Gaussian primes in almost all narrow sectors
aa r X i v : . [ m a t h . N T ] M a r GAUSSIAN PRIMES IN ALMOST ALL NARROW SECTORS
BINGRONG HUANG, JIANYA LIU, AND ZE´EV RUDNICK
Abstract.
We study Gaussian primes lying in narrow sectors, and show that almost allsuch sectors contain the expected number of primes, if the sectors are not too narrow. Introduction
Let p be a prime ideal in the ring of Gaussian integers Z [ i ]. Assuming the norm N( p ) = p = 1 mod 4 is a split prime, we can write p = a + b with α = a + ib a generator of p ,unique up to multiplication by one of the units ± , ± i . This gives us an angle θ p , so that α = a + ib = √ pe iθ p , which is unique if we fix it to lie in [0 , π/ θ p are equidistributed in [0 , π/
2) as p varies over primeideals of Z [ i ]. This means that if we fix a subinterval I ⊆ [0 , π/ X → ∞ , { p ⊂ Z [ i ] : X < N( p ) ≤ X, θ p ∈ I } ∼ | I | π/ X log X . (1.1)Recall that by the Prime Ideal Theorem, { p ⊂ Z [ i ] : X < N( p ) ≤ X } ∼ X/ log X .Kubilius [8] and others studied the existence of prime angles in narrow sectors. Ricci [10]proved that (1.1) remains valid for any interval I ⊂ [0 , π/
2) of length | I | > X − / ε . Bya sieve method, Harman and Lewis [5] proved the existence of prime angles in somewhatnarrower sectors without an asymptotic expression. Assuming the generalized RiemannHypothesis (GRH), it is known that (1.1) is valid for any interval I ⊂ [0 , π/
2) of length | I | > X − / ε . It is important to point out that one cannot do better because of theexistence of “forbidden regions”, for instance the interval (0 , / √ X ) does not contain anyprime angle of norm less that X (in fact angles for any such integer ideals). This is a strikingdifference from the well studied problem of (rational) primes in short intervals, where oneexpects that any interval [ x, x + y ] will contain ∼ y/ log x primes as soon as y ≫ x ε , for any ε >
0, while similarly to our problem, the Riemann Hypothesis only gives the existence inintervals of length y ≫ x / o (1) .Instead of asking about all short sectors, one can ask for the existence of prime anglesin “typical” sectors, that is in almost all short sectors. Assuming GRH, Parzanchevski andSarnak [9], and Rudnick and Waxman [11] showed that almost all sectors contain a primeangle, in fact that the asymptotic formula (1.1) (at least in a smooth form) remains validfor almost all I if | I | > X − ε , which is the most we can expect (up to log factors) sincethe number of prime ideals with norm about X is roughly X/ log X , hence almost all sectors Date : March 12, 2019.
Key words and phrases.
Angle, almost all, Gaussian prime, narrow sector, variance.The research of B.H. and Z.R. was supported by the European Research Council under the EuropeanUnion’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n o shorter than 1 /X will contain no prime angles θ p with N( p ) ≈ X . Rudnick and Waxman [11]gave a precise conjecture about the asymptotic behavior of the number variance, supportedby a theorem for a function field analogue of the problem.In this paper, we prove an unconditional result on Gaussian primes in almost all narrowsectors. We will say that a property holds for almost all narrow sectors I = ( β, β + γ ] (where γ ≈ X − ρ , 0 < ρ <
1, and
X < N( p ) ≤ X with X → ∞ ) if it holds for all β ∈ [0 , π ] excepton an exceptional set of measure o (1). Theorem 1.
