Dirichlet characters and low-lying zeros of L-functions

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Journal of Number Theory www.elsevier.com/locate/jnt

General Section

Dirichlet characters and low-lying zeros of L-functions Peter J. Cho a,∗,1 , Jeongho Park b,2 a

Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan, Republic of Korea b Department of Mathematics, Pohang University of Science and Technology, Pohang, Republic of Korea

a r t i c l e

i n f o

Article history: Received 6 June 2019 Received in revised form 13 November 2019 Accepted 10 December 2019 Available online xxxx Communicated by S.J. Miller Keywords: Dirichlet character One-level density n-level density Ratios conjecture

a b s t r a c t Let r be a positive integer ≥ 2. We consider a family of primitive Dirichlet characters of order r with conductor coprime to r. For this family, we compute the one-level density with explicit lower order terms in two ways, using Weil’s explicit formula and the Ratios conjecture. Also, the n-level density for the family twisted by a fixed cuspidal automorphic representation π of GLM (AQ ) is obtained. It turns out that, when r ≥ 3, the symmetry type for our family is always unitary. © 2019 Elsevier Inc. All rights reserved.

* Corresponding author. E-mail addresses: [email protected] (P.J. Cho), [email protected] (J. Park). This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A1B03935186). 2 The author was supported by the National Research Foundation of Korea (NRF) funded by the Ministry of Education, under the Basic Science Research Program (2017R1D1A1B03028670). 1

https://doi.org/10.1016/j.jnt.2019.12.001 0022-314X/© 2019 Elsevier Inc. All rights reserved.

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1. Introduction Katz and Sarnak’s n-level density conjecture claims that low-lying zeros of L-functions in a natural family behaves differently depending on the symmetry type of the family. Let F be a natural family of automorphic L-functions (or automorphic representations) such as a family of Dirichlet L-functions or a family of L-functions associated with modular forms of weight k. For L(π, s) ∈ F, denote its analytic conductor by cπ .3 Let F(X) be the set of L-functions such that their membership is determined by some condition n depending on the parameter X. Let Φ(x1 , x2 , · · · , xn ) = i=1 φi (xi ) be the product of n even Schwartz functions φi whose Fourier transforms are compactly supported. Let ρ denote a non-trivial zero of L(π, s), and let γ = −i(ρ − 1/2). Then, we can order γ’s by comparing their real parts. Note that the zeros are not necessarily simple, and thus we have for distinct integers i, j with i < j, (γi ) ≤ (γj ). For a self-dual representation π, the index set is the whole set Z of integers or Z \ {0} by the parity of multiplicity at the central point 1/2. For a single automorphic representation π ∈ F, we define    log cπ log cπ log cπ , γ j2 , · · · , γ jn D(n) (π, Φ) = Φ γ j1 2π 2π 2π j1 ,j2 ,··· ,jn :distinct

if π is not self-dual,   log cπ log cπ log cπ , γ j2 , · · · , γ jn Φ γ j1 = 2π 2π 2π j ,j ,··· ,j

(1.1)

∗

1

2

n

if π is self-dual, where

∗ j1 ,j2 ,··· ,jn

(1.2)

is over ji ∈ Z with ja = ±jb for a = b if the root number is −1,

and if the root number is 1 then ji runs over Z \ {0}. There are five possible symmetry types of families of L-functions: U , SO(even), SO(odd), O, and Sp. The corresponding density functions W (G) are determined in [18]. They are W (n) (U )(x) = det(K0 (xj , xk ))1≤j≤n, , W (n) (SO(even))(x) = det(K1 (xj , xk ))1≤j≤n, 1≤k≤n

1≤k≤n

W (n) (SO(odd))(x) = det(K−1 (xj , xk ))1≤j≤n, + 1≤k≤n

n 

δ(xν ) det(K−1 (xj , xk ))1≤j=ν≤n,

ν=1

1≤k=ν≤n

W (n) (Sp)(x) = det(K−1 (xj , xk ))1≤j≤n, , W (n) (O)(x) 1≤k≤n

=

W

(n)

(SO(even))(x) + W (n) (SO(odd))(x) . 2

3 In this context, analytic conductor means the value q(f, 0) of the “usual” analytic conductor at s = 0 [16, (5.7), p. 95].

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where K (x, y) =

sin π(x−y) π(x−y)

3

π(x+y) +  sinπ(x+y) .

Conjecture 1.1 (Katz and Sarnak’s n-level density conjecture). Assume Φ is a Schwartz  is compactly supported. Then, we have class function on Rn whose Fourier transform Φ   1 (n) lim D (π, Φ) = Φ(x)W (n) (G(F))(x)dx, X→∞ |F(X)| π∈F(X)

Rn

where W (n) (G(F)) is the n-level scaling density function corresponding to the symmetry type G(F). For various families, the n-level density conjecture has been tested and all the results have supported it. Especially, in [17], the 1-level densities for modular L-functions L(f, s)  and symmetric square L-functions L(sym2 (f ), s) are obtained for a larger support of Φ, and as a corollary it was shown that a positive proportion of the L-functions do not vanish at the central point 1/2. In [25], the n-level density conjecture for quadratic Dirichlet  satisfies |x1 | + |x2 | + · · · + |xn | < 1. Since it L-functions is verified when the support of Φ is impossible to give a complete list, we name just a few of them [1,3,13,21,25,26]. Those who are interested in this problem may take a look at the references therein. Let us introduce the family we consider in this work. In previous papers [4,5], we considered the family of primitive cubic Dirichlet L-functions and the family of cubic Dirichlet L-functions twisted by a fixed automorphic representation. In this paper, we extend the results to primitive Dirichlet characters of order r for any positive integer r > 1. When r is a prime, we consider all primitive characters of order r with conductors co-prime to r. When r is a composite, for technical reasons we do not cover all primitive characters of order r. We consider primitive characters χ of order r with conductors l q = i=1 pi co-prime to r such that for all i, the restriction of χ modulo pi is still of order r, instead of a proper divisor of r. Please see Sec. 2. In this paper, we first compute the one-level density precisely with lower order terms. Katz and Sarnak’s conjecture predicts only the limit of n-level density as the parameter X goes to infinity. This implies that there is a room to make the conjecture more accurate. For example, it is expected that the symmetry type for the family of quadratic Dirichlet L-functions and that for the family of Artin L-functions ζK (s)/ζ(s) for Sn -fields K are both Sp [3,25]. Nevertheless, the lower order terms for these two families should be different, reflecting arithmetic features of the families. We compute the one-level density of the family in two ways. The first way is via Weil’s explicit formula. The other way is via the Ratios conjecture, which is a powerful recipe developed by Conrey, Farmer and Zimbauer [6] to predict answers for important problems in L-functions such as k-th moments of the Riemann zeta function. The Ratios conjecture can be applied to compute one-level densities for many different families of L-functions [8,10,14,22]. We show that the Weil explicit formula and the Ratios conjecture give the same result when the support of φ is within (−1, 1).

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The one-level density in our consideration is a weighted version by a smooth function. Let w(t) be an even Schwartz function that is nonnegative and is nonzero. The total weight is defined by W ∗ (X) =

∗ 

w(qχ /X)

χ

∗ where χ is the sum over primitive rth order Dirichlet characters χ with modulus qχ > 1, (qχ , r) = 1 and the restriction of χ modulo p for any p|qχ has order r. For an even Schwartz class function φ whose Fourier transform is compactly supported, let (1) DX (χ; φ)

where γ = −i(ρ − 1/2) and L = log



X 2πe

  L , = φ γ 2π γ

. The 1-level density is defined to be

  q (1) 1 w DX (χ; φ). W ∗ (X) χ X ∗

D(1) (φ; X) =

Let ϕ be the Euler totient function. Throughout this paper, we let

a(n) =

 a(p),

where a(p) =

p|n

p p+ϕ(r)

if p ≡ 1 mod r,

1

otherwise.

