Remarks on Landau-Siegel zeros
Debmalya Basak Jesse Thorner Alexandru Zaharescu


Dedicated to Dorian Goldfeld on the 50th anniversary of his elegant proof of Siegel's theorem


For certain families of L-functions, we prove that if each L-function in the family has only real zeros in a fixed yet arbitrarily small neighborhood of s=1, then one may considerably improve upon the known results on Landau–Siegel zeros.  Sarnak and the third author proved a similar result under much more restrictive hypotheses.



Channel-State duality with centers
    Daniele Oriti
    April 29, 2024
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§ INTRODUCTION

Building on the seminal work of Hadamard and de la Vallée Poussin for the Riemann zeta function ζ(s), it is known that if χq_χ is a primitive  Dirichlet character, then there exists an absolute and effectively computable constant c_1>0 such that the Dirichlet L-function L(s, χ) has at most one zero β (necessarily real and simple) in the region

    Re(s) ≥ 1-c_1 / log (q_χ(|Im(s)|+3)).

If β exists, then χ is real and nontrivial.  It follows from Siegel's lower bound on L(1, χ) <cit.> that for any ε>0, there exists a constant c_2(ε)>0 such that if β exists, then

    β≤ 1-c_2(ε) q_χ^-ε.

See Goldfeld <cit.> for a concise proof of Siegel's lower bound on L(1,χ).  Unfortunately, no known proof provides an effective determination of c_2 in terms of ε.

Define

    𝒮={χq_χ}.

It follows from the refinement of Siegel's theorem by Tatuzawa <cit.> that for all 0<ε<1/2, there exists an effectively computable constant c_3(ε)>0 such that

    #{}≤ 1.


Let Hypothesis H denote the hypothesis that if χ∈𝒮, then all zeros of L(s,χ) lie on Re(s)=1/2 or Im(s)=0.  In other words, the generalized Riemann hypothesis is assumed to hold only for the non-real zeros, and Landau–Siegel zeros are permitted to exist.  Assuming Hypothesis H, Sarnak and Zaharescu <cit.> improved Tatuzawa's theorem as follows:  For all ε>0, there exists an effectively computable constant c_4(ε)>0 such that

    #{}≤ 1.

This “exponentiates” the quality of the zero free region at the cost of a strong hypothesis for the non-real zeros of Dirichlet L-functions.

Here, we will use Turán's power sum method to prove that (<ref>) holds under a much weaker hypothesis.  In order to state our hypothesis, we fix 0 < δ < 1/10.

[H_δ]





Fix 0 < δ < 1/10.  If H_δ is true, then for all ε>0, there exists an effectively computable constant q_0=q_0(δ, ε)>0 such that

    #{}≤ 1.



Our proof provides an explicit permissible expression for q_0.  See (<ref>).


It follows from Heath-Brown's zero density estimate in <cit.> that if the χ∈𝒮 are ordered by conductor q_χ, then a density 1 subset of χ∈𝒮 satisfy H_δ.  In contrast, Hypothesis H has not been verified for any χ∈𝒮 yet.



Our next result, which is ineffective, is an immediate corollary of Theorem <ref>.


Fix 0 < δ < 1/10. If H_δ is true, then for all ε>0, there exists an ineffective constant c(δ, ε)>0 such that if χ∈𝒮, then

    L(σ,χ)≠ 0,    σ≥ 1-c(δ,ε)(log q_χ)^-ε.



Variants of the hypothesis H_δ for other families of L-functions lead to results for those families similar to Theorem <ref>.  After proving Theorem <ref>, we will sketch some examples.



§ PRELIMINARIES


Fix 0 < δ < 1/10.  It suffices to let 0<ε<1.  We define

    q_0 = exp(exp(10 000/δ^3 ε^2)).

Let q_1 and q_2 be positive integers such that

    q_0≤ q_2≤ q_1.

For j∈{1,2}, let χ_jq_j be distinct primitive real Dirichlet characters.  Let ψ be the primitive Dirichlet character that induces χ_1 χ_2.  It follows that ψ is real, with conductor at most q_1^2.  We assume H_δ, which implies that L(s,χ_1), L(s,χ_2), and L(s,ψ) have no non-real zeros in the disk |z-1|<δ.  The Riemann zeta function ζ(s) provably has no zero in this disk, so it follows that

    D(s)=ζ(s)L(s,χ_1)L(s,χ_2)L(s,ψ)

has no non-real zeros in the disk |z-1|<δ.

