Question

Let $\Lambda(n)$ be the von Mangoldt function. The prime number theorem is equivalent to the statement that $\sum_{n \leq N} \Lambda(n) \approx N$. Defining $\lambda_{*}(n)= \Lambda(n)-1$ we may rewrite this as $S(N) = \sum_{n \leq N} \Lambda_{*}(n) =o(N)$. Now it is known that $|S(N)| \gg |N|^{1/2}$ infinitely often. Moreover, on the RH we have that $|S(N)| \ll N^{1/2}ln^2(N)$. Not that these estimates differ by a factor of $ln^2(N)$.

My question is the following: What do probabilistic considerations suggest the correct error term to be?

Let me suggest a model: Let $X_n$ be a sequence of independent random variables such that $X_n = \ln(n)-1$ with probability 1/ln(n) and $-1$ with probability $1-1/ln(n)$, and form the sum $T(N)= \sum_{n=1}^{N} X_n$. Is there an elementary function $E(N)$ such that $ \lim sup_N |T(N)|/E(n) = 1 $ holds almost surely?

Notice that if the primes had positive density in the integers and we adjusted our model accordingly the law of the iterated logarithm would allow us to take $E(N)$ to be a multiple of $|N|^{1/2}\ln\ln(N)$.

(More generally, I'm interested in understanding sums of the above form (that is independent random variables with slowly increases variance) if you know of an appropriate reference.)

Answer 1

Let $P_n$ be independent variables which are 1 with probability $1/\log n$ and $0$ with probability $1-1/\log n$ and let $$ \Pi(x) = \sum_{n\leq x} P_n.$$

Then Cram\'{e}r showed that, almost surely,

$$ \limsup_{x\rightarrow \infty} \frac{|\Pi(x)-\ell i(x)|}{\sqrt{2x}\sqrt{\frac{\log\log x}{\log x}}} = 1 $$

where

$$\ell i (x) = \int_2^x \frac{dt}{\log t}.$$

See page 20 here: http://www.dms.umontreal.ca/~andrew/PDF/cramer.pdf

Edit: H. L. Montgomery has given an unpublished probabilistic argument that suggests

$$ \limsup_{x\rightarrow \infty} \frac{|\psi(x)-x|}{\sqrt{x} (\log\log\log x)^2} = \frac{1}{2\pi}.$$

This is announced in: H.L. Montgomery, "The zeta function and prime numbers," Proceedings of the Queen's Number Theory Conference, 1979, Queen's Univ., Kingston, Ont., 1980, 1-31.

Answer 2

Cramer's model is well-known to be (provably) a little bit off from the truth (See Maier's work on short gaps between primes). There is a different probabilistic model that you can use. It is motivated by the following equivalent form of the Riemann Hypothesis: $$ \text{RH true} \iff \sum_{n \leq X} \mu(n) \ll_{\varepsilon} X^{1/2 + \varepsilon} $$ for every fixed $\varepsilon > 0$, where $\mu$ is the the Moebius function. Motivated by this equivalence, we consider the following problem: Let $X_p$ be a sequence of independent random variable with $\mathbb{P}(X_p = 1) = \mathbb{P}(X_p = -1) = 1/2$ and for squarefree $n = p_1 \ldots p_k$ define $$ X_n = X_{p_1} \ldots X_{p_k} $$ What almost-sure bounds for $$ \sum_{n \leq X} X_n $$ can we obtain? It turns out that this is a difficult problem in it's own right. But a few results are known. It was known for a long time (since Wintner) that $$ \sum_{n \leq X} X_n \ll_{\varepsilon} X^{1/2 + \varepsilon} $$ almost surely and Halasz improved this to bound to an $\ll X^{1/2} \exp(C \sqrt{\log\log X})$ (Halasz proved a slightly weaker bound but his method can be used to obtain the bound stated here). So you could say that this model suggests a bound of $X^{1/2} \exp(C \sqrt{\log\log X})$ for partial sums of the Moebius function, but the difference between $X_n$ and $\mu(n)$ turns out surprisingly to be too stricking to believe that this is the truth. However, the model itself certainly deserves more study!

As a post-scriptum, let me mention that the best known bound (assuming RH) for the Partial sums of $\mu(n)$ is $C \cdot X^{1/2} \exp(\sqrt{\log X} (\log\log X)^{14})$ (see http://arxiv.org/abs/0705.0723)