Under GRH,4 we have the one-level density for the family of primitive Dirichlet characters of order r.  ⊂ (−σ, σ), σ < 1. Theorem 1.2. Let φ be an even Schawartz class function with supp(φ) Under GRH, we have D(1) (φ; X) =

∗ q    log π φ(0) φ(0) w log q − ∗ LW (X) χ X L

2 − L

∞ φ(τ ) −∞

1 + 4L

∞ −∞

 ≥1

p

a(p) log p pr/2+2rπτ i/L

dτ

     1 πiτ 1 πiτ − +Ψ + dτ φ(τ ) Ψ 4 L 4 L

4 We need GRH for the following families of L-functions: ζ(s), Dirichlet L-functions with χ mod 2r, Hecke L-functions over K = Q(ζr ) with characters of order r, and ζK (s).

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1 + 4L

∞ −∞

5

     πiτ 1 + 2δ πiτ 1 + 2δ − +Ψ + dτ φ(τ ) Ψ 4 L 4 L



+ O X −1/2+σ/2+ , where Ψ(z) =

Γ Γ (z)

1+(−1)r . 2

is the digamma function and δ =

The above expression has an estimation as follows. Note that the Mellin transform ∞ w(t)ts−1 dt

w(s) = 0

is positive and is differentiable at s = 1. We can determine the coefficient of every power order term of 1/L. Corollary 1.3. Assume GRH. For any integer M ≥ 1, we have ⎞ ⎛   2a(p) log p  (1 − δ) φ(0) (1) w  + ⎠ ⎝ + 1 − γ − 2 log 2 − π− D(1) (φ; X) = φ(0) r/2 L w(1) 2 p p ≥1

+

M −1  k=1

Ik φ(k) (0) + OM,φ k!Lk+1





1 + X −1/2+σ/2+ , LM +1

where Ik = −

  2a(p)(r )k (log p)k+1 ≥1

pr/2

p

∞ −2

k+1 k+1

π

0

xk (e−πx + e−(1+2δ)πx ) dx. 1 − e−4πx

In particular, we have the following result, which shows that the symmetry type for our family of L-functions is unitary for r ≥ 3. It is known that the possible symmetry type for quadratic characters is symplectic.  < 1, we have Corollary 1.4. Assume GRH. For σ := sup(supp(φ)) lim D

X→∞

(1)

  φ(0) (φ; X) =  − φ(0)

for r ≥ 3, φ(0) 2

for r = 2.

Proof. For the case of r = 2, the term −φ(0)/2 follows from the first equality in Lemma 3.7. 2 Remark 1.5. In [11], Gao and Zhao showed a similar result for the case r = 3, 4 and σ < 3/7 without assuming GRH.

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Now, we state the one-level density via the Ratios conjecture, which agrees with Theorem 1.2 up to the error term (X −1/2+σ/2+ ). Theorem 1.6 (One level density prediction via the Ratios conjecture). Assume Conjecture 4.1. Let φ be an even Schwartz function and assume that φ is compactly supported. The one-level density for the family of L-functions L(χ, s) of primitive Dirichlet characters of order r is given by the following expression.

D(1) (φ; X) =

∗ q    log π φ(0) φ(0) w log q − ∗ LW (X) χ X L

2 − L

∞ φ(τ ) −∞

1 + 4L 1 + 4L

∞ −∞

∞ −∞

 ≥1

p

a(p) log p dτ pr/2+2rπτ i/L

     1 πiτ 1 πiτ − +Ψ + dτ φ(τ ) Ψ 4 L 4 L      1 + 2δ πiτ 1 + 2δ πiτ − +Ψ + dτ φ(τ ) Ψ 4 L 4 L



+ O X −1/2+ . Lastly, we return to the n-level density problem. Corollary 1.4 says that the possible symmetry type for the family of primitive characters of order r is unitary for r ≥ 3, and is symplectic for r = 2. A natural question to raise up is whether the symmetry type of a family may change or not when twisted by a fixed automorphic representation, In [3], this question was investigated for the family of certain Artin L-functions. The original symmetry type for the family of Artin L-functions is Sp but when the family is twisted by a fixed self-dual representation, the symmetry type turns to SO(even), SO(odd) or Sp. For the family of primitive quadratic characters, Rubinstein [25] computed the n-level density for the family of quadratic Dirichlet L-functions twisted by a fixed self-dual representation. So we do not include this case in this paper. In [4], the authors answered for this question for the family of cubic Dirichlet characters. They verified that the symmetry type remains as U after the twist. They proposed a conjecture that the symmetry type would be always U if the representations in a family are not self-dual. We confirm this phenomenon for the family of primitive characters of order r. Let’s fix a cuspidal automorphic representation π of GLM (AQ ). We compute the n-level density for the family of Rankin-Selberg L-functions L(π × χ):

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  q (n) 1 w DX (π × χ; Φ). W ∗ (X) χ X ∗

D(n) (Φ; X) =

Theorem 1.7. Assume that either r ≥ 3 or r = 2 and π is not self-dual. Let π be an irreducible cuspidal automorphic representation of GLM (AQ ) for which the Petersson is Ramanujan conjecture is satisfied. Let Φ be a Schwartz class function for which Φ(x) supported in |x1 | + |x2 | + · · · + |xn | < σ/M for some positive constant σ < 1. Then, under GRH,  lim D(n) (Φ, X) =

Φ(x)W (U )(x)dx,

X→∞

(1.3)

Rn

 where W (U )(x) = det

sin π(xj −xk ) π(xj −xk )

1≤j≤n, . 1≤k≤n

In Sec. 2, we review primitive Dirichlet characters of order r and discuss their parametrization in terms of power residue symbols. In Sec. 3, we compute the onelevel density via Weil’s explicit formula. In Sec. 4, we recover the one-level density via the Ratios conjecture. In Sec. 5, the n-level density for the twisted family is obtained. Acknowledgment We thank the anonymous referee for his/her careful reading of our paper and many helpful comments and suggestions. 2. Dirichlet characters of order r 2.1. Dirichlet characters In this section, we study the structure of primitive Dirichlet characters of order r for a given r > 1. First, we determine what integer values are allowable to be conductors for primitive characters. In this section we use pi , qi to denote rational primes. Let r = pe11 · · · pet t , and let qχ = q1a1 · · · qsas be the conductor of a primitive character χ of order r. For a while we assume gcd(r, qχ ) = 1. Observe that χ is a product of s (lifted) primitive character χi with conductor qiai and order li such that lcm(l1 , l2 , · · · , ls ) = r. (In particular, li > 1.) We want to show that ai = 1. By our assumption, li is co-prime to qi and qi ≡ 1 mod li . (In particular, qi = 2.) Assume k ≤ ai . It is easy to see that, in Z/qiai Z, every number m ≡ 1 mod qik is a (qi − 1)qik−1 -th power, so χi (m) = 1. Hence qi is always an induced modulus of χi , and χi is not primitive unless ai = 1. Conversely, if qi ≡ 1 mod li , there exist ϕ(li ) primitive characters of order li modulo qi . We have shown that

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Lemma 2.1. Let gcd(r, q) = 1, q = q1a1 · · · qsas . Then q is the conductor of a primitive Dirichlet character of order r if and only if a1 = · · · = as = 1 and there is a sequence l = (l1 , l2 , · · · , ls ) of positive integers li > 1 with lcm(l1 , · · · , ls ) = r and qi ≡ 1 mod li for all i. Now we need to deal with the case gcd(r, qχ ) > 1. Let l > 1 be a divisor of r and χ a primitive character of order l with conductor pb . Then  l|pb−2

if p = 2, b ≥ 3,

l|(p − 1)p

b−1

otherwise.

