Assume that $x,y > 2$, and that $x<y$. Then the second Hardy–Littlewood conjecture states that

$$\pi(x + y) - \pi(y) \leq \pi(x).$$

We can easily justify this heuristically, since

$$ \textrm{li}(x+y)-\textrm{li}(y) < \textrm{li}(x), $$ where we have applied the prime number theorem on the form

$$\pi(x) \sim \textrm{li}(x):=\int_2^x \frac{1}{\log t}dt.$$

As stated on Wikipedia, the conjecture is believed to be false, as it would be inconsistent with the more confident first Hardy–Littlewood conjecture on prime k-tuples, but it is expected that the first violation will only occur at very large $x$.

Now, as $x$ grows large, we have that $$ \lim_{x\rightarrow \infty} \textrm{li}(x) - [\textrm{li}(x+y)-\textrm{li}(y)] \rightarrow \infty, $$ while asymptotically $$ \lim_{x\rightarrow \infty} \frac{\textrm{li}(x)}{\textrm{li}(x+y)-\textrm{li}(y)} \rightarrow 1. $$

From the first of these expressions it appears naively that the larger $x$ is, the less probability for the conjecture to fail; even more so if $y$ also becomes large relative to $x$. I would very much like to know whether there are other heuristic arguments for understanding why the conjecture likely is false, besides the clash with the first Hardy–Littlewood conjecture.

  • $\begingroup$ You may find this and this interesting. $\endgroup$ – user57432 Oct 15 '14 at 4:47

These are the heuristics.

A. The Hardy - Littlewood conjecture for the numbers $\pi_k(x) =c_kx \log^{-k} x\left (1+o(1)\right )$ of $k$-tuples of primes $n+a_1,n+a_2,\ldots,n+a_k$ as $n \to \infty$.

It is valid for all admissible $k$ tuples $a_1,a_2, \ldots,a_k$, with $k\ll xe^{\sqrt{\log x}}$. The upper bound for $k$ follows from the prime number theorem, a sharper bound was proved in [1].

B. The Hardy - Littlewood conjecture for $\pi(x+y)-\pi(x)< \pi(y)$ for primes in short intervals as $x \to \infty$.

It is valid for all short intervals $[x,x+y]$ such that $y\gg xe^{\sqrt{\log x}}$.

The lower bound for $y$ follows from Maier's theorem about the numbers of primes in short intervals , see [2], and conjecture A.

Thus, the conjectures are true within their domain of definition. These conjectures (and perhaps better bounds) will be unexpectedly proved, any time, any day, by any author or authors.

  1. Christian Elsholtz, Upper bounds for prime $k$-tuples of size $\log N$ and oscillations, Arch. Math. 82 (2004), 33-39.
  2. Maier, Helmut, Primes in short intervals. Michigan Math. J. 32 (1985), no. 2, 221-225.
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    $\begingroup$ How does this address the question "are there other heuristic arguments for understanding why the conjecture likely is false"? $\endgroup$ – Yemon Choi Jan 13 '19 at 16:10

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