ON PRIMES IN QUADRATIC PROGRESSIONS & THE SATO-TATE CONJECTURE

Transcription

1 ON PRIMES IN QUADRATIC PROGRESSIONS & THE SATO-TATE CONJECTURE LIANGYI ZHAO INSTITUTIONEN FÖR MATEMATIK KUNGLIGA TEKNISKA HÖGSKOLAN (DEPT. OF MATH., ROYAL INST. OF OF TECH.) STOCKHOLM SWEDEN

2 Dirichlet showed that f(n) = an + b is prime for infinitely many n N if and only if gcd(a, b) = 1. If f(n) is a polynomial of degree 2 or higher, no similar statement is known. Set Λ(n) = { log p if n = p k ; 0 otherwise. G. H. Hardy and J. E. Littlewood conjectured that (1) where n x Λ(n 2 + k) S(k)x, S(k) = p>2 1 ( ) k p p 1 with ( k p ) being the Legendre symbol. 1

3 Theorem 1 (Baier, Z. 2006). Given A, B > 0, we have, for x 2 (log x) A y x 2, k y µ(k) 0 n x Λ(n 2 + k) S(k)x 2 = O ( yx 2 (log x) B ). From the theorem, we have the following. Corollary 1 (Baier, Z. 2006). Given A, B, C > 0 and S(k) as defined in the theorem, we have, for x 2 (log x) A y x 2, that n x Λ(n 2 + k) = S(k)x + O ( x (log x) B holds for all square-free k not exceeding y with at most O ( y(log x) C) exceptions. ) The theorem and corollary verify that the Hardy- Littlewood conjecture (1) is the correct conjecture.

4 Proof. Circle method, we split the integration interval [0,1] into major arcs and minor arcs. n x 1 = Λ(n 2 + k) 0 Λ(m)e(αm) e ( α(n 2 + k) ) dα, m z n x where e(z) = exp(2πiz), and z = x 2 + y. In brief, the major arcs are the part of the interval [0,1] that are close to a rational number of with small denominator and the minor arcs are the rest. The contribution of the major arcss will give raise to the main term. The rest will be error terms.

5 Theorem 1 has recently been improved. The theorem holds true for much smaller values of y, i.e. x 1+ε y x 2 ε. Moreover, the unfortunately restriction that k must be square-free is also removed. In fact, the improvment follows from the follow theorem for short segments of quadratic progressions on average. Theorem 2 (Baier, Z. 2007). Suppose that z 3, z 2/3+ε z 1 ε and z 1/2+ε K z 1 ε. Then, given B > 0, we have (2) where and 2z z 1 k K S Λ (t,, k) S(k)S 1 (t,, k) 2 dt S Λ (t,, k) = 2 K (log z) B, S 1 (t,, k) = t<n 2 +k t+ t<n 2 +k t+ Λ(n 2 + k), 1.

6 Here our approach is a variant of the dispersion method of J. V. Linnik. Expanding the modulus square in (2), we will get three terms U(t), V (t) and W(t), the mean-values of which we develop asymptotic formulas. The main terms will cancel out at the end, giving the desired result. Under the generalized Riemann hypothesis(grh) for Dirichlet L-function, may be taken from a wider range z 1/2+ε z 1 ε. It would be highly desirable to have the results in which K = o( z) in Theorem 2 since in that situation the quadratic progressions under consideration would be completely disjoint. We should highly value any suggestion toward this end.

7 Theorem 3 (Baier, Z. 2006). For any ε > 0, there exist infinitely many primes of the form p = am 2 +1 with a p 5/9+ε. This is an approximation to the n problem as the latter is equivalent to the statement of the theorem with a = 1. Hence in this sense, the smaller the majorant we can have for a, the closer we are to the n problem. A more quantitatively, for any ε > 0, #{p x : p = am 2 + 1, a p 5/9+ε } x 7/9 ε. The result is to pick out primes that are congruent to 1 modulo a large square, which was given by Bombieri-Vinogradov Theorem for Square moduli which follows from large sieve for square moduli.

8 The method would also give similar statements for a variety of polynomials such as am 2 1. But the n 2 1 problems needs no approximation. We also note that the set of integers of the form m is very sparse. Note that #{n x : n = m 2 + 1} = O( x). It is generally very difficult to produce primes out of such sparse sets. Friedlander and Iwaniec proved the celebrated result of the infinitude of primes of the form m 2 +n 4 (with an asymptotic formula). #{k x : k = m 2 + n 4 } = O(x 3/4 ). Heath-Brown proved a similar statement for m 3 + 2n 3. #{k x : k = m 3 + 2n 3 } = O(x 2/3 ). The set of natural numbers n = am with a n 5/9+ε is also very sparse. #{n x : n = am 2 + 1, a n 5/9+ε } = O(x 7/9+ε ).

9 Theorem 4 (Baier, Z. 2006). For any ε > 0 and fixed A > 0, we have q x 2/9 ε q where max a gcd(a,q)=1 ψ(x; q, a) = ψ(x; q2, a) x ϕ(q 2 ) Λ(n). n x n a mod q x (log x) A, q does not run over the correct range. This would be a complete analogue of the classical Bombieri-Vinogradov theorem if the range for q is 0 < q x 1/4 ε. The theorem improve a result of H. Mikawa, T. Peneva which is an improvement of a result of P. Elliott. The theorem follows from large sieve for square moduli.

