TY - JOUR

T1 - Hyperbolic prime number theorem

AU - Friedlander, John B.

AU - Iwaniec, Henryk

N1 - Funding Information:
The first author is supported in part by NSERC grant A5123. The second author is supported in part by NSF grant DMS-03-01168.

PY - 2009/3

Y1 - 2009/3

N2 - We count the number S(x) of quadruples \left( x-1 ,x-2 ,x-3 ,x-4 \right) \in \mathbbZ4 for which p = x2-1 + x2-2 + x2-3 + x2-4 \leqslant x is a prime number and 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)∈∈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)∈∈x/log x.

AB - We count the number S(x) of quadruples \left( x-1 ,x-2 ,x-3 ,x-4 \right) \in \mathbbZ4 for which p = x2-1 + x2-2 + x2-3 + x2-4 \leqslant x is a prime number and 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)∈∈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)∈∈x/log x.

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U2 - 10.1007/s11511-009-0033-z

DO - 10.1007/s11511-009-0033-z

M3 - Article

AN - SCOPUS:61349166437

SN - 0001-5962

VL - 202

SP - 1

EP - 19

JO - Acta Mathematica

JF - Acta Mathematica

IS - 1

ER -