The polynomial \(X^2+Y^4\) captures its primes.

*(English)*Zbl 0926.11068The question of showing that a specified polynomial over the integers represents infinitely many primes is a very old one, but until now the only successes had been in instances where \(N(x)\), the number of values of the polynomial not exceeding a large number \(x\), is very nearly as large as \(x\). Besides the very classical cases of linear polynomials \(an+b\) and binary quadratic forms \(am^2+bmn+cn^2\) one might mention the general quadratic polynomial in two variables discussed by H. Iwaniec [Acta Arith. 24, 435-459 (1974; Zbl 0271.10043)] and the more recent result of E. Fouvry and H. Iwaniec [Acta Arith. 79, 249-287 (1997; Zbl 0881.11070)] concerning primes of the form \(n^2+q^2\) where \(q\) is also prime.

The authors’ result is an asymptotic formula for the number of primes not exceeding \(x\) gives by values of the polynomial \(f(a,b)=a^2+b^4\), for which the number \(N(x)\) is now of the much smaller order \(x^{3/4}\). The result actually given is the formula \[ \mathop{\sum\sum}_{a^2+b^4 \leq x}\Lambda\bigl( a^2+b^4 \bigr) = {4 \over \pi}\kappa x^{3/4}\biggl\{ 1+O\biggl( {\log\log x \over \log x} \biggr) \biggr\} . \] Here \(\Lambda\) denotes the von Mangoldt function. The number \(4/\pi\) is the value of the appropriate infinite product over all primes and \(\kappa\) is a constant arising from the area of the region \(a^2+b^4 \leq x\).

The authors indicate that their approach should apply to other polynomials \(f\bigl( a,b^2 \bigr)\) in which \(f\) is a quadratic form. Thus the entry \(b^4\) in their theorem may be thought of \((b^2)^2\) (a biquadrate, as opposed to a fourth power), and their approach does not seem to apply to the polynomial \(a^2+b^3\), for example, which might otherwise have been thought of as providing an easier problem.

The key weapon in the authors’ attack is their new asymptotic sieve for primes (see the preceding review Zbl 0926.11067). Thus their principal task is now to verify the necessary estimates for sums of the type \(\sum_l\bigl| \sum_m \beta_m a_{lm} \bigr| \), where \(a_n\) is the number of representations of \(n\) as \(a^2+b^4\). Their treatment of this question, which occupies almost all of their long paper, is a veritable tour de force which would be impossible for the reviewer even to summarise adequately here. There is a nice summary in the first seven pages of the paper. Included in the treatment one will find a treatment by harmonic analysis of the lattice point problem for the quartic region \(c^4-2\gamma c^2d^2+d^4\leq x\), where \(0<\gamma<1\), and the values are constrained to lie in an arithmetic progression to a (large) modulus \(\Delta\). An extension to grössencharacters of Siegel’s theorem is obtained by adapting a method of D. Goldfeld [Proc. Natl. Acad. Sci. USA 71, 1055 (1974; Zbl 0287.12019)]. Some of the other ideas used are drawn from earlier papers of the authors. The reader is not required to be familiar with automorphic forms, but it is suggested that it was essential that the authors were, as otherwise the paper could not have been written in its present form.

One of the by-products of the authors’ work is given by them as a separate theorem, as follows. They show \[ \mathop{\sum\sum}_{r^2+s^2=p\leq x}\biggl( {s \over r} \biggr) \ll x^{76/77} , \] in which the summand is a Jacobi symbol in which \(r\) is odd. Thus there is a degree of equidistribution among the “spins” of primes \(p\) given by the Jacobi symbols under consideration.

The authors’ result is an asymptotic formula for the number of primes not exceeding \(x\) gives by values of the polynomial \(f(a,b)=a^2+b^4\), for which the number \(N(x)\) is now of the much smaller order \(x^{3/4}\). The result actually given is the formula \[ \mathop{\sum\sum}_{a^2+b^4 \leq x}\Lambda\bigl( a^2+b^4 \bigr) = {4 \over \pi}\kappa x^{3/4}\biggl\{ 1+O\biggl( {\log\log x \over \log x} \biggr) \biggr\} . \] Here \(\Lambda\) denotes the von Mangoldt function. The number \(4/\pi\) is the value of the appropriate infinite product over all primes and \(\kappa\) is a constant arising from the area of the region \(a^2+b^4 \leq x\).

The authors indicate that their approach should apply to other polynomials \(f\bigl( a,b^2 \bigr)\) in which \(f\) is a quadratic form. Thus the entry \(b^4\) in their theorem may be thought of \((b^2)^2\) (a biquadrate, as opposed to a fourth power), and their approach does not seem to apply to the polynomial \(a^2+b^3\), for example, which might otherwise have been thought of as providing an easier problem.

The key weapon in the authors’ attack is their new asymptotic sieve for primes (see the preceding review Zbl 0926.11067). Thus their principal task is now to verify the necessary estimates for sums of the type \(\sum_l\bigl| \sum_m \beta_m a_{lm} \bigr| \), where \(a_n\) is the number of representations of \(n\) as \(a^2+b^4\). Their treatment of this question, which occupies almost all of their long paper, is a veritable tour de force which would be impossible for the reviewer even to summarise adequately here. There is a nice summary in the first seven pages of the paper. Included in the treatment one will find a treatment by harmonic analysis of the lattice point problem for the quartic region \(c^4-2\gamma c^2d^2+d^4\leq x\), where \(0<\gamma<1\), and the values are constrained to lie in an arithmetic progression to a (large) modulus \(\Delta\). An extension to grössencharacters of Siegel’s theorem is obtained by adapting a method of D. Goldfeld [Proc. Natl. Acad. Sci. USA 71, 1055 (1974; Zbl 0287.12019)]. Some of the other ideas used are drawn from earlier papers of the authors. The reader is not required to be familiar with automorphic forms, but it is suggested that it was essential that the authors were, as otherwise the paper could not have been written in its present form.

One of the by-products of the authors’ work is given by them as a separate theorem, as follows. They show \[ \mathop{\sum\sum}_{r^2+s^2=p\leq x}\biggl( {s \over r} \biggr) \ll x^{76/77} , \] in which the summand is a Jacobi symbol in which \(r\) is odd. Thus there is a degree of equidistribution among the “spins” of primes \(p\) given by the Jacobi symbols under consideration.

Reviewer: G.Greaves (Cardiff)

##### MSC:

11N35 | Sieves |

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

11N36 | Applications of sieve methods |

11P21 | Lattice points in specified regions |