2\log\log n, showing that a few exceptionally highly divisble numbers are able to make the average deviate substantially).

Now of course on the primes \tau and \omega collapse violently, being stuck at respectively 2 and 1. The question is whether just after such a shocking value, at n=p+1, 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 p+1 isn’t absolutely any number, for instance it’s surely even; but then is (p+1)/2 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 p+1\equiv 1\text{ or } 0\text{ mod }3 whith equal probability.

From the point of view of \omega

There the answer is: yes, perfectly. Indeed, Heini Halberstam established that the Erdös-Kac theorem holds when n is constrained to range in the shifted primes p+1. That is, \omega(p+1) lies within M\sqrt{\log\log p} of \log\log p for a positive proportion p(M) of primes, the density being again gaussian.

From the point of view of \tau

Not quite. For this consider the Titchmarsh divisor problem, consisting in estimating \sum_{p\leq x} \tau(p+1). If \tau(p+1) was replaceable by \log(p+1) as we suppose, then this sum would be asymptotic to \sum_{p\leq x}\log p\sim x by Mertens formula. It turns out that it is in fact asymptotic to \frac{\zeta(2)\zeta(3)}{\zeta(6)}x, the constant prefactor being well over 1,9, so that it can be said that p+1 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 \sum_{n\leq x,n\equiv [0] 2}\tau(n)\equiv 2/3x\log x, so that in average \tau(2n) is close to 4/3\log n, but 1,9 is still much larger sensibly larger than 4/3. Here the higher chances of p+1 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 \log \tau(p+1) 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 1/\zeta(2)=\prod_{q \text{ prime}}(1-q^{-2}). It is then reasonable to wonder if among numbers of the forme n=p+1, the proportion of square free is the same. It’s clear to see that it can’t quite be so: indeed, p+1 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 q(q-1) chance of being divisibly by the prime q. So one guesses that p+1 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 p+1 is \prod_q (1-1/(q(q-1)), to compare with \prod_{q \text{ prime}}(1-q^{-2}).

The property of being squarefree can be replaced by the property of being k-free for instance, the density among shifted primes being then \prod_q (1-1/(q^{k-1}(q-1)), to compare with \prod_{q \text{ prime}}(1-q^{-k}).

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

We know that

\displaystyle{x\log x\sim\sum_{n\leq x}\tau(n)=\sum_{\substack{n\equiv 0\mod [2]\\n\leq x}}\tau(n)+\sum_{\substack{n\equiv 0\mod [2]\\n\leq x}}\tau(n)=S_0(x)+S_1(x)}

And let us suppose that \sum_{n\equiv 0\mod [2],n\leq x}\tau(n)\sim Ax\log x and \sum_{n\equiv 1\mod [2],n\leq x}\tau(n)\sim Bx\log x.

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 m odd, so

\displaystyle{\sum_{v_2(n)=k,n\leq x}\tau(n)=(k+1)\sum_{n\leq x/2^k,n\equiv 1[2]}\tau(n)\sim(k+1)Bx/2^k\log x} for fixed k. Pretending this asymptotic also holds when k is allowed to grow with x, we infer from S_0(x)=2S_1(x/2)+(3-2)S_1(x/4)+(4-3)S_1(x/8)\cdots that C=D+D/2+D/4\cdots=2D. Hence S_0(x)\sim 2x/3\log x and S_1(x)\sim x/3\log x.

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

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