When p = 2, by the argument preceding Lemma 2.1, we easily see that pb−1 |l and p ≡ 1 mod l/pb−1 . When p = 2, l is a power of 2. If l = 2, then b = 2 or 3. If l = 2e with e ≥ 2, then b = e + 2. Hence, we have Proposition 2.2. Let χ be a primitive character of order r with conductor qχ . Then qχ is of the form qχ = q1b1 · · · qsbs such that there is a sequence l = (l1 , l2 , · · · , ls ) of positive integers li > 1 with, writing li = qivi li , qi  li , (1) lcm(l1 , · · · , ls ) = r, (2) qi ≡ 1 mod li for all i, (in particular, if qi = 2 then li = 2k for some k), (3) ⎧ ⎪ ⎪ ⎨2 or 3 bi =

e+2 ⎪ ⎪ ⎩v + 1 i

if qi = 2, li = 2, if qi = 2, li = 2e with e ≥ 2, if qi is odd.

Conversely, if q is of this form, there exists a primitive character χ of order r with conductor q. In particular, when r is a prime, we have Proposition 2.3. Assume that r is a prime. (1) When r = 2, qχ is the conductor of a primitive quadratic character χ if and only if qχ = 2b m where m is an odd square-free integer and b = 0, 2 or 3. (2) When r > 2, qχ is the conductor of a primitive character of order r if and only if qχ = r b

finite

q≡1

mod r

q,

b = 0 or 2.

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(3) Let q be the conductor of a primitive character of order r with gcd(q, r) = 1. Then, the number of primitive characters of order r with conductor q is ϕ(r)u(q) , where u(n) is the number of distinct prime divisors of n. 2.2. Power residue symbols By Proposition 2.3, we can see that once we determine all ϕ(r) primitive Dirichlet characters of order p with prime conductor q ≡ 1 mod p, we can parametrize all primitive characters of order p with conductors coprime to p. We obtain such a parametrization using power residue symbols. When r is a composite, the parametrization of primitive characters of order r becomes messy due to (1) in Proposition 2.2. However, if we restrict the characters to satisfy (r, qχ ) = 1 and l1 = · · · = ls = r, we can parametrize those primitive characters χ of order r. Now, let’s find the ϕ(r) primitive characters of order r with a prime conductor p, p ≡ 1 mod r. Weclaim that they are in one-to-one correspondence with the r-th power residue symbols p· restricted to rational integers coprime to r. r

Let K = Q(ζr ) where ζr is a primitive r-th root of unity. Let p be a rational prime that is not ramified in K/Q, and let p ⊂ OK be a prime ideal over p. The r-th power residue symbol, viz.   · : OK −→ ζr  ∪ {0}, (2.1) p r is defined by  N (p)−1   ζrm ≡ α r α = p r 0 if α ∈ p.

mod p

for a unique m ∈ {0, 1, · · · , r − 1} if α ∈ / p, (2.2)

Recall that p splits completely if and only  if p ≡ 1 mod r. We claim that in case p ≡ 1 mod r, the r-th power residue symbols { p·i : i = 1, · · · , ϕ(r)}, when restricted to r

Z \ pZ, form the complete set of primitive Dirichlet characters of order r with conductor p. p−1 Let c ∈ Z be a primitive element modulo p. Then c r is a primitive r-th root of p−1 1 modulo p, which implies that c r ≡ ζ mod p for some primitive r-th root ζ of 1. Since p splits completely, G(K/Q) acts transitively on (i) the set of prime ideals above p−1 r p ≡ ζ σ mod pσ , and  and (ii) on the set of primitive r-th roots of 1. But then c c = ζ = ζ σ = pcσ . p r r A remark about primitive Dirichlet characters of order r is that the subgroup H <  ∗ of order r is cyclic. This means that any element of H with exact order r, or, (Z/pZ) a primitive character of order r, generates the other primitive elements as well. We can  k  thus write p·j = p·i for some k coprime to r. r

r

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∗   In the sequal we use χ and a interchangeably, where a is the sum over integral ideals a of K = Q(ζr ), satisfying (1) (N (a), r) = 1, (2) N (a) is square free, (3) N (a) > 1. (1) implies that no prime factors of N (a) are ramified, and (2) implies that the inertia degree of any prime factor of N (a) is 1. In other words, a is a proper integral ideal of K such that the norm of each prime factor of a splits completely and no two prime factors

of a are conjugate to each other. We define χ(n) = na r , and in this case qχ = N (a). In  addition, for technical reasons, we will write a for the sum over a satisfying (1) and (2). We also define a characteristic function ν on the set of integral ideals of K, by

ν(a) =

 1

if a satisfies (1) and (2),

0

otherwise.

Let n, p, r, s, t ∈ Z, and p be a prime ideal above p. Let ψ(·) = ψn (·) =



abuse of notation, when χ = a· r we write ψ(χ) = ψn (χ) = na r .

n

·

r

. With

We collect several claims to state the situation clearly. Claim 1. ψn (p1 ) + · · · + ψn (pϕ(r) ) =  σ Proof.

n p

r

 =

nσ pσ



r

=

n pσ

r

.

 σ∈G(K/Q)

ψnσ (p1 ).

2

Claim  2. Assume r is an odd prime. If p  r, p ≡ 1 mod r, and p is above p, then n = 1. p r

Proof. Note that, since r is an odd prime, every ζrs for 0 < s < r is a primitive r-th root of unity. G(K/Q) acts transitively on the set of primitive r-th roots of unity, so N (p)−1 (ζrs )σ = ζrs for all σ ∈ G(K/Q), σ = id. But (n  r − ζrs )σ ∈ pσ , so if the inertia degree of p is greater than 1 (i.e. p ≡ 1 mod r), then np = 1; otherwise, np would not be r r well-defined. 2 Claim 3. Let q(χ) be the conductor of χ and f (p) the inertia degree of p. Let p be a prime ideal above p. We can write  ψ(χ) = s q(χ) a

p≡1 mod r p=p1 ···pϕ(r)

 1+

ψ(pϕ(r) ) ψ(p1 ) + ··· + s N (p1 ) N (pϕ(r) )s



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=

1+

p N (p) splits completely

=



 ψ σ (p) 1+ N (p)s

ψ σ1 (p) ψ σϕ(r) (p) + ··· + s N (p) N (p)s

1/ϕ(r) ·

p pr,f (p)>1

σ∈G

pr



11

1/ϕ(r)

η(p) 1 + f (p)·s p

−1/ϕ(r) ,

where |η(p)| ≤ ϕ(r). The value of η(p) needs not be ϕ(r) exactly. For this, we have the following constructive example. Claim 4. Let r > k > 2, r = kr , K = Q(ζr ). Then  there exist n ∈ Z and a prime ideal pK of OK with inertia degree f > 1, such that pnK = 1. r

Proof. Let E = Q(ζrk ). Choose a rational prime p ≡ 1 mod r , p ≡ 1 mod k, pf ≡ 1 ∗ mod k where f > 1 is the order of p in (Z/kZ) . Let pE be the ideal of E above p and let pK be above pE . Then there is a nonidentity automorphism σ ∈ G(K/Q) that fixes ζrk , and E is a proper subfield of K. Since p ≡ 1 mod r , p splits completely in E/Q, and N (pE ) = p. There exists a rational integer n that is primitive modulo p, so that p−1 n r ≡ ζrk mod pE . But then n pf −1 r