Let β_j be the greatest real zero of L(s,χ_j).  We suppose to the contrary that

    β_1≥ 1-(log q_1)^-ε        β_2≥ 1-(log q_2)^-ε.

It follows from (<ref>), (<ref>), and (<ref>) that

    max{1-β_1,1-β_2}≤ (log q_0)^-ε≤δ/10.

We define

    η = δ/e+β_2-1.

It follows from (<ref>), (<ref>), and (<ref>) that

    δ/2e≤η≤δ/e





§ PROOF OF THEOREM <REF>


Let D(s) be as in (<ref>).  We begin with a relation for β_1, β_2 and all non-real zeros of D(s).



Let k≥ 1 and ℓ≥ 2 be integers.  If ω denotes a zero of D(s) and η is as in (<ref>), then

    1/η^kℓ-1/(1+η-β_1)^kℓ≥1/(1+η-β_2)^kℓ + Re∑_Im(ω) ≠ 01/(1+η-ω)^kℓ.



If Re(s)>1, then we have the Dirichlet series expansion

    -D'/D(s)=∑_∑_m=1^∞(1+χ_1(p^m)+χ_2(p^m)+ψ(p^m))log p/p^ms=∑_n=1^∞a_D(n)/n^s.

Since D(s) is the Dedekind zeta function of a biquadratic extension of ℚ, we have that a_D(1)=1 and a_D(n)≥ 0.  On the other hand, D(s) has a Hadamard product factorization.  In particular, there exist a_D,b_D∈ℂ such that

    (s-1)D(s) = s^ord_s=0D(s)e^a_D+b_D s∏_ω≠ 0 
     D(ω)=0(1-s/ω)e^s/ω.

We emphasize that ω ranges over the trivial and nontrivial zeros of each factor of D(s).


When Re(s)>1, we equate the Dirichlet series expansion of -(D'/D)(s) with the logarithmic derivative of the Hadamard product of D(s), thus obtaining

    ∑_n=1^∞a_D(n)/n^s=1/s-1-b_D-ord_s=0D(s)/s-∑_ω≠ 0(1/s-ω+1/ω).

Let k≥ 1 and ℓ≥ 2 be integers.  We take the real part of the (kℓ-1)-th derivative of both sides, arriving at

    1/(kℓ-1)!Re∑_n=1^∞a_D(n)(log n)^kℓ-1/n^s=Re(1/(s-1)^kℓ-∑_ω1/(s-ω)^kℓ).

Now, the possible trivial zero ω=0 is now included in the sum over ω.

We let s=1+η.  Since a_D(n)≥ 0 uniformly, it follows that

    Re∑_ω1/(1+η-ω)^kℓ<1/η^kℓ.

Note that if ω is a real zero of D(s), then ω<1 and (1+η-ω)^kℓ≥ 0.  Since β_1 and β_2 are real zeros of D(s), we conclude via nonnegativity that

    1/(1+η-β_1)^kℓ+1/(1+η-β_2)^kℓ+Re∑_ω
    Im(ω)≠ 01/(1+η-ω)^kℓ≤Re∑_ω1/(1+η-ω)^kℓ<1/η^kℓ.

The desired result follows.


We define a sequence of complex numbers {z_n}_n=1^∞ as follows.  Let

    z_1 = (1+η-β_2)^-ℓ.
 
Choose an ordering {ω_j}_j=2^∞ of the zeros ω with Im(ω)≠ 0 such that if

    z_j = (1+η-ω_j)^-ℓ,

then |z_2|≥ |z_3|≥⋯≥ |z_j|≥|z_j+1|≥⋯.  It follows from (<ref>), (<ref>), and our assumption of H_δ that if ω is a non-real zero of D(s), then

    |1+η-ω|≥√(η^2+δ^2)≥η+δ/2≥ 1+η-β_2.

In particular, we have that |z_1|≥ |z_2|≥ |z_3|≥⋯.



