Is there a progress on a solution of the inequality $\pi (m+n) \leq \pi (m) + \pi (n)$

Question

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

Answer 1

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.

Answer 2

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 k-tuplets conjecture

2004 to today: Green-Tao theorem and later developments lead to proofs of statements similar to the k-tuplets conjecture, giving hope that "moral certainty that H-L 1923 is false" can be replaced by "a proof that k-tuplets 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/k-tuples.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 $\pi(2K+1)$.

Answer 3

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 $\delta \sim \theta^2$.

Answer 4

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.

Answer 5

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/the-quest-for-narrow-admissible-tuples/