N (pK )−1 r

=n

pf −1 r

=n

p−1 pf −1 +···+p+1 k r

f −1

≡ ζrp

+···+p+1

mod pE .

pf −1 r

≡ 1 mod pE . But pf −1 + · · · + p + 1 ≡ f mod r , and pf −1 n 1 < f < k < r , which implies that n r ≡ 1 mod pK . In other words, pK = 1. 2

If n

≡ 1 mod pK , then n

r

The Weil explicit formula depends on the parity of χ. If r is odd, −1 = (−1)r , so every character of order r is even. When r is even, the parity of a primitive character of order r is the parity of the remainder when q is divided by 2r. Claim 5. Let r be an even integer, and χ a primitive Dirichlet character of order r with conductor q. Then  χ(−1) =

if q ≡ 1 mod 2r,

1 −1

if q ≡ 1 + r

mod 2r.

 p−1 Proof. Assume χ has a prime conductor p. Then −1 ≡ (−1) r mod p, so writing p r   p = kr + 1, −1 = (−1)k . Thus p· is even if p ≡ 1 mod 2r, and is odd if p ≡ 1 + r p r

r

mod 2r. The set {1, r + 1} is a multiplicative group of order 2 modulo 2r, and hence the claim can be extended to composite conductors as well. 2

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12

Claim 6 ([23, VIII.5.5]). Suppose (n, r) = 1. Let Tn be the multiplicative group of fracr−1 tional ideals  of K = Q(ζr ) coprime to nr. Then Ψn : Tn → {1, ζr , · · · , ζr } defined by n Ψn (p) = p is a Hecke character of K. r

For convenience we recall the following form of reciprocity law. Proposition 2.4 (Eisenstein’s reciprocity). Let n be a rational integer relatively prime to r, and α ∈ Z[ζr ] relatively prime to nr and congruent to a rational integer modulo (1 − ζr )2 . Then n α

r

=

α n

. r

Proof. For proof, we refer to [15, Chap. 14.5]. 2 3. One-level density We use the well-known Weil explicit formula in the following form, which is from [24]. Theorem 3.1 (Weil’s explicit formula). Let F (x) be a measurable function such that ∞

1

e( 2 +δ0 )2π|x| |F (x)|dx < ∞, −∞

and ∞

1

e( 2 +δ0 )2π|x| |dF (x)| < ∞ −∞

where δ0 > 0 is fixed. Suppose that F (x) = 12 (F (x− ) + F (x+ )) for all x (i.e., F (x) is piecewise smooth), and that F (x) + F (−x) = 2F (0) + O(|x|). Put ∞ Ψ(s) =

F (x)e−(s−1/2)2πx dx

−∞

for −δ0 < σ < 1 + δ0 . Let χ be a Dirichlet character and qχ its conductor. Let ρ denote the nontrivial zeros of L(χ, s), and γ = −i(ρ − 1/2). Then, lim

T →∞

 |γ|≤T

     ∞ 1  Λ(n) −1 1 χ(n)F log n + log n (3.1) χ(n)F 2π n=1 n1/2 2π 2π   q F (0) 1 Γ 1 a χ + log + F (0) + 2π π 2π Γ 4 2

Ψ(ρ) = −

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∞ + 0

where a =

13

e−4π(1/4+a/2)x (2F (0) − (F (−x) + F (x))) dx, 1 − e−4πx

1−χ(−1) . 2

tL

Now, let F (t) = Ψ(1/2 + it) := φ 2π , where φ is an even Schwartz function whose Fourier transform is compactly supported. Corollary 3.2. Let χ be a Dirichlet character and qχ its conductor. Let ρ denote the nontrivial zeros of L(χ, s), and γ = −i(ρ − 1/2). Then,

lim

T →∞





φ

|γ|≤T

γL 2π

 =−

     ∞ 1  Λ(n)  log n + χ(n)φ log n χ(n) φ L n=1 n1/2 L L

(3.2)

 q φ(0)   Γ  1 a  φ(0) χ log + + + L π L Γ 4 2 4π + L

∞ 0

e−4π(1/4+a/2)x 1 − e−4πx

   − φ 2πx dx. φ(0) L



Recall that χ is always even when the order r is odd. When r is even, Claim 5 suggests that a half of the characters is even and another half is odd. The following lemma is helpful in simplifying the Weil explicit formula. Lemma 3.3. Assume r is even. Under GRH, for j ∈ {1, r + 1}, ∗  qχ ≡j

w

χ mod 2r

q χ

X

 1   qχ w + O X 1/2+ . 2 χ X ∗

=

Proof. Let θ1 , · · · , θϕ(2r) be the Dirichlet characters modulo 2r. Then 



ϕ(2r)

θi (k)θi (m) =

i=1

ϕ(2r)

if k ≡ m mod 2r,

0

otherwise.

Since the multiplicative order of qχ modulo 2r is at most 2, θi (qχ ) = ±1, and θi (qχ ) = −1 is possible only when θi (r + 1) = −1. We write ∗  χ qχ ≡1 mod 2r

w

q χ

X

ϕ(2r) ∗ 1    qχ w = θi (qχ ) ϕ(2r) i=1 χ X

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14

1   qχ 1 w + 2 χ X ϕ(2r) ∗

=

∗ 



w

q

χ i θi (r+1)=−1

χ

X

θi (qχ ).

Write θi (a) for θi (N (a)). When θi (r + 1) = −1, using Mellin inversion,  w

1 X

 +

∗ 

w

q

χ

χ

X

θi (qχ ) =

1 2πi

  ν(a)θi (a) s X w(s)ds, N (a)s a

(2)

and we write  ν(a)θi (a) N (a)s

a

=



p≡1 mod r (p=p1 ···pϕ(r) )

ϕ(r)θi (p) 1+ ps

 =



1+

θi (p)

 ξ

ξ(p)

ps

p=r

 ,

where ξ runs over Dirichlet characters modulo r. Because θi (r + 1) = −1, θi ξ is not principal for any ξ, and by Proposition 3.6 and under GRH, the above product has an analytic continuation to the region Re(s) > 1/2. Pushing the contour of integration to (1/2 + ), we get the lemma. 2 In [12], Gauss obtained the following expression for digamma function. For (a, q) = 1, Γ Γ

  a π πa = −γ − log 2q − cot +2 q 2 q

 0
 cos

2πaj q

 log sin

πj , q

(3.3)

where γ = 0.5772156649 · · · is the Euler-Mascheroni constant. In particular, we have Γ Γ

  1 π = −γ − 3 log 2 − , 4 2

  3 π = −γ − 3 log 2 + . 4 2

Γ Γ

We then have the following Lemma 3.4.   ∗ q    1−δ φ(0) φ(0) χ γ + 3 log 2 + log π + π D (φ; X) = w log qχ − LW ∗ (X) χ X L 2 ∗

−

∗ q  log p  m log p    (χ(pm ) + χ(pm )) w φ LW ∗ (X) p,m>0 pm/2 L X χ

1

2π + L

∞



0

e−πx + e−(1+2δ)πx 1 − e−4πx

+ O X −1/2+ .