Fix 0 < δ < 1/10.  Let η be as in (<ref>), and let ℓ = ⌈loglog q_1 ⌉.  If H_δ is true, then

    ∑_j=1^∞|z_j|/|z_1|≤ 7.001



Write ω_j=β_j+iγ_j.  Recall that our assumption of H_δ implies that D(s) has no non-real zeros in the disk |z-1|<δ.  Consequently, H_δ implies that if  |γ_j|≤ 1, then |z_j|≤δ^-ℓ.  Therefore, if T≥ 0 and

    N_D(T)=#{ω D(ω)=0, |Im(ω)|≤ T, 0≤Re(ω)≤ 1},

then the sum to be estimated is at most

    1+δ^-ℓ/|z_1|∑_|γ_j|≤ 11+1/|z_1|∫_1^∞dN_D(t)/t^ℓ≤1+δ^-ℓ/|z_1|N_D(1)+ℓ/|z_1|∫_1^∞N_D(t)/t^ℓ+1dt.

Recall that q_2≤ q_1 and q_ψ≤ q_1 q_2.  It follows from <cit.> that if T≥ 5/7, then

    |N_D(T)-T/πlog(q_1 q_2 q_ψ(T/2π e)^4)|≤log(q_1 q_2 q_ψT^4)+28.

Thus, if T≥ 1, then N_D(T)≤ 2Tlog(q_1 T)+4log q_1+25.  By (<ref>) and (<ref>), (<ref>) is

    ≤ 1+25(δ/e)^ℓ+25e^-ℓ+(6e^-ℓ+4(δ/e)^-ℓ)log q_1+2(δ/e)^ℓℓ/(ℓ-1)^2(1+(ℓ-1)log q_1).

The desired result now follows from our choice of ℓ, our range of δ, and the bounds (<ref>) and (<ref>).




Fix 0 < δ < 1/10.  If H_δ is true, then there exists an integer 1≤ k≤ 169 such that

    1/(1+η-β_2)^kℓ + Re∑_Im(ω) ≠ 01/(1+η-ω)^kℓ≥1/8(1+η-β_2)^kℓ.



Let {y_j}_j=1^∞ be a sequence of complex numbers such that |y_1|≥ |y_2|≥ |y_3|≥⋯. Set

    K=∑_j ≥ 1|y_j|/|y_1|.

Turán proved that there exists 1 ≤ k ≤ 24 K such that

    Re∑_j ≥ 1 y_j^k ≥1/8|y_1|^k

(see <cit.>).  We apply Turán's result to {z_j}_j=1^∞, bounding K from above using Lemma <ref>.

It follows from Lemmata <ref> and <ref> that

    1/η^kℓ-1/(1+η-β_1)^kℓ≥1/8(1+η-β_2)^kℓ.

Multiplying through by η^kℓ, and observing that ⌈loglog q_1⌉≤ 2loglog q_1, we find that

    1-(1- 1-β_1/1+η-β_1)^2k loglog q_1≥1/8( 1-1-β_2/1+η-β_2)^2k loglog q_1.

By (<ref>) and (<ref>), we have that

    1-β_1/1+η-β_1,1-β_2/1+η-β_2∈(0,1).



If 0<a<1 and b≥ 1, then 1-(1-a)^b<ab.


If 0<a<1 and b≥ 1, then 1-a<1 and (1-a)^b-1<1.  It follows that ((1-a)^b-1-1)b< 0.  In particular, if 0<a<1 and b≥ 1, then d/da(1-(1-a)^b-ab)< 0.  Since lim_a→ 0^+(1-(1-a)^b-ab)=0, the desired result follows.



Applying Lemma <ref> to (<ref>) with

    a=1-β_1/1+η-β_1,     b = 2kloglog q_1,

we obtain the bound

    2k(1-β_1) loglog q_1/1+η-β_1   ≥1/8( 1-1-β_2/1+η-β_2)^2k loglog q_1=1/8(log q_1)^2k log(1-1-β_2/1+η-β_2).

(We have that b≥ 2 by (<ref>) and (<ref>).)  Dividing through by (2kloglog q_1)/(1+η-β_1), we find that

    1-β_1≥1+η-β_1/16kloglog q_1(log q_1)^2k log(1-1-β_2/1+η-β_2)≥η/2704 loglog q_1(log q_1)^338 log(1-1-β_2/1+η-β_2).

