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I was thinking about the Second Hardy-Littlewood conjecture for quite sometime (some of my posts are related to this). In one of my earlier post I conjectured that the inequality ($\pi(x)$ denotes the prime counting function), $$\pi(x)+\pi(y)>2\pi\left(\dfrac{x+y}{2}\right)$$ is true. However, R. Israel showed that this inequality falls infinitely often. Basing upon the counter example that he gave I put forward the following two conjectures,

Conjecture 1 $\pi(x)+\pi(y)\le2\pi\left(\dfrac{x+y}{2}\right)$ for all sufficiently large $x$ and $y$.

It seems that the former conjecture is true if the quantity $\dfrac{x}{y}$ be sufficiently large (assume $x\ge y$).

So my question is that,

Is there any proof of Conjecture 1?

Assuming there is no such proof at present can anyone suggest me some relevant literature which shed some light on proving this conjecture?

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  • $\begingroup$ If you think about how such a conjecture could fail, you could imagine a dense constellation of primes between x and (x+y)/2 and very few primes between (x+y)/2 and y In fact, it is likely that both conjectures fail infinitely often, and conjecture 2 certainly fails for x=y. You might consider running a computer program to check your conjecture for a few million pairs (x,y), to see how it fails. Gerhard "Ask Me About Prime Gaps" Paseman, 2015.06.30 $\endgroup$
    – Gerhard Paseman
    Commented Jul 1, 2015 at 4:28
  • $\begingroup$ I think that I was too occupied regarding the truth of the conjecture. well, thanks for your suggestion, I will surely try that when I get some time. However, it would be great to have some literature regarding this conjecture. $\endgroup$
    – user57432
    Commented Jul 1, 2015 at 4:32
  • $\begingroup$ It might be great, but it is unlikely. There might be more refined versions of the conjecture in connection with Goldbach's binary conjecture. However, primes fluctuate wildly enough that I would only expect either relation to hold with x and y separated by a significant power of x. For a start, check out texts by Hans Riesel and by Paulo Ribenboim on primes. That and much experimentation may give you a feel for plausible conjectures on distribution of primes. Gerhard "Feels His Way Around Primes" Paseman, 2015.06.30 $\endgroup$
    – Gerhard Paseman
    Commented Jul 1, 2015 at 4:36

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Look for a large gap in the distribution of primes. For this conjecture, the gap between $n!+2$ and $n!+n$ will suffice. Set $y = n!+2$ (which is composite) and set $m$ (which will be $\frac{x+y}{2}$, so $x$ will be $2m-y$ eventually) to be the largest composite so that there are no primes between $m$ and $y$. Then there will be primes between $x$ and $m$ and none between $m$ and $y$.

So $\pi(x) + \pi(y) = \pi(x) + \pi\left(\frac{x+y}{2}\right) > 2 \pi\left(\frac{x+y}{2}\right)$. So your conjecture will fail for infinitely many pairs $x$ and $y$.

Gerhard "This Belongs On Another Forum" Paseman, 2015.06.30

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  • $\begingroup$ Note that if you allow y to be prime, you can extend the argument above to any gap. Put another way, for any prime (and also for any composite) y there is an x such that conjecture 1 fails for that x and y, and x will often differ from y by at most a constant times log y . $\endgroup$
    – The Masked Avenger
    Commented Jul 1, 2015 at 15:15

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