Second conjecture of Hardy-Littlewood

In Theory of the numbers, the second conjecture of Hardy-Littlewood relates to the number of prime numbers.

Either π ( X ) the number of prime numbers p such as p ≤ X , the conjecture postulates that

π ( X + there ) - π ( X ) ≤ π ( there )

for all X , there ≥ 2.

What means that the number of prime numbers between X + 1 and X + is there always lower or equal to the number of prime numbers between 1 and there . This is probably false in general and incompatible with the first conjecture of Hardy-Littlewood as showed it Richards in a nonobvious way in 1974. It is probable that the conjecture will be cancelled for very great values of X and there .

there would not be a problem with this formulation. If one poses X = there, that gives:

$\backslash \; pi\; (x+y)\; -\; \backslash \; pi\; (X)\; \backslash \; Leq\; \backslash \; pi\; (there)\; \backslash \; mbox\; \{either\}\; \backslash \; pi\; (x+x)\; -\; \backslash \; pi\; (X)\; \backslash \; Leq\; \backslash \; pi\; (X)\; \backslash \; mbox\; \{or\}\; \backslash \; pi\; (2x)\; \backslash \; Leq\; 2.\; \backslash \; pi\; (X)$

For example:

$\backslash \; pi\; (1000)\; \backslash \; Leq\; 2.\; \backslash \; pi\; (500)$

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