   − φ 2πx dx φ(0) L

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15

Proof. The proof of [9, Lemma 2.1] works the same. A trifling modification can be made integrating Lemma 3.3 and the fact that in case r is odd, χ is always even. 2 To compute the one-level density for our family, we need to know the average of the first term and the third term of Lemma 3.4. These parts are carried out in Sec. 3.1 to Sec. 3.2. In Sec. 3.3, we find a different expression for the fourth term so that we can compare the one-level density with the Ratios conjecture prediction. 3.1. The first term of Lemma 3.4:

 φ(0) LW ∗ (X)

∗ χ

w

q

X

log q

  (α) To estimate the first term in Lemma 3.4 we compute the sum α w NX log N (α), for which we follow the proof of [8, Lemma 2.5 and 2.8]. In this section we prove the following lemma. Lemma 3.5. Under GRH, ∗ 

w

χ

  q  w (1) + log 2πe W ∗ (X) + O X 1/3 log X . log q = LW ∗ (X) + X w(1)

The following proposition proves useful. Proposition 3.6 ([5, Proposition 2.3]). For Dirichlet characters χ1 , · · · , χm , we have



p

=

1+

m  χi (p) i=1



ps

L(χi , s) ·

i

·

L(χ2i , 2s)−1 ·

i

L(χ2i χj , 3s) ·

  p

L(χi χj , 2s)−1 ·

i
L(χ4i , 4s) ·

i

i=j

where H(χ1 , · · · , χm ; s) =





L(χi χj χk , 3s)2

i
L(χ2i χ2j , 4s) · H(χ1 , · · · , χm ; s),

i
 1+O

1 p4s

converges absolutely for Re(s) > 1/4.

Proof of Lemma 3.5. Observe that

1+

∗    ν(a) 1 1 = = = s s qχ N (a) N (a)s χ a a

p=p1 ···pϕ(r) : split

 1+

1 1 + ··· + N (p1 )s N (pϕ(r) )s

= p≡1

mod r

 1+

ϕ(r) ps



 .

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Using the orthogonality of Dirichlet characters, we write 

p≡1

mod r

ϕ(r) 1+ s p

 =





1+

p=r

ξ

ξ(p)

ps

 ,

(3.4)

where ξ runs over all Dirichlet characters modulo r. By Proposition 3.6, under GRH we have an analytic continuation of (3.4), say,

I(s) := analytic continuation of p≡1

mod r

  ϕ(r) 1 1+ s to the region Re(s) > . p 4 (3.5)

Under GRH, I(s) has a simple pole at s = 1, and a pole of some finite order at s = 1/3. Now the proof of [5, Lemma 2.2] carries over verbatim, and we have Lemma 3.5. 2 3.2. The third term of  Lemma 3.4:

 q log p  m log p ∗ 1 m m φ ∗ p,m pm/2 χ w X (χ(p ) + χ(p )) LW (X) L In this section we prove the following lemma. Lemma 3.7. Under GRH, for any integer M ≥ 1, ∗ q  log p  m log p    w φ (χ(pm ) + χ(pm )) m/2 L X p χ p,m>0    2a(p) log p  r log p  ∗ 1/2+σ/2+  = W (X) + O X φ L pr/2 p ≥1

= W ∗ (X)

M −1  k=0

 + OM,φ

⎛

⎞ (k) (0)   2a(p)(r )k (log p)k+1 φ ⎝ ⎠ k!Lk pr/2 p ≥1

 W ∗ (X) 1/2+σ/2+ . +X LM

We start with the following estimations. Lemma 3.8. Under GRH, for n = an r-th power, ∗  χ

w

q χ

X

 ψn (χ) = a(n)W ∗ (X) + O X 1/3 .

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17

Proof. As before, we write 

 w

a

N (a) X

 ψn (a) =

1 2πi

  ν(a)ψn (a) (2)

N (a)s

a

X s w(s)ds.

(3.6)

 ν(a) a Because n is an r-th power, the sum over a in the integral becomes s = (N (a),n)=1 N (a)  s p I(s) p|n ps +ϕ(r) . Taking out the residue at s = 1, the proof of [5, Lemma 2.5] p≡1 mod r

works exactly the same.

2

Lemma 3.9. Under GRH, for n = an r-th power, ∗ 

w

q

χ

χ

X

ψn (χ) w, n X 1/2+ .

(3.7)

Proof. Let σ1 , · · · , σϕ(r) be the automorphisms of K, f (p) the inertia degree of p in K/Q, and p any prime ideal above p. We have  ν(a)ψn (a) a

N (a)s



=

p≡1 mod r p=p1 ···pϕ(r)

ψn (pϕ(r) ) ψn (p1 ) 1+ + ··· + N (p1 )s N (pϕ(r) )s

=



p N (p) splits completely

=

pr



 ψ σ (p) n 1+ N (p)s σ∈G



σ

1+

ψnσ1 (p) ψnϕ(r) (p) + · · · + N (p)s N (p)s

1/ϕ(r) ·

p pr,f (p)>1



1/ϕ(r)

η(p) 1 + f (p)·s p

−1/ϕ(r) ,

where η(p) is some complex number satisfying |η(p)| ≤ ϕ(r). Then, generalizing Proposition 3.6 to the Dedekind zeta function of K = Q(ζr ) and  n (a) Hecke characters for K, under GRH, the series a ν(a)ψ can be analytically continued N (a)s to the region Re(s) > 1/2. We move the contour of integration in (3.6) from (2) to (1/2 + ).5 By well-known bounds on L-functions (for example, [16, Theorem 5.19] or [5, Lemma 4.2]) and the fast decaying property of w(s) [8, Lemma 2.1], we obtain the lemma. 2 Let ∗ q  log p  m log p    χ(pm ). Sm (X) = w φ L X pm/2 p χ

5

Under GRH, LK (ψn , s)1/2 has branching points at Re(s) = 1/2.

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18

Lemma 3.10. Under GRH, 

   a(p) log p  r log p  1/3  + O X φ , L pr/2 ≥1 p  Sm (X)  X 1/2+ . S1 (X)  X 1/2+σ/2+ ,

Sr (X) = W ∗ (X)

≥1

m≡0 mod r m>1

Proof. Note that ψpr = ψpr for any ≥ 1. The first assertion follows directly from Lemma 3.8. For m ≡ 0 mod r, we observe that ψpm = ψpm has conductor  pϕ(r) . By Lemma 3.9 we have  log p  log p  p X 1/2+  X 1/2+σ/2+ , S1 (X)  √ φ p L p and 



Sm (X) 

m≡0 mod r m>1

p
log p p



1 1 − p−1/2

 p X 1/2+  X 1/2+ . 2

Proof of Lemma 3.7. The first half of the lemma immediately comes from Lemma 3.10. The last expression is obtained by exactly the same computation as in [5, Lemma 2.4]. 2 3.3. The fourth term of Lemma 3.4:

2π L

∞ 0

e−πx +e−(1+2δ)πx 1−e−4πx



 − φ φ(0)

2πx L

dx

Lemma 3.11. We have 2π L

∞ 0

e−πx + e−(1+2δ)πx 1 − e−4πx =



  2πx   dx φ(0) − φ L

   (1 − δ)π φ(0) + 3 log 2 + γ L 2 ∞       1 πiτ 1 1 πiτ − +Ψ + dτ + φ(τ ) Ψ 4L 4 L 4 L −∞

