Let $\mathcal{H}_k = (h_1,h_2,\cdots,h_k)$ be an admissible k-tuple.

The k-tuple conjecture predicts that the number of primes $(b+h_1,b+h_2,\cdots,b+h_k)\in \mathbb{P}^k$ with $b+h_k \leq x$ is:

$$\pi_{\mathcal{H}_k}(x) \sim \left(\prod_{\text{p prime}}\frac{1-\frac{w(\mathcal{H}_k, p)}{p}}{(1-\frac1p)^{k}} \right)\dfrac{x}{\log(x)^k}$$

Where $w(\mathcal{H}_k, p)$ is the number of distinct residues $\pmod p$ in $\mathcal{H}_k$.

We can deduce this formula from Cramer's random model, but Hardy and Littlewood get the same result using the circle method.

How did Hardy and Littlewood formulate this conjecture using the circle method?

  • 3
    $\begingroup$ The question has been asked and answered on Math SE and Math Overflow. Here is a link to Some Problems of Partitio Numerorum III, On the Expression of a Number as a Sum of Primes, Acta Math. 44, 1-70, 1922, where it is formulated. $\endgroup$
    – Conifold
    Aug 11, 2019 at 13:07
  • $\begingroup$ @Conifold, this article about Goldbach conjecture and some other types of primes sets (not including k-tuple conjecture) $\endgroup$
    Aug 11, 2019 at 13:50
  • 1
    $\begingroup$ This is the reference that Green and Tao, p.6 give. Perhaps, it is in other parts of Some Problems of Partitio Numerorum? They also mention Dickson's 1904 paper. $\endgroup$
    – Conifold
    Aug 11, 2019 at 22:39
  • $\begingroup$ @Conifold, this article didn't use the circle method, and use a complex notations and methods.. $\endgroup$
    Aug 12, 2019 at 11:29
  • 2
    $\begingroup$ The $k$-tuple conjecture is in the paper mentioned by Conifold. You need to look through the whole paper, not just the first few pages. :) See Theorem X1 on page 61. It depends on Hypothesis X (unproved) on p. 56. $\endgroup$
    – KCd
    Aug 14, 2019 at 23:32


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