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The conjecture that $\pi(x+y) \leq \pi(x) + \pi(y)$, with $\pi$ the counting function for prime numbers, is customarily attributed to Hardy and Littlewood in their 1923 paper, third in the Partitio Numerorum series on additive number theory and the circle method. For example, Richard Guy's book Unsolved Problems in Number Theory cites the paper and calls the inequality a "well-known conjecture ... due to Hardy and Littlewood".

Partitio Numerorum III is one of the most widely read papers in number theory, detailing a method for writing down conjectural (but well-defined) asymptotic formulas for the density of solutions in additive number theory problems such as Goldbach, twin primes, prime $k$-tuplets, primes of the form $x^2 + 1$, etc. This formalism in the case of prime $k$-tuplets, with the notion of admissible prime constellations and an asymptotic formula for the number of tuplets, came to be known as the "Hardy-Littlewood [prime k-tuplets] conjecture" and indeed is one of several explicit conjectures in the paper.

However, the matter of whether $\pi(x+y) \leq \pi(x) + \pi(y)$, for all $x$ and $y$, does not actually appear in the Hardy-Littlewood paper. They discuss the inequality only for fixed finite $x$ and for $y \to \infty$, relating the packing density of primes in intervals of length $x$ to the $k$-tuplets conjecture. (The lim-sup density statements that H & L consider were ultimately shown inconsistent with the k-tuplets conjecture by Hensley and Richards in 1973).

The questions:

  1. Are there other works of Hardy or Littlewood where the inequality on $\pi(x+y)$ is discussed, or stated as a conjecture?

  2. Where does the inequality first appear in the literature as a conjecture?

  3. Is there any paper that suggests the inequality (for all finite $x$ and $y$, not the asymptotic statement considered by Hardy-Littlewood) is likely to be correct?

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Found it, section A9 in Guy (second edition), page 24. He does not give any reference to H-L in that section, although in A1, page 5, he mentions your same part number 3, then part number 6 in section D4, page 151. This could go back a very long way, essentially folklore with decades of authors misquoting the paper and then saying they do not believe the conjecture. Do Hensley and Richards correctly report on the H-L paper? Your language suggests that they do. –  Will Jagy Jul 7 '10 at 1:30
    
Richards' paper from 1973 does both. It shows that the k-tuples conjecture rules out $\pi(n)$ being the (lim sup) densest packing of primes in an interval of length $n$. However, it refers to both the lim sup statement, and the stronger, non-asymptotic $\pi(x+y)$ inequality, as "conjectures" due to Hardy and Littlewood, citing Partitio Numerorum III. There is evidence both for and against the idea that H + L offered the limsup statement as a conjecture in PN3, but the nonasymptotic inequality doesn't appear anywhere in that paper. –  T.. Jul 7 '10 at 21:05
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1 Answer

I took a look at Schinzel and Sierpinski, Sur certaines hypotheses concernant les nombres premiers, Acta Arith IV (1958) 185-208, reprinted in Volume 2 of Schinzel's Selecta, pages 1113-1133. In the Selecta, there is a commentary by Jerzy Kaczorowski, who mentions "the G H Hardy and J E Littlewood conjecture implicitly formulated in [33] that $\pi(x+y)\le\pi(x)+\pi(y)$ for $x,y\ge2$." [33] is Partitio Numerorum III. Schinzel and Sierpinski (page 1127 of the Selecta) define $\rho(x)=\limsup_{y\to\infty}[\pi(y+x)-\pi(y)]$, and point to that H-L paper, pp 52-68. They then write (page 1131), "$\bf C_{12.2}.$ L'hypothese de Hardy et Littlewood suivant laquelle $\rho(x)\le\pi(x)$ pour $x$ naturels $\gt1$ equivaut a l'inegalite $\pi(x+y)\le\pi(x)+\pi(y)$ pour $x\gt1,y\gt1$." It should be said that the proof that the first inequality implies the second relies on Hypothesis H, which essentially says that if there is no simple reason why a bunch of polynomials can't all be prime, then they are, infinitely often.

Schinzel and Sierpinski express no opinion as to any degree of belief in the conjecture under discussion.

I don't suppose this actually answers any of the questions, although Kaczorowski's use of the word "implicitly" may be significant.

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