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Prime values of \(a^2 + p^4\). (English) Zbl 1405.11124

A difficult problem in number theory is to establish the infinitude of primes represented by polynomials with integer coefficients (provided there are no natural reasons which restrict these primes to a finite set). For polynomials in one variable, this problem is solved only for linear polynomials by celebrated work of Dirichlet. More is known for polynomials in two variables. Famously, Fermat considered the quadratic form \(F(x,y)=x^2+y^2\) and claimed that it represents precisely the primes congruent to 1 modulo 4 and the prime 2, which was proved by Euler who also considered quadratic forms \(F(x,y)=x^2+ny^2\) for \(n\)’s greater than 1 (for detailed information, see the book [D. A. Cox, Primes of the form \(x^2+ny^2\). Fermat, class field theory, and complex multiplication. 2nd ed. Hoboken, NJ: John Wiley & Sons (2013; Zbl 1275.11002)]. The set \(S\) of positive integers represented by a binary quadratic form as above satisfies \[ \sharp\{n\in S : n\leq x\} \gg x^{1-\delta} \] for every \(\delta>0\). The problem of detecting primes represented by a given polynomial becomes much harder if there exists \(\lambda<1\) such that the set \(S\) of positive integers represented by this polynomial satisfies \[ \sharp\{n\in S \;:\;n\leq x\} \ll x^{\lambda}. \] For example, if \(F(x,y)=x^2+y^4\), then the above bound holds for \(\lambda=3/4\), and if \(F(x,y)=x^3+2y^3\), then it holds for \(\lambda=2/3\). Asymptotic formulas for the numbers of primes represented by these particular polynomials were established in groundbreaking works by J. Friedlander and H. Iwaniec [Ann. Math. (2) 148, No. 3, 945–1040 (1998; Zbl 0926.11068)] and, subsequently, by the first-named author of the paper under review [Acta Math. 186, No. 1, 1–84 (2001; Zbl 1007.11055)]. General cubic forms in two variables were treated by the first-named author and B. Z. Moroz [Proc. Lond. Math. Soc. (3) 84, No. 2, 257–288 (2002; Zbl 1030.11046)].
Now the authors look at the polynomial \(F(x,y)=x^2+y^4\) investigated by Friedlander and Iwaniec [loc. cit.] but restrict the variable \(y\) to primes, which presents an additional difficulty. They establish the following formula. \[ \sharp\{a^2+p^4\leq x\;:\;a>0 \text{ and } a^2+p^4 \text{ is prime}\}= \nu \frac{4Jx^{3/4}}{\log^2 x} \left(1+O_{\varepsilon}\left(\frac{1}{(\log x)^{1-\varepsilon}}\right)\right), \] where \[ J=\int\limits_{0}^{1} \sqrt{1-t^4} dt \] and \[ \nu=\prod\limits_{p}\left(1-\frac{\chi_4(p)}{p-1}\right), \] for \(\chi_4\) the non-principle character modulo 4. As in the previous works mentioned above, the main tasks are to establish a level of distribution for the set of integers represented by the polynomial in question, and to estimate certain special bilinear sums which allows to overcome the parity barrier (this constitutes the most significant part of the method). The work of Friedlander-Iwaniec uses the regularity of squares of integers, which allows them to extract certain main terms in their bilinear sums and to show that the corresponding errors are small on average. To this end, they use a delicate harmonic analysis argument. This part of their method does not carry over to the modified problem of representing primes in the form \(a^2+p^4\), where \(p\) is a prime itself. The reason is that prime squares are not regular in the above sense. The authors replace these arguments by a method which applies to more general situations and is similar in spirit to a general form of the Barban-Davenport-Halberstam theorem on primes in arithmetic progressions. The significant point here is that this equidistribution result holds when the modulus goes up to nearly the square of the length of the sum in question, in contrast to the classical Barban-Davenport-Halberstam theorem. This result is of independent interest.

MSC:

11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11N37 Asymptotic results on arithmetic functions
11N36 Applications of sieve methods
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References:

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[3] Friedlander, J., Iwaniec, H.: The polynomial \[X^2+Y^4\] X2+Y4 captures its primes. Ann. Math. (2) 148(3), 945-1040 (1998) · Zbl 0926.11068 · doi:10.2307/121034
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[8] Heath-Brown, D.R., Moroz, B.Z.: On the representation of primes by cubic polynomials in two variables. Proc. Lond. Math. Soc. (3) 88(2), 289-312 (2004) · Zbl 1099.11050 · doi:10.1112/S0024611503014497
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