1 + 4L

∞ −∞

     πiτ 1 + 2δ πiτ 1 + 2δ − +Ψ + dτ. φ(τ ) Ψ 4 L 4 L

Proof. As in the proof of [20, Lemme I.2.1], we have ∞ 0

∞   1 e−θx   t  (Ψ(θ + iat) − Ψ(θ)) d, φ φ(0) − φ(ax) dx = 1 − e−x 2π 2π −∞

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19

where Ψ(x) = Γ (x)/Γ(x) is the digamma function. We put θ = (1 + 2δ)/4, a = write ∞ 4π 0

e−(1+2δ)πx 1 − e−4πx ∞

= −∞

From (3.3), for δ =

2π L

∞ 0

=

1 2L

∞ −∞

and

  2πx   dx φ(0) − φ L

     πiτ 1 + 2δ 1 + 2δ + −Ψ dτ. φ(τ ) Ψ 4 L 4

1+(−1)r , 2

e−πx + e−(1+2δ)πx 1 − e−4πx



1 2L ,



we can write Ψ

1+2δ

4

=

(2δ−1)π 2

− 3 log 2 − γ. Then

  2πx   dx φ(0) − φ L

        1 + 2δ πiτ (1 − δ)π 1 πiτ φ(0) + +Ψ + + γ + 3 log 2 + . φ(τ ) Ψ 4 L 4 L L 2

But φ(τ ) is even, and hence 

∞ φ(τ )Ψ −∞

1 = 2

∞ −∞

1 + 2δ πiτ + 4 L

 dτ

     πiτ 1 + 2δ πiτ 1 + 2δ − +Ψ + dτ. φ(τ ) Ψ 4 L 4 L

2

Lemma 3.12 ([5, Lemma 2.9]). For any M ≥ 1,

2π L

   ∞ −πx e + e−(1+2δ)πx   2πx dx φ(0) − φ 1 − e−4πx L 0

=−

M −1  k=1

  ∞ k −πx 1 x (e + e−(1+2δ)πx ) φ(k) (0)2k+1 π k+1 . dx + O M,φ k!Lk+1 1 − e−4πx LM +1 0

Proof of Theorem 1.2 and Corollary 1.3. Theorem 1.2 is a mere consequence of Lemma 3.4, Lemma 3.7 and Lemma 3.11. Corollary 1.3 also follows from Lemma 3.4, Lemma 3.5, Lemma 3.7 and Lemma 3.12. 2

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20

4. The Ratios conjecture In this section, following the exposition of Conrey and Snaith [7], we want to get a precise expectation of D∗ (φ; X) based on the Ratios conjecture for L-functions of Dirichlet characters of order r. Let f be an even Schwartz function. Following the convention of [7, Section 3], with 1/2 + 1/ log X < c < 3/4, we let S(f ) =



 w

a

=



 w

a

N (a) X N (a) X



f (γa )

γa



⎛



⎞



1 ⎜ ⎝ 2πi

⎟ L (χa , s) f ⎠ L(χa , s)

−

(c)





1 −i s − 2

 ds.

(1−c)

The integral over (c) is 1 2π



∞ f −∞



1 t−i c− 2

 

 w

a

N (a) X





L χa , 12 + c − 12 + it

dt, L χa , 12 + c − 12 + it

and assuming RH we have L /L  log((|s| + 3)N (a)). Put Rw (ν ; ν) =



 w

a

N (a) X



L χa , 12 + ν

, L χa , 12 + ν

so that  a

 w

N (a) X





 L χa , 12 + c − 12 + it ∂ 

= Rw (ν ; ν) 1 1  ∂ν L χa , 2 + c − 2 + it

.

(4.1)

ν  =ν=c−1/2+it

For an r-th power n ∈ Z, by Lemma 3.8, we have ∗ 

w

χ

q χ(n) = a(n)W ∗ (X) + error. X

∗

When n is not an r-th power we assume χ w Xq χ(n) = small. We expect the following conjecture to hold. Conjecture 4.1. For Re((r − 1)ν + ν) >

Rw (ν , ν) =

 a

 w

N (a) X

1−r 2 ,



Re(ν ) > − 12 ,

L χa , 12 + ν

L χa , 12 + ν

(4.2)

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21



 A(ν ; ν)ζ r2 + rν

W ∗ (X) + O X 1/2+ , = r ζ 2 + (r − 1)ν + ν where A(ν ; ν) = =

ζ

 

ν  −ν

+ (r − 1)ν + ν 1 − p 2

1 + a(p) r/2+rν  p −1 ζ 2r + rν p 

  1 + O p−r/2−1−rν + p−r/2−1−(r−1)ν −ν + p−3 .

r

p≡1

mod r

Derivation of Conjecture 4.1. We only consider the main term of the conjecture in this recipe. For the numerator of Rw (ν ; ν), we use the approximate functional equation. For a primitive Dirichlet character χ of order r, [19, Teopema 1] becomes  1     Γ  4 + 1 q −ν χ(n) L χ, + ν = + ε(χ)  2 π n1/2+ν Γ 14 + n≤x

a 2 a 2

− +

ν 2 ν



2

 χ(n) + Remainder, n1/2−ν  n≤y (4.3)

1−χ(−1) . 2

where ε(χ) is the root number of L(χ, s), and a = [7, Section 2.2] we write

For the denominator, as in

∞

 μ(h)χ(h) 1 = . L(χ, s) hs

(4.4)

h=1

We put (4.3) and (4.4) into Conjecture 4.1. If we sum all the contributions of the first term of (4.3), we will asymptotically get W ∗ (X)



μ(h)a(hm) 1/2+ν m1/2+ν  h hm=r-th power ⎛

= W ∗ (X)

⎜ ⎜ ⎝ p



h,m h+m≡0 mod r

⎞ ⎟ ⎟.

μ(ph )a(ph+m )

ph(1/2+ν)+m(1/2+ν  ) ⎠

The term μ(ph ) implies that it suffices to consider h = 0 or 1. For h = 0, we have  a(p) a(pm ) m≡0 mod r pm(1/2+ν  ) = 1 + pr/2+rν  −1 , and for h = 1,  m≡r−1

−a(ph+m ) mod r

The Euler factor at p becomes

p

1/2+ν+m(1/2+ν  )

=−

a(p)pν p



r/2+rν 

−ν

−1

.

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22

1 + a(p) ·

1 − pν p



r/2+rν 

−ν

−1

.

For p ≡ 1 mod r, for which a(p) = 1, the above Euler factor is 1 − pν

1+

p



r/2+rν 

−ν

1 − p−r/2−(r−1)ν = −1 1 − p−r/2−rν 



−ν

.

For p ≡ 1 mod r, because Re(rν ) > −1 − r/2 and Re((r − 1)ν + ν) > −r/2 under our 1 assumption, we can use the expression 1− = 1 +  + 2 + · · · repeatedly to have 

1+

p − p1+ν −ν (p + ϕ(r))(pr/2+rν  − 1) 

=1+ = = =



p−r/2−rν − p−r/2−(r−1)ν −ν (1 + ϕ(r)p−1 )(1 − p−r/2−rν  )

(1 − ϕ(r)p−1 + ϕ(r)2 p−2 + · · · )(1 + ϕ(r)p−1 − p−r/2−(r−1)ν 1 − p−r/2−rν  

−ν

+ O(p−3 + p−r/2−1−rν + p−r/2−1−(r−1)ν 1 − p−r/2−rν 

1 − p−r/2−(r−1)ν



−ν

+ O(p−3 + p−r/2−1−rν + p−r/2−1−(r−1)ν 1 − p−r/2−rν 

1 − p−r/2−(r−1)ν 1 − p−r/2−rν 





−ν

−ν



)(1 + p−r/2−(r−1)ν





−ν

1 − p−r/2−(r−1)ν

× (1 − p−r/2−(r−1)ν =



− ϕ(r)p−r/2−1−rν )





−ν

)





−ν

)

−ν

+ p−r−2(r−1)ν



−2ν

+ ···)



1 + O(p−3 + p−r/2−1−rν + p−r/2−1−(r−1)ν



−ν

) .