Let ≤ ρ < / . Then (1.1) holds for almost all short sectors I ⊂ [0 , π/ oflength | I | > X − ρ . Our result is a consequence of an upper bound of the number variance estimated usingzero-density theorems.We recall that under RH, Selberg [13] showed (in 1943) that almost all intervals ( x, x + φ ( x ) log x ] contain primes for any function φ ( x ) tending to infinity. See [12], [7], [3, Chap.9]) for unconditional results.The problem of Gaussian primes in small balls is similar in flavour to that of primes inshort intervals, and was studied by Coleman [1] who established individual results, both onGRH and unconditionally (see also [4]), and a result about almost all balls.We briefly discuss the analogous problems for real quadratic fields in Section 4.2. Preliminaries
Hecke characters and their L-functions.
For a non-zero ideal a = ( α ) ⊆ Z [ i ], withgenerator α , Hecke defined characters Ξ k ( α ) = ( α/ ¯ α ) k , k ∈ Z , which give well definedfunctions on the ideals of Z [ i ]. In terms of the angles associated to ideals, we have e i kθ a =Ξ k ( a ).To each such character Hecke [6] associated its L-function L ( s, Ξ k ) := X = a ⊆ Z [ i ] Ξ k ( a )N( a ) s = Y p prime (1 − Ξ k ( p ) N( p ) − s ) − , Re( s ) > . Note that L ( s, Ξ k ) = L ( s, Ξ − k ). Hecke showed that if k = 0, these functions have an analyticcontinuation to the entire complex plane, and satisfy a functional equation: ξ ( s, k ) := π − ( s +2 | k | ) Γ( s + 2 | k | ) L ( s, Ξ k ) = ξ (1 − s, k ) . (2.1)We denote L ∞ ( s, k ) := π − ( s +2 | k | ) Γ( s + 2 | k | ) . The zero-free region.
We will need a “non-standard” zero free region for L ( s, Ξ k ): Theorem 2 (Kubilius [8]) . For k > , and V := p ( T + 2) + (2 k ) , if ρ k,n = β k,n + iγ k,n isa zero of L ( s, Ξ k ) , then we have − β k,n ≫ V log log V ) / , | γ k,n | < T. AUSSIAN PRIMES IN ALMOST ALL NARROW SECTORS 3
The zero-density estimate.
We now introduce a zero-density theorem. We set N ( σ ; T, K ) := { ρ k,n = β k,n + iγ k,n : 0 < k ≤ K, | γ k,n | < T, β k,n ≥ σ } . In his thesis (1976), Ricci [10] showed
Theorem 3 ([10]) . For σ ≥ / , K, T ≥ , and T = o ( K ) , we have N ( σ ; T, K ) ≪ T K (1 − σ ) (log K ) B for some B > . The number variance
We wish to get an unconditional result on angles of Gaussian primes in almost all narrowsectors. We recall some definitions: Pick f ∈ C ∞ c ( R ), which is even and real valued, and for K ≫ F K ( θ ) := X j ∈ Z f (cid:16) Kπ/ (cid:0) θ − j π (cid:1)(cid:17) . (3.1)Let Φ ∈ C ∞ c (0 , ∞ ) and set ψ K,X ( θ ) := X a Φ (cid:16) N( a ) X (cid:17) Λ( a ) F K ( θ a − θ ) , the sum over all powers of prime ideals, with the von Mangoldt function Λ( a ) = log N( p ) if a = p r is a power of a prime ideal p , and equal to zero otherwise.By [11, Lemma 3.1], the expected value of ψ K,X ( θ ) is E ( ψ K,X ) = h ψ K,X i ∼ XK Z ∞−∞ f ( x )d x Z ∞ Φ( u )d u. Here for any function H of θ , we define the mean value h H i := 1 π/ Z π/ H ( θ )d θ. We wish to study the number varianceVar( ψ K,X ) = 1 π/ Z π/ (cid:12)(cid:12)(cid:12) ψ K,X ( θ ) − E ( ψ K,X ) (cid:12)(cid:12)(cid:12) d θ. Theorem 4. If K = X τ with τ < / , then Var( ψ K,X )( E ( ψ K,X )) ≪ (log X ) − A , as X → ∞ . Consequently we get, that in this range, for almost all θ , there is a prime ideal p withN( p ) ≍ X so that | θ − θ p | < /K . BINGRONG HUANG, JIANYA LIU, AND ZE´EV RUDNICK
The proof of Theorem 4.
