Primes represented by \(x^3+ 2y^3\).

*(English)*Zbl 1007.11055This 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 \[ {\mathcal B}^{(K)} = \{ J :3 X^3<N(J)\leq 3X^3(1+\eta)\}. \]

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 \[ {\mathcal B}^{(K)} = \{ J :3 X^3<N(J)\leq 3X^3(1+\eta)\}. \]

Reviewer: Daniel Goldston (San Jose)

##### MSC:

11N32 | Primes represented by polynomials; other multiplicative structures of polynomial values |

11N36 | Applications of sieve methods |

Full Text:
DOI

##### References:

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