Taking the product over all p, we get −1 r + rν ζ + (r − 1)ν + ν A(ν , ν), 2 2 

−r/2−1−(r−1)ν  −ν  −r/2−1−rν  −3 conwhere A(ν , ν) = + p + p 1 + O p p≡1 mod r ζ

r

verges absolutely for Re((r − 1)ν + ν) > (1 − r)/2 and Re(ν ) > −1/2. As in [5], we assume that the average of the root numbers ε(χ) goes to 0 as qχ → ∞, and neglect the contribution of the second term in (4.3). 2 Proof of Theorem 1.6. To compute (4.1), we define C1 (t) as follows: ∂ ∂ν

   −1 r r  ζ + rν ζ + (r − 1)ν + ν A(ν , ν)   2 2

We can also write

ν  =ν=t

=: C1 (t).

(4.5)

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23

    ∂ 1 − pν −ν C1 (z) = 1 + a(p) r/2+rν  ∂ν p p −1 =



1−1 1 + a(p) r/2+rz p −1

p

=

ν  =ν=z



 ·



1 + a(p )

p

1−1

−1

p r/2+rz − 1

B(p , z)

 −a(p) log p , pr/2+rz − 1 p

where B(p, z)  =



a(p) log p

2 −p

ν  −ν+r/2+rν 

+p

pr/2+rν  − 1

ν  −ν

− rp

r/2+rν 

+ rp

ν  −ν+r/2+rν 



 . ν  =ν=z

∗ ∗  Put fˆ(x) = 2π L φ(x). Then D (φ; X) = SX (f )/W (X). Theorem 1.6 follows from the following proposition.

Proposition 4.2. Assuming Conjecture 4.1, W ∗ (X)   q w π X χ ∗

SX (f ) =



∞

−∞

f (t)C1 (it)dt +

∗ q fˆ(0)   q w log 2π χ X π

+ C2 W ∗ (X) + O X 1/2+ ,

(4.6)

where

C2 =

1 8π



∞ f (−t) −∞



Γ + Γ

Γ Γ



     1 t Γ 1 t Γ 1 + 2δ t − i + + i + − i 4 2 Γ 4 2 Γ 4 2

 1 + 2δ t + i dt. 4 2



Proof. Cauchy’s differentiation formula implies that the error term O X 1/2+ in

  Rw (ν ; ν) remains O X 1/2+ after taking the derivative. Writing SX (f ) = (c) − (1−c) , we shall have  =

W ∗ (X) 2π

(c)

Let aχ =

1−χ(−1) . 2

f −∞

For

     1 1 C1 c − + it dt + O X 1/2+ . t−i c− 2 2



∞

 (1−c)

as in [7, (3.8), (3.9)], we write

(4.7)

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24



       1 1 L (1 − s, χ) −i s − ds = f i s− ds 2 L(1 − s, χ) 2

L (χ, s) f L(χ, s)

(1−c)



=

(c)

         q 1 Γ 1 + aχ − s 1 Γ aχ + s L (χ, s) 1 − log − − − ds. f i s− 2 π 2Γ 2 2Γ 2 L(χ, s)

(c)

Since f is even, using Lemma 3.3 we have SX (f ) 1  q = w 2πi χ X ∗

 +



  1 L (χ, s) ds −i s − 2 L(χ, s)

 f

(c)

          q 1 Γ 1 + aχ − s 1 Γ aχ + s L (χ, s) 1 log + + + ds f i s− 2 π 2Γ 2 2Γ 2 L(χ, s)

(c)

1  q w πi χ X ∗

=

  L (χ, s) 1 ds −i s − 2 L(χ, s)



 f

(c)

+

1 2πi

∗  χ

    q 1 q ds f i s− log X π 2 (c)

1   q w 8πi a=0 χ X 1

+

w

∗

 + O X 1/2+ .

 f

        Γ 1+a−s Γ a + s 1 + ds i s− 2 Γ 2 Γ 2

(c)

In the first term of the last expression, by (4.7) we move the contour of integration (c) to c = 1/2. Then Proposition 4.2 follows, and we obtain Theorem 1.6. 2 5. The n-level density From now on we consider the n-level density of low lying zeros of our family of  is supported in |x1 | + · · · + |xn | < σ/M for any L-functions L(π × χ, s). We assume Φ(x) fixed σ < 1. Note that π × χ is non-selfdual unless r = 2 and π is self-dual. The case of r = 2 and π is self-dual is studied by Rubinstein [25]. Throughout this section, we assume that π×χ is non-selfdual. For a single automorphic representation π × χ, we define the n-level density to be (n)

DX (π × χ; Φ) =

 j1 ,j2 ,··· ,jn :distinct

  L L L . Φ γj1 , γj2 , · · · , γjn 2π 2π 2π

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25

Let Q be the conductor of π. The total weight apart from Q is defined by ∗∗ †  N (a)  WQ∗ (X) = w(q/X) = w X χ a

∗∗ † where χ = a is a sum over primitive characters of order r χ : n → na r parameterized by integral ideals a of Q(ζr ) satisfying (1) (N (a), rQ) = 1, (2) N (a) is square-free. Since qχ (= N (a)) is coprime to Q, by a work of Barthel and Ramakrishnan [2], the conductor of π × χ is QqχM . Since w(t) is compactly supported away from 0, log cπ×χ = R + O(1), where R = log X M . The n-level density we are interested in is ∗ DQ (Φ; X) =

∗∗

1 WQ∗ (X)

χ

w

q (n) DX (π × χ; Φ). X

We show that the n-level density agrees with the unitary type U . Theorem 5.1. Assume GRH. Let π be an irreducible cuspidal automorphic representation of GLM (AQ ) for which the Petersson-Ramanujan conjecture is satisfied. Assume that  Φ(x) is supported in |x1 | + |x2 | + · · · + |xn | < σ/M for some positive constant σ < 1. Then  ∗ lim DQ (Φ, X) = Φ(x)W (U )(x)dx, (5.1) X→∞

Rn

 where W (U )(x) = det

sin π(xj −xk ) π(xj −xk )

1≤j≤n, . 1≤k≤n

In [4], we obtained the same result for the case r = 3. A proof for the n-level density result consists of three steps, which are computation of LHS, that of RHS, and the comparison of the LHS and the RHS in (5.1). Since in [4] we already carried out the second step and the third step, it is enough to show that our LHS coincides with the LHS in [4]. In n-level density for the family of Rankin-Selberg L-functions L(π × χ, s), we count Dirichlet characters whose conductors are coprime to Q. Hence, we state modified versions of Lemma 3.8 and Lemma 3.9 without proof. Put  p if p ≡ 1 mod r and p  Q, aQ (p) = p+ϕ(r) 1 otherwise.

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26

Lemma 5.2. Under GRH, when an integer n = an r-th power, † α

where aQ (n) =

 p|n

 w

N (α) X



 ψn (α) = aQ (n)WQ∗ (X) + O X 1/3 ,

aQ (p).

Lemma 5.3. Under GRH, when an integer n = an r-th power, † α

 w

N (α) X

 ψn (α) w, n X 1/2+ .