We have [11, Corollary 4.4] ∗ ψ K,X ( θ ) − h ψ K,X i = − X k =0 e − i kθ K b f ( kK ) X n ˜Φ( ρ k,n ) X ρ k,n + O (cid:16) X / log K (log X ) (cid:17)! , the inner sum over all nontrivial zeros ρ k,n = β k,n + iγ k,n of L ( s, Ξ k ).Computing the mean square over θ and dividing by the square of the expected value,namely by ( X/K ) , givesVar( ψ K,X )( E ( ψ K,X )) ≪ X X k =0 | b f ( kK ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ˜Φ( ρ k,n ) X ρ k,n + O (cid:16) X / log K (log X ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Below we will see that the contribution of the first term is O ((log X ) − A ). The contributionof the second term O (cid:16) X / log K (log X ) (cid:17) is O ( X − ǫ ) if K < X − ǫ , so that modulo this fact we haveVar( ψ K,X )( E ( ψ K,X )) ≪ X X k =0 | b f ( kK ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ˜Φ( ρ k,n ) X ρ k,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X − ǫ . Next, we note that up to a negligible error, we can truncate the sum over k to 0 < k 1, where we take δ > K < X (1 − δ ) . Indeed, since b f is rapidly decaying, using the trivial bound Re ρ k,n ≤ 1, we see that the tailof the sum is bounded by1 X X k>K δ | b f ( kK ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ˜Φ( ρ k,n ) X ρ k,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ X k>K δ | b f ( kK ) | (cid:16) X n | ˜Φ( ρ k,n ) | (cid:17) For each k = 0, we use the rapid decay of the Mellin transform | ˜Φ( ρ k,n ) | ≪ (1 + | γ k,n | ) − and the bound { n : T ≤ | γ k,n | < T + 1 } ≪ log(2 | k | ( T + 2))to bound the sum over the zeros by X n | ˜Φ( ρ k,n ) | ≤ ∞ X T =0 log(2 | k | ) log( T + 2)(1 + T ) ≪ log(2 | k | )Hence the tail of the sum is bounded by1 X X k>K δ | b f ( kK ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ˜Φ( ρ k,n ) X ρ k,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ X k>K δ | b f ( kK ) | (log k ) ≪ K − on using | b f ( y ) | ≪ | y | − /δ for | y | > 1. ThereforeVar( ψ K,X )( E ( ψ K,X )) ≪ X X We may also restrict the sum to zeros with | γ k,n | < K ε , at the cost of another negligibleerror, again by using the rapid decay of ˜Φ. Thus we haveVar( ψ K,X )( E ( ψ K,X )) ≪ W + X − ǫ where W := 1 X X 1, there are no zeros of L ( s, Ξ k ) with real part β k,n > − ∆ M , by Theorem 2.We use a dyadic decomposition to bound W : W ≪ log KX X T < K ε , N ( 12 + m M ; T, K δ ) ≪ T K (1+ δ ) ( − m M ) (log K ) B . BINGRONG HUANG, JIANYA LIU, AND ZE´EV RUDNICK Hence W ≪ X /M (log K ) B +1 X M − ∆ X m =0 X mM X ≤ T K < X (1 − δ ) (here δ > XK (1+ δ ) > X − (1+ δ ) (1 − δ ) = X δ +2 δ > X δ we have M − ∆ X m =0 (cid:16) XK (1+ δ ) (cid:17) mM ≪ (cid:16) XK (1+ δ ) (cid:17) − ∆ M so that we obtain W ≪ X /M (log K ) B +1 (cid:16) XK (1+ δ ) (cid:17) − ∆ M ≪ (log K ) B +1 X δ ∆ − M . Since we took ∆ = ⌈ /δ ⌉ , we have δ ∆ − M ≥ M , so that W ≪ (log K ) B +1 X M ≪ (log K ) B +1 exp( − log X/ (log K ) / ) ≪ exp( − 12 (log X ) / ) = O ((log X ) − A ) . Thus we see that if K < X (1 − δ ) ,Var( ψ K,X )( E ( ψ K,X )) ≪ (log X ) − A . This completes the proof of Theorem 4.3.2. Primes vs prime powers. Now set ψ prime K,X ( θ ) := X p Φ (cid:16) N( p ) X (cid:17) F K ( θ p − θ ) log N( p ) , the sum over all prime ideals. LetVar( ψ prime K,X ) = 1 π/ Z π/ (cid:12)(cid:12)(cid:12) ψ prime K,X ( θ ) − E ( ψ prime K,X ) (cid:12)(cid:12)(cid:12) d θ. Corollary 5. If K = X τ with τ < / , then Var( ψ prime K,X )( E ( ψ prime K,X )) ≪ (log X ) − A , as X → ∞ . AUSSIAN PRIMES IN ALMOST ALL NARROW SECTORS 7 Proof. By [11, Lemma 3.1] again, the mean value of ψ prime K,X ( θ ) is E ( ψ prime K,X ) = h ψ prime K,X i ∼ h ψ K,X i ∼ XK Z ∞−∞ f ( x )d x Z ∞ Φ( u )d u. (3.2)Moreover, |h ψ prime K,X i − h ψ K,X i| ≪ X / K . And by the prime ideal theorem, we have X a Φ (cid:16) N( a ) X (cid:17) Ξ k ( a )Λ( a ) = X p Φ (cid:16) N( p ) X (cid:17) Ξ k ( p ) log N( p ) + O ( X / ) . Then by the same proof as in [11, Lemma 4.8], we have h| ψ K,X − ψ prime K,X | i ≪ XK . Using the triangle inequality as in [11, § (cid:3) Proof of Theorem 1. By Chebyshev’s inequality and Corollary 5, we find that ψ prime K,X ( β ) ∼ E ( ψ prime K,X ) ∼ XK Z ∞−∞ f ( x )d x Z ∞ Φ( u )d u. (3.3)for almost all β ∈ [0 , π/ f + ε ( y ) be a smooth function dependingon ε satisfying that f + ε ( x ) = 1 if x ∈ [0 , f + ε ( x ) ∈ [0 , 1] if x ∈ [ − ε, ∪ [1 , ε ], and f + ε ( x ) = 0 otherwise. Here ε is a small positive constant. Define F + K,ε ( θ ) as in (3.1), that is F + K,ε ( θ ) = X j ∈ Z f + ε (cid:16) Kπ/ (cid:0) θ − j π (cid:1)(cid:17) . Similarly, we can choose Φ + ε to be a smooth function depending on ε satisfying that Φ + ε ( u ) = 1if u ∈ [1 , + ε ( u ) ∈ [0 , 1] if u ∈ [1 − ε, ∪ [2 , ε ], and Φ + ε ( u ) = 0 otherwise.Let A = { p ⊆ Z [ i ] : θ p ∈ I := [ β, β + π/ K ] , X < N( p ) ≤ X } . Note that for X > /ε , X p ∈A ≤ X p Φ + ε (cid:16) N( p ) X (cid:17) F + K,ε ( θ p − β ) ≤ ε log X X p Φ + ε (cid:16) N( p ) X (cid:17) F + K,ε ( θ p − β ) log N( p ) . For β ∈ [0 , π/ 2) satisfying (3.3) with f = f + ε and Φ = Φ + ε , we have X p ∈A ≤ (1 + 2 ε ) XK log X = (1 + 2 ε ) | I | π/ X log X , for X > X ( ε ) sufficiently large. BINGRONG HUANG, JIANYA LIU, AND ZE´EV RUDNICK Similarly, we can obtain a lower bound of P p ∈A β ∈ [0 , π/ 2) satisfying (3.3) with f = f − ε and Φ = Φ − ε , we have X p ∈A ≥ (1 − ε ) | I | π/ X log X , for X > X ( ε ). Taking the limit ε → 0, we obtain an asymptotic formula for P p ∈A β . This completes the proof of Theorem 1.4. Prime angles for real quadratic fields One can also study questions about angles related to the representation of primes as normsin a real quadratic field, similar to those for Gaussian primes.Let E be a real quadratic field, and ǫ = ǫ E > {± ǫ n : n ∈ Z } . For simplicity, we will assume that the fundamental unit has negativenorm, and that the (narrow) class number of E is one, so that all