in 1923 Hardy and Littlewood proposed the conjecture $\pi (m+n) \leq \pi (m) + \pi (n)$. Is there any progress towards solving this conjecture?

14$\begingroup$ It's known that this contradicts the general prime $k$tuple conjecture primes.utm.edu/glossary/page.php?sort=PrimeKtupleConjecture and many number theorists find the $k$tuple conjecture the more plausible. $\endgroup$– Robin ChapmanJul 5, 2010 at 7:08

2$\begingroup$ You mean Littlewood not Wright (who was born 1906): see en.wikipedia.org/wiki/… $\endgroup$– Charles MatthewsJul 5, 2010 at 11:11

$\begingroup$ re: "many number theorists", is there ANY number theorist who finds this conjecture credible, given that it contradicts the $k$tuplets conjecture (and so is also is challenged by the supporting evidence for the $k$tuplets conjecture, such as the probability heuristics that apply to many other problems, or the GreenTao proof that the heuristic is correct in many cases similar to the ktuples problem). $\endgroup$– T..Jul 5, 2010 at 22:55

3$\begingroup$ T, about 30 years ago I went to a philosophy of math talk, where the speaker was interested in what it says about mathematics, that our instincts are so good. He pointed to our uncanny ability to make conjectures that turn out to be true. I told him about the HensleyRichards work showing that the $\pi(m+n)$ conjecture and the prime $k$tuple conjecture couldn't both be true. His immediate reply was that the $\pi(m+n)$ conjecture must be correct! But he doesn't count as an answer to your question, as he was a philosopher of math, not a number theorist. $\endgroup$– Gerry MyersonJul 6, 2010 at 1:43

2$\begingroup$ Is it plausible that there are only finitely many counterexamples with $n\ge m\ge cn$ and for any constant $c>0$? $\endgroup$– Brendan McKayAug 5, 2013 at 16:50
5 Answers
As Robin Chapman points out, this conjecture is probably false. Nonetheless, similar results have been obtained, usually using sieve theory. Montgomery and Vaughan have proven $$\pi(x+y)\leq\pi(x)+\frac{2y}{\log y}.$$
Combined with a standard Chebyshev estimate, this gives $$\pi(x+y)\leq\pi(x)+16\pi(y),$$ say (for all $x,y\geq2$). Erdos conjectured that $$\pi(x+y)\leq\pi(x)+(1+o_{y\to\infty}(1))\frac{y}{\log y}$$ which, combined with the prime number theorem, would give $$\pi(x+y)\leq\pi(x)+(1+o_{y\to\infty}(1))\pi(y).$$
This may be as close as possible, and some (possibly including Selberg, see comments) believe even this to be false, and the constant 2 in the first result mentioned is the best possible.

2$\begingroup$ Where does Selberg state his belief that the last inequality is false? $\endgroup$ Aug 5, 2013 at 19:37

2$\begingroup$ Good question. Searching my memory, I think I got that impression, and indeed the content of this answer (including the Erdos conjecture), from 'Lectures on sieves' by HeathBrown (arxiv.org/abs/math/0209360). The relevant quote is "Erdos apparently believed that the constant may be taken as 1, while Selberg is reputed to have suggested that no constant below is admissible.", without references. $\endgroup$ Aug 5, 2013 at 20:14
The historical highlights for this conjecture are:
1923 : Hardy and Littlewood's classic paper, Partitio Numerorum III, elevates an obvious question to a conjecture. More precisely, H & L note that $\pi(x+y) \leq \pi(x) + \pi(y)$ (for large enough $y$) is "forcibly suggested" by the data for $x,y \leq 200$; prove some upper and lower bounds on the (lim sup) densest packing of primes in an interval of length $x$; calculate the densest packing for $x=35, 59$ and $97$; and finish with the remark that "beyond $x=97$ it would seem that [the densest packing] falls further below $\pi(x)$, at least within any range in which calculation is practicable". These speculative comments from the paper become known as a conjecture.
1973 : Ian Richards and his doctoral student Douglas Hensley explode the conjecture by showing that it contradicts the (much more plausible) prime ktuplets conjecture
2004 to today: GreenTao theorem and later developments lead to proofs of statements similar to the ktuplets conjecture, giving hope that "moral certainty that HL 1923 is false" can be replaced by "a proof that ktuplets conjecture is true (and HL1923 is therefore false)".
The $\pi(x+y)$ conjecture never had much evidence for it. Hensley and Richards set out to disprove it by a computer calculation. They ended up proving the existence of subsets $D$ of {1,2,...,$n$} denser than the first $n$ primes ($D > \pi(n)$) and with no congruence obstruction to $x + D$ being a set of primes for infinitely many $x$ (for no prime $p$ do the elements of $D$ fill all congruence classes modulo $p$). Sets free of congruence obstructions are called "admissible prime constellations" and the other more credible conjecture of Hardy and Littlewood is that for any admissible constellation, infinitely many copies exist in the primes. For $D$ this implies that for infinitely many $x$,
$\pi(x+D)  \pi(x) \geq D > \pi (D)$.
The optimal example (according to http://www.opertech.com/primes/ktuples.html ) is with $n = 3159, D = 447 > \pi(n) = 446$, so conjecturally,
$\pi(x+3159) > \pi(x) + \pi(3159)$ infinitely often, with the number of examples less than $N$ being of order $cN/(\log N)^{447}$
Hensley and Richards' technique for constructing dense constellations of primes is to take all the primes and their negatives in a symmetrical interval $[K,K]$, and delete enough of the first few primes to prevent congruence obstructions. They proved that this construction is, infinitely often, denser than $\pi(2K+1)$.

$\begingroup$ Does language like "elevate ... obvious" and "explode" indicate some hostility to Hardy or Littlewood? $\endgroup$ Jul 6, 2010 at 8:19

1$\begingroup$ I added more on Hardy and Littlewood's contribution. The question is correctly described as being obvious and lacking evidence, it was elevated (intentionally or not) to conjecture status as a result of Hardy and Littlewood's paper, and any plausibility of that conjecture was shattered by the Hensley and Richards results. "Exploded" is a better descriptor than "refuted" or "resolved", since they did not fully disprove the conjecture, but reduced its falsity to a wellaccepted family of hypotheses on the distribution of primes. $\endgroup$– T..Jul 6, 2010 at 20:00
As has already been pointed out, Montgomery and Vaughan have proven that $$\pi(x+y) \leq \pi(x)+ 2 \frac{y}{\log(y)} $$ from which, with some care, one can derive that: $\pi(x+y) \leq \pi(x)+ 2\pi(y)$ (this is worked out in the original paper of Montgomery and Vaughan).
The natural question is thus to further refine the constant $2$. There seems to be no unrestricted result on this problem, however Friedlander and Iwaniec (in their recent book on sieve theory) have shown that
$$\pi(x+y) \leq \pi(x) + (2\delta) \frac{y} {\log(y)} $$
holds for $x^{\theta} < y < x$ where $\delta:= \delta(\theta)$ is a function of $\theta$. In other words, one can improve the constant $2$ as long as $y$ isn't too small compared to $x$.
Recently Bourgain and Garaev have refined this result to give the quantitative relationship of $\delta \sim \theta^2$.
Some partial results, as well as the proof of the claim mentioned by Chapman are given in the book The Little Book of Bigger Primes by Ribenboim.
Zhang's work on bounded gaps between primes connects with this conjecture and there is a candidate counterexample. See http://sbseminar.wordpress.com/2013/07/02/thequestfornarrowadmissibletuples/