(5.2)

Lemma 5.4 (Weil’s explicit formula, [4, Corollary 2.2]). Let φ be an even Schwartz function for which φ is compactly supported. Assume χ is primitive, and qχ  X. Then, (1)

DX (π × χ, φ)

       ∞  Λ(m) 1 log m log m    − 1 a + a + O . (m) φ (m) φ = φ(0) π×χ π ˆ ×χ 1/2 R m=1 m R R R

Let F run through set partitions of {1, · · · , n} into nonempty subsets, and ν(F) be the number of parts in F. We mostly follow the argument in [25, pp. 151-152], but do (j) not take positive γd or recover 2−ν(F) . We have ∗ DQ (Φ; X)

  ν(F)

∗∗  qχ    R 1 n−ν(F) . = ∗ w (−1) φi γ (|F | − 1)! WQ (X) χ X 2π γ i∈F =1 F

Let Φ (x) =  γ

 i∈F

φi (x). By Lemma 5.4, we have

  R Φ γ 2π

       ∞ 1  Λ(m) log m log m 1   a + a + O (m) Φ (m) Φ π×χ  π ˆ ×χ  R m=1 m1/2 R R R   1  , π × χ) + O = Φ (0) + D (π × χ) + D (ˆ R

  (0) − =Φ

where   ∞ 1  Λ(m)aπ×χ (m)  log m √ Φ D (π × χ) = − R m=1 R m and D (ˆ π × χ) is obtained from D (π × χ) by replacing π × χ with π ˆ × χ.

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27

Then we write

∗ DQ (Φ; X) =

  F

(−1)n−ν(F)

S,A,B

⎛   (0) · ⎝ Φ

∈S

·

⎞

ν(F)

(|F | − 1)!⎠

=1

∗∗

w

χ

q χ

X

D (π × χ) ·

∈A

1 WQ∗ (X)

 D (ˆ π × χ) + O

∈B

1 R

 ,

where we sum over S  A  B = {1, 2, · · · , ν(F)}.6 Let a = |A|, b = |B|. Recall that aπ×χ (pk ) = aπ (pk )χ(pk ). We want to compute ∗∗

1 ∗ (X) WQ

=

w

q χ

χ

X

D (π × χ) ·

∈A

Λ(m1 ) · · · Λ(ma )aπ (m1 ) · · · aπ (ma )Λ(r1 ) · · · Λ(rb )aπˆ (r1 ) · · · aπˆ (rb ) √ m1 · · · ma r1 · · · rb

1 a =1 r1 ,··· ,rb =1

i=1 i ∈A

 Φ i



log mi R

 ·

b

D (ˆ π × χ)

∈B

a+b  ∞  1 − R m ,··· ,m

a



  Φ j

j=1 j ∈B



log rj R

 ·

∗∗ q  1 χ w χ(m1 · · · ma )χ(r1 · · · rb ). ∗ WQ (X) χ X

(5.3)

Write χ(m1 · · · ma )χ(r1 · · · rb ) = χ(m1 · · · ma r1r−1 · · · rbr−1 ). When m1 · · · ma r1r−1 · · · rbr−1 = an r-th power, by Lemma 5.3, the innermost sum over χ is  X 1/2+ (m1 · · · rb ) . It is easy to derive that the contribution of such combinations of m1 , · · · , rb to (5.3) is  X −1/2+σ/2+ . We now focus on the cases m1 · · · ma r1r−1 · · · rbr−1 = an r-th power, which happens if and only if vp (m1 · · · ma ) ≡ vp (r1 · · · rb ) mod r for all p. Besides, the cases vp (m1 · · · ma r1 · · · rb ) ≥ 3 contribute to (5.3) by at most 1   pe 1  . e/2 R R p p e≥3

We thus consider only the cases that m1 · · · ma = r1 · · · rb are squarefree, and by the factor Λ(m1 ) · · · Λ(rb ), the cases that mi , rj are all primes. The redundances that arise when we take the same prime for mi , mi , also contribute a negligible amount (O(R−1 )) to (5.3). We simply take the sum over all rational primes m1 , · · · , ma , r1 , · · · , rb in this sense. In particular, only the case a = b survives; if a = b, we have (5.3)  R−1 . 6

 denotes a disjoint union.

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28

When m1 · · · ma r1r−1 · · · rbr−1 = an r-th power, by Lemma 3.8, under GRH, we have7  q 1 w χ(m1 · · · ma )χ(r1 · · · rb ) = aQ (m1 · · · ma ) + O(X −2/3 ). W ∗ (X) χ X ∗

 Here we easily see that aQ (m1 · · · ma ) can be replaced by 1, because aQ (p) = 1 + O p1 ,  and O p1 contributes to (5.3) by O(R−1 ). The Rankin-Selberg L-function L(π × π ˆ , s) has a simple pole at s = 1. By [16, (5.52)] we see that 

|aπ (p)|2 log p = x + Oπ (xe−cπ

√ log x

).

p≤x

The proof of [25, p. 159, Subclaim 3.2] now works the same using this estimate for the   is even, we can write summation by parts. Since Φ        log p 1 1 1  |aπ (p)|2 (log p)2  log p    Φ (u)Φ (u)|u|du + O Φ Φ  = . 2 R p p R R 2 R R

(5.4) What we have proved so far is as follows: Assume GRH. Let A and B be two subsets of {1, 2, . . . , ν(F )}. If |A| = |B| = m, ∗∗ q

 1 χ w D (π × χ) · D (ˆ π × χ) ∗ WQ (X) χ X ∈A

∈B

   m  m 

1 1   Φaj (u)Φbj (u)|u|du + O . = 2 R j=1 (A:B)

R

Otherwise, ∗∗ q 

1 1 χ w D (π × χ) · D (ˆ π × χ)  . ∗ WQ (X) χ X R ∈A

∈B

Note that we can replace the sum disjoint 



S,A,B⊂{1,··· ,ν(F)} (A:B) |A|=|B|

7

aQ (m1 · · · ma r1r−1 · · · rbr−1 ) = aQ (m1 · · · ma ).

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29

by 

2|S2 |/2

S2 ⊂{1,··· ,ν(F)} |S2 |: even



,

(A:B)

 where (A:B) runs over all ways of dividing elements of S2 {{aj , bj }}j=1,··· ,|S2 |/2 . For example, 

1 = (|S2 | − 1)(|S2 | − 3) · · · 1 =

(A:B)

into pairs

|S2 |! . (|S2 |/2)!2|S2 |/2

We thus have the following lemma. Lemma 5.5. Assume σ < 1. Under GRH, we have

D(n) (Φ : X) =



⎛

ν(F)

(−1)n−ν(F) ⎝

F ×

=1 2 |/2   |S

⎞ (|F | − 1)!⎠

 S2

⎛ ⎝

⎞   (0)⎠ Φ

∈S2c

 a (u)Φ  b (u)|u|du Φ j j

(A:B) j=1 R



+O

1 R



,

where S2 ranges over all subsets of {1, 2, . . . , ν(F)} of even cardinality and over all ways of pairing up the elements of S2 .

 (A:B)

is

Since Lemma 5.5 is the same with Lemma 5.6 [4], Theorem 1.7 follows. References [1] L. Alpoge, S.J. Miller, Low-lying zeros of Maass form L-functions, Int. Math. Res. Not. (10) (2015) 2678–2701. [2] L. Barthel, D. Ramakrishnan, A nonvanishing result for twists of L-functions of GL(n), Duke Math. J. 74 (1994) 681–700. [3] P.J. Cho, H.H. Kim, n-level densities of Artin L-functions, Int. Math. Res. Not. (17) (2015) 7861–7883. [4] P.J. Cho, J. Park, n-level densities for twisted cubic Dirichlet L-functions, J. Number Theory 196 (2019) 139–155. [5] P.J. Cho, J. Park, Low-lying zeros of cubic Dirichlet L-functions and the Ratios conjecture, J. Math. Anal. Appl. 474 (2019) 876–892.

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