ideals are principal (Gaussconjectured that this occurs infinitely often). Given a prime p which splits (completely)in E , to any solution of Norm E/ Q ( α ) = p , that is to any generator of an ideal p | ( p ), weassociate an angle variable t ( p ) ∈ R / (2 log ǫ ) Z ≃ S by exp( iπt ( p )log ǫ ) = | α ˜ α | iπ log ǫ (where α ˜ α is the Galois involution of E ), which is independent on the choice of generatorof the ideal p . Note that for the Galois conjugate ideal we have t (˜ p ) = − t ( p ) mod 2 log ǫ .For example, take E = Q ( √ O E = Z [ √ ǫ = 1 + √ p = ± ± p as a norm, that is solve a − b = ± p . The corresponding angle parameters describe therelative size of the solution coordinates a and b .Hecke [6] showed that as p varies over split primes, the corresponding angle parametersbecome uniformly distributed in R / (2 log ǫ ) Z ≃ S .Using ideas similar to those for the case of Gaussian primes, one can show results analogousto Theorem 1. The non-standard zero-free region for the corresponding L-functions are dueto Coleman [2]. The zero-density theorem needed can be proved along the lines of Theorem 3. References [1] M.D. Coleman, The distribution of points at which binary quadratic forms are prime. Proc. Lond. Math.Soc. (3) , 433–456, 1990.[2] M.D. Coleman, A zero-free region for the Hecke L-functions. Mathematika , no. 2, 287–304, 1990.[3] G. Harman, Prime-Detecting Sieves.(LMS-33) . Princeton university press, 2012.[4] G. Harman, A. Kumchev and P.A. Lewis, The distribution of prime ideals of imaginary quadratic fields, Trans. Amer. Math. Soc. Mathematika, , no. 1-2, 119–135, 2001.[6] E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. I. Math. Z. 1, 357–376, 1918. II, Math. Z. 6, 11–51, 1920.[7] C. Jia, Almost all short intervals containing prime numbers, Acta Arith. :1, 21–84, 1996.[8] J. Kubilius, On a problem in the n -dimensional analytic theory of numbers. (Lithuanian) Vilniaus Valst.Univ. Mokslo Darbai. Mat. Fiz. Chem. Mokslu Ser. 4, 5–43, 1955. AUSSIAN PRIMES IN ALMOST ALL NARROW SECTORS 9 [9] O. Parzanchevski and P. Sarnak, Super-Golden-Gates for P U (2). arXiv:1704.02106 , 2017.[10] S.J. Ricci, Local distribution of primes. Thesis (Ph.D.)-University of Michigan , 1976.[11] Z. Rudnick and E. Waxman, Angles of Gaussian primes. To appear in the Israel Jour. of Math. arXiv:1705.07498 , 2017.[12] B. Saffari and R. C. Vaughan. On the fractional parts of x/n and related sequences. II. Ann. Inst.Fourier (Grenoble) 27 (1977), no. 2, 1–30.[13] A. Selberg, On the normal density of primes in small intervals, and the difference between consecutiveprimes. Arch. Math. Naturvid. , (6):87–105, 1943. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel E-mail address : [email protected] School of Mathematics, Shandong University, Jinan, Shandong 250100, China E-mail address : [email protected] School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel E-mail address ::