We establish, via a heuristic Fourier inversion calculation, that the Hardy-Littlewood twin prime conjecture is equivalent to an asymptotic formula for the two-point correlation function of Riemann zeros at a height E on the critical line. Previously it was known that the Hardy-Littlewood conjecture implies the pair correlation formula, and we show that the reverse implication also holds. An averaged form of the Hardy-Littlewood conjecture is obtained by inverting the E → ∞limit of the two-point correlation function and the precise form of the conjecture is found by including asymptotically lower order terms in the two-point correlation function formula.
|Number of pages||12|
|Journal||Journal of Physics A: Mathematical and Theoretical|
|Early online date||13 Aug 2019|
|Publication status||Published - 13 Aug 2019|
- number theory
- quantum chaos
- random matrix theory