10 Various versions of large sieve for square moduli were obtained both jointly and indepdently by Baier and Z., via very intricate techniques. Detecting Farey fractions with square denomenators in short intervals via combinatorial arguments. Exploring cancellation of main term coming from stationary phase with weights. Exploring cancellation due to change of arguement of quadratic Gauss sums. We don t have the complete analogue of classical Bombieri-Vinogradov theorem for sqaure moduli because we did not obtain the complete analogue of large sieve for square moduli.

11 Conjecturally(Z. 2004), we expect Farey fractions with square denomenators to distribute in a nice way. The conjecture was tested numerically. If such fractions are distributed nicely, then large sieve for square moduli is nice. If we have a nice large sieve for square moduli, then Theorem 4 has the correct range. Theorem 4 with the correct range gives infinitude of primes Note p = am with a p 1/2+ε. #{n x : n = am 2 + 1, a n 1/2+ε } = O(x 3/4+ε ). The GRH gives the same result as above. The Elliott-Halberstam conjecture for square moduli would give infinitude of primes p = am with a p ε. The the number of n = am with n x and a n ε is O(x 1/2+ε ).

12 Let E be an elliptic curve over Q. For any prime number p of good reduction, the number of points on the reduced curve modulo p equals #E(F p ) = p + 1 λ E (p). By Hasse s theorem, there exists a unique angle 0 θ π such that λ E (p) = p ( e iθ E(p) + e iθ E (p) ) = 2 pcos θ E (p). If E has complex multiplication(cm), Deuring (1941) showed that half of the primes satisfy λ E (p) = 0 ( supersingular primes ), and for the remaining half of the primes the angles θ E (p) are distributed uniformly in the interval [0, π]. If E admits no CM, Sato and Tate (1965) formulated the following conjecture. For any 0 θ 1 θ 2 π, and x 1, let π θ 1,θ 2 E (x) := #{p x : θ 1 θ E (p) θ 2 }. Then, with π(x) = #{p x : p P}, lim x π θ 1,θ 2 E π(x) (x) θ 2 = 2 sin 2 θ dθ. π θ 1

13 Modified Sato-Tate Conjecture: Suppose E is an elliptic curve over Q without complex multiplication. For 1 α β 1, x 1, let Then Θ E (α, β; x) := p x α λ E (p)/(2 p) β log p. lim x Θ E (α, β; x) x = F(α, β) := 2 π β α 1 t 2 dt.

14 Relation to symmetric power L-functions Langlands and Serre have related the Sato-Tate conjecture to symmetric power L-functions associated to the curve E. Sato-Tate conjecture holds if each symmetric power L-function associated to E L m (E; s) := p (E) m j=0 ( 1 e (m 2j)iθ E (p) p s ) 1 can be extended meromorphically to the complex plane, satisfy the expected functional equation and are holomorphic and non-zero for Rs 1. In a recent preprint R. E. Taylor (2006) proved the afore-mentioned conditions hold, thus proving the Sato-Tate conjecture.

15 In the sequel, we set γ := β α. Theorem 5 (Baier, Z. 2006). Let x 1, 0 < α β 1. Assume that x ε 5/12 γ/β x ε and F(α, β) x 1/2+ε. Then, if A, B > x 1/2+ε and AB > x 1+ε /F(α, β), we have, for every c > 0, 1 4AB a A b B = xf(α, β) Θ E(a,b) (α, β; x) ( ( )) O log c, x where the implied O-constant depends only on c. We also have the following almost-all result. Theorem 6 (Baier, Z. 2006). Let x 1, 0 < α β 1. Assume that x ε 5/12 γ/β x ε, F(α, β) x 1/2+ε, A, B > x 1+ε, x 2+ε /F(α, β) 2 < AB, and AB < exp(exp(x 1 ε )). Then, for any c, d > 0, we have ( ( )) 1 Θ E(a,b) (α, β; x) = xf(α, β) 1 + O log c x for all a A, b B with at most O ( AB/(log x) d) exceptions. Here the implied O-constant depends only on c and d.

16 Remarks on the Sato-Tate results Taylor s work does not imply any result on average due to the lack of uniformity with respect to the coefficients a and b and also the angles α and β. Using results due to Duering, the expression of our interest follow an asymptotic formula whose main term involves the Kronecker class number. Applying the Dirichlet class number formula, the said main term is studied by considering the resulting Dirichlet L-functions. We need a Barban-Davenport-Halberstam type theorem for short intervals for the main term. The error terms are transformed into character sums via considerations for isomorphism classes of elliptic curves over F p. Characterizing the isomorphism classes will give raise to congruence relations on the coefficients of the elliptic curves which are detected using character sums.

17 The result follows by careful studies of the resulting characters sums, using fourth moment estimate of character sums of Frieldander-Iwaniec and Polya- Vinogradov estimate for character sums. These results improve some recent results of K. James and G. Yu(2006). These results have recently been improved by W. D. Banks and I. E. Shparlinski (2006). The connection between primes in quadratic progressions and Sato-Tate conjecture lies in a much more general conjecture attributed to S. Lang and H. Trotter. Lang-Trotter conjecture implies Sato-Tate conjecture and gives the Hardy-Littlewood asymptotics (1) for a large family of quadratic progressions.