It follows from (<ref>), (<ref>), and (<ref>) that

    ε>2log(2704 η^-1loglog q_1)/loglog q_1,

so

    η/2704 loglog q_1(log q_1)^338 log(1-1-β_2/1+η-β_2)≥ (log q_1)^338 log(1-1-β_2/1+η-β_2)-ε/2.

Therefore, by (<ref>), (<ref>), and (<ref>), we find that

    (log q_1)^-ε≥ (log q_1)^338log(1-1-β_2/1+η-β_2)-ε/2.

Recalling the definition of η in (<ref>), we solve for β_2 in (<ref>), thus obtaining

    β_2≤ 1-δ/e(1-e^-ε/676).

However, by (<ref>) and (<ref>), the bound (<ref>) contradicts (<ref>), as desired.




§ EXTENSIONS OF THEOREM <REF>


We briefly describe a way to extend Theorem <ref> to Dirichlet characters whose order exceeds 2.  Let m≥ 2 be an integer, let  j∈{1,2}, let q_j≥ 3 be an integer, and let χ_jq_j be a primitive Dirichlet character of order m.  Note that L(s,χ_1) and L(s,χ_1) have the same real zeros, so we assume that χ_2∉{χ_1,χ_1}.  Let G=⟨χ_1,χ_2⟩ be the group of Dirichlet characters generated by χ_1 and χ_2, and let G^* be the set of primitive Dirichlet characters that induce the characters in G.  The Dirichlet series D(s) = ∏_χ∈ G^*L(s,χ) is the Dedekind zeta function of the compositum of two cyclic extensions of degree m over ℚ.  In particular, the Dirichlet coefficients of D(s) are nonnegative.  Also, the conductor of D(s) is bounded by (q_1 q_2)^m^2.  The identity element of a group is unique, so D(s) has a simple pole at s=1 arising from the factor of ζ(s).  Note that if χ_2∈⟨χ_1⟩-{χ_1,χ_1}, then D(s) is the product of L-functions of primitive characters that induce the characters in {χ_1^j 0≤ j≤ m-1}.

Fix 0 < δ < 1/10.  Consider the set 𝒮_m of all primitive Dirichlet characters whose order divides m.  In order to ensure that all zeros of D(s) in the disk |z-1|<δ are real, it suffices to assume that for all ν∈𝒮_m, all zeros of L(s,ν) in the disk |z-1|<δ are real.  Now, small modifications to our proof of Theorem <ref> yield the following result.


Fix an integer m ≥ 2, and let 𝒮_m be the set of primitive Dirichlet characters of order dividing m. Fix 0 < δ < 1/10.  Assume that if ν∈𝒮_m, then all zeros of L(s,ν) in the disk |z-1|<δ are real.  For all ε>0, there exists an effectively computable constant q_0=q_0(δ,ε,m)>0 such that

    #{}≤ 1.



Theorem <ref> recovers Theorem <ref> when m=2.


Let ϕ(x+iy) be a Hecke–Maaß cusp form on the modular surface SL_2(ℤ)\ℍ with Laplace eigenvalue λ_ϕ>0, and let Sym^2ϕ be its symmetric square lift.  Gelbart and Jacquet <cit.> proved that there exists a cuspidal automorphic representation of GL_3 whose L-function is L(s,Sym^2ϕ).  In <cit.>, Goldfeld, Hoffstein, and Lieman proved that there exists an absolute and effectively computable constant c_5>0 such that

    L(σ,Sym^2ϕ)≠ 0,    σ≥ 1-c_5/(logλ_ϕ).

By modifying our proof of Theorem <ref> in a manner similar to the work of Hoffstein and Lockhart in <cit.>, one can prove the following result.


Fix 0 < δ < 1/10.  Let 𝔖 be the set of Hecke–Maaß cusp forms on SL_2(ℤ)\ℍ.  Suppose that for all ϕ_1,ϕ_2∈𝔖 satisfying ϕ_1≠ϕ_2, all zeros of

    ζ(s)L(s,Sym^2ϕ_1)L(s,Sym^2ϕ_2)L(s,Sym^2ϕ_1×Sym^2ϕ_2)

in the disk |z-1|<δ are real.  For all ε>0, there exists an effectively computable constant λ_0=λ_0(δ,ε) such that

    #{}≤ 1.





abbrv

