Hyperbolic prime number theorem. (English) Zbl 1278.11089

Friedlander, John B.; Iwaniec, Henryk

Acta Math. 202, No. 1, 1-19 (2009).

Summary: We count the number \(S(x)\) of quadruples \((x_1,x_2,x_3,x_4)\in\mathbb Z^4\) for which \[ p = x^2_1 + x^2_2 + x^2_3 + x^2_4 \leq x \] is a prime number and the \(x_i\) satisfying the determinant condition: \(x_1 x_4 - x_2 x_3 = 1\). By means of the sieve, one shows easily the upper bound \(S(x) \ll x / \log x\). Under a hypothesis about prime numbers, which is stronger than the Bombieri-Vinogradov theorem but is weaker than the Elliott-Halberstam conjecture, we prove that this order is correct, that is \(S(x) \gg x / \log x\).

MSC:

11N05	Distribution of primes
11N36	Applications of sieve methods

References:

[1]	Bombieri, E., On twin almost primes. Acta Arith., 28 (1975/76), 177–193. · Zbl 0319.10051
[2]	Bombieri, E., Friedlander, J. B. & Iwaniec, H., Primes in arithmetic progressions to large moduli. Acta Math., 156 (1986), 203–251. · Zbl 0588.10042
[3]	Bourgain, J., Gamburd, A. & Sarnak, P., Sieving and expanders. C. R. Math. Acad. Sci. Paris, 343 (2006), 155–159. · Zbl 1217.11081
[4]	Chamizo, F. & Iwaniec, H., On the sphere problem. Rev. Mat. Iberoamericana, 11 (1995), 417–429. · Zbl 0837.11054
[5]	Fouvry, É., Autour du théorème de Bombieri–Vinogradov. Acta Math., 152 (1984), 219–244. · Zbl 0552.10024
[6]	Heilbronn, H., On the averages of some arithmetical functions of two variables. Mathematika, 5 (1958), 1–7. · Zbl 0125.02604
[7]	Iwaniec, H., The half dimensional sieve. Acta Arith., 29 (1976), 69–95. · Zbl 0327.10046
[8]	– Spectral Methods of Automorphic Forms. Graduate Studies in Mathematics, 53. Amer. Math. Soc., Providence, RI, 2002. · Zbl 1006.11024
[9]	Iwaniec, H. & Kowalski, E., Analytic Number Theory. American Mathematical Society Colloquium Publications, 53. Amer. Math. Soc., Providence, RI, 2004. · Zbl 1059.11001
[10]	Smith, R. A., On (n)r(n + a). Proc. Nat. Inst. Sci. India Sect. A, 34 (1968), 132–137. · Zbl 0191.05002
[11]	– The circle problem in an arithmetic progression. Canad. Math. Bull., 11 (1968), 175–184. · Zbl 0164.05002