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Primes represented by \(x^3+ 2y^3\). (English) Zbl 1007.11055
This paper proves the remarkable result that there are infinitely many primes of the form \(x^3 +2 y^3\), which in particular proves a long-standing conjecture of Hardy and Littlewood that there are infinitely many primes which are the sum of three nonnegative cubes. Specifically, the author proves that the number of such primes with \(X< x,y\leq X(1+\eta)\) with \(\eta =(\log X)^{-c}\) for some positive constant \(c\), is \[ \sigma_0{\eta^2 X^2\over 3 \log X}\left(1+O((\log\log X)^{-1/6})\right) \] as \(X\to \infty\), where \[ \sigma_0 = \prod_{p}\left(1-{\nu_p-1\over p}\right) \] and \(\nu_p\) is the number of solutions of \(x^3\equiv 2 \pmod p\).
This result follows and was inspired by the landmark result of J. Friedlander and H. Iwaniec [Ann. Math. (2) 148, 945-1040 (1998; Zbl 0926.11068)] that proved the existence of infinitely many primes of the form \(x^2+y^4\). However the author develops his proof essentially from scratch rather then trying to modify the general machinery of Friedlander and Iwaniec. The proof itself is much too complicated to describe in any detail here, the paper itself being 83 pages, and involves intricate sieve results both for primes and for ideals in the field of \(\mathbb{Q}(\root 3 \of {2})\).
The author provides two sections of the paper for a broad outline of the proof, which helps to establish the main ideas of the proof. One considers the sets, for integers \(x\) and \(y\), \[ {\mathcal A} = \{ x^3+2 y^3 : X<x,y\leq X(1+\eta),\;(x,y)=1\}, \] and for \(J\) the integral ideals of \(K=Q(\root 3 \of {2})\) and \(N\) the norm from \(K\) to \(Q\), \[ {\mathcal B} = \{ N(J) :3 X^3<N(J)\leq 3X^3(1+\eta)\}. \] Denoting the number of primes in \({\mathcal A}\) by \(\pi({\mathcal A})\), and the number of primes in \({\mathcal B}\) by \(\pi({\mathcal B})\), by the prime ideal theorem one can easily evaluate \(\pi({\mathcal B})\) asymptotically, and the theorem reduces to proving that \[ \pi({\mathcal A}) = {\sigma \eta \over 3 X}\pi({\mathcal B}) +O \Biggl({\eta^2X^2\over \log X (\log \log X)^{1/6}}\Biggr). \] This procedure thus avoids the need to explicitly calculate the main term in \(\pi({\mathcal A})\), a major simplification, and the proof now reduces to comparing \(\pi({\mathcal A})\) with \(\pi({\mathcal B})\). To make this comparison, identical sieve decompositions are performed on both sequences by a Buchstab identity argument. One needs Type I estimates for the \` level of distribution’ of \({\mathcal A}\) and \({\mathcal B}\), Type II estimates, and Vaughan’s or similar identities. It should be mentioned that much of the analysis of the paper is performed in the field \(K\) on the sets \[ {\mathcal A}^{(K)} = \{ (x+ y \root 3 \of {2}): X<x,y\leq X(1+\eta), (x,y)=1\}, \] and \[ {\mathcal B}^{(K)} = \{ J :3 X^3<N(J)\leq 3X^3(1+\eta)\}. \]

11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11N36 Applications of sieve methods
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[1] Daniel, S., On the divisor-sum problem for binary forms.J. Reine Angew. Math., 507 (1999), 107–129. · Zbl 0913.11041
[2] Friedlander, J. &Iwaniec, H., The polynomialX 2+Y 4 captures its primes.Ann. of Math., 148 (1998), 945–1040. · Zbl 0926.11068
[3] —-, Asymptotic sieve for primes.Ann. of Math., 148 (1998), 1041–1065. · Zbl 0926.11067
[4] Goldston, D. A., On Bombieri and Davenport’s theorem concerning small gaps between primes.Mathematika, 39 (1992), 10–17. · Zbl 0758.11037
[5] Greaves, G. R. H., Large prime factors of binary forms.J. Number Theory, 3 (1971), 35–59. · Zbl 0214.30301
[6] Halberstam, H. &Richert, H.-E.,Sieve Methods. London Math. Soc. Monographs, 4. Academic Press, London-New York, 1974. · Zbl 0298.10026
[7] Hardy, G. H. &Littlewood, J. E., Some problems of ’Partitio Numerorum’; III: On the expression of a number as a sum of primes.Acta Math., 44 (1923), 1–70. · JFM 48.0143.04
[8] Heath-Brown, D. R., Diophantine approximation with square-free numbers.Math. Z., 187 (1984), 335–344. · Zbl 0539.10026
[9] –, The ternary Goldbach problem.Rev. Mat. Iberoamericana, 1 (1985), 45–59. · Zbl 0599.10041
[10] –, The number of primes in a short interval,J. Reine Angew. Math., 389 (1988), 22–63. · Zbl 0646.10032
[11] –, The solubility of diagonal cubic Diophantine equations.Proc. London Math. Soc. (3), 79 (1999), 241–259. · Zbl 1029.11010
[12] Hooley, C., On a problem of Hardy and Littlewood.Acta Arith., 79 (1997), 289–311. · Zbl 0877.11052
[13] Huxley, M. N., The large sieve inequality for algebraic number fields.Mathematika, 15 (1968), 178–187. · Zbl 0174.08201
[14] Iwaniec, H., Frimes represented by quadratic polynomials in two variables.Acta Arith., 24 (1974), 435–459. · Zbl 0271.10043
[15] Mitsui, T., Generalized prime number, theorem.Japan. J. Math., 26 (1956), 1–42. · Zbl 0126.27503
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