On the density hypothesis for $L$-functions associated with holomorphic cusp forms

We study the range of validity of the density hypothesis for the zeros of $L$-functions associated with cusp Hecke eigenforms $f$ of even integral weight and prove that $N_{f}(\sigma, T) \ll T^{2(1-\sigma)+\varepsilon}$ holds for $\sigma \geq 1407/1601$. This improves upon a result of Ivi\'{c}, who had previously shown the zero-density estimate in the narrower range $\sigma\geq 53/60$. Our result relies on an improvement of the large value estimates for Dirichlet polynomials based on mixed moment estimates for the Riemann zeta function. The main ingredients in our proof are the Hal\'{a}sz-Montgomery inequality, Ivi\'{c}'s mixed moment bounds for the zeta function, Huxley's subdivision argument, Bourgain's dichotomy approach, and Heath-Brown's bound for double zeta sums.


Introduction
Zero-density estimates for the Riemann zeta function and L-functions play a central role in analytic number theory.They have important arithmetic consequences; see for instance [21,Chap. 12] and [28,Chap. 15] for an overview of applications in prime number theory.
Key words and phrases.Density hypothesis; Zero-density estimates; Riemann zeta function; Holomorphic modular forms.
B. Chen gratefully acknowledges support by the China Scholarship Council (CSC).G. Debruyne gratefully acknowledges support by Ghent University through a postdoctoral fellowship (grant number 01P13022).
The work of J. Vindas was supported by the Research Foundation-Flanders, through the FWOgrant number G067621N.
has become known as the density hypothesis.While a proof that the density hypothesis holds uniformly for 1/2 ≤ σ ≤ 1 seems to be out of reach by present methods, there has been substantial progress towards maximizing its range of validity.Montgomery showed [27] that the density hypothesis (1.2) holds in the range σ ≥ 9/10.The range of validity was subsequently improved (cf. [15,29,8,17,24]), and the current record is due to Bourgain [3], who showed that (1.2) is valid for σ ≥ 25/32 = 0.78125.
It is also natural to consider zero-density estimates for Dirichlet L-functions [2,12,13,16,18,23,24] and for L-functions associated with modular forms [14,22,25,30,34].We are interested in studying the density hypothesis for the latter case.So, let f (z) = ∞ n=1 a f (n)e 2πiz be a holomorphic cusp form of even integral weight κ for the full modular group SL(2, Z).We assume that f is a Hecke eigenform [1,7], and that it is normalized, i.e. a f (1) = 1.We set λ f (n) = a f (n)n − κ−1 2 and notice that this multiplicative function satisfies |λ f (n)| ≤ d(n), where d(n) is the divisor function, an inequality that was shown by Deligne [6,Thm. 8.2,p. 302] as a consequence of his proof of Weil's conjectures.The L-function L(s, f ) associated with f is defined as A classical result of Hecke establishes that L(s, f ) extends to the whole complex plane as an entire function of s.
Denote by N f (σ, T ) the number of zeros ρ = β + it of L(f, s) in the rectangle σ ≤ β ≤ 1, |t| ≤ T .In 1989, Ivić [22] showed that N f (σ, T ) ≪ T 2(1−σ)+ε holds for σ ≥ 53/60.The successful establishment of the density hypothesis in the range σ ≥ 1/2 should be one of the key ingredients for obtaining estimates for the asymptotic distribution of λ f (p) for primes in short intervals that are as good as if one were to assume the grand Riemann hypothesis.
The main goal of this paper is to improve Ivić's result by showing 1 : Theorem 1.1.We have the bound for σ ≥ 1407/1601.
Here 1407/1601 ≈ 0.8788, while 53/60 = 0.8833....We now describe the general strategy for our proof of Theorem 1.1.The first step, which is standard, is to apply the zero-detection method to divide the zeros of L(s, f ) into two categories, the so-called class-I zeros and class-II zeros.The number of class-II zeros is directly estimated by using Good's second moment estimate for L-functions associated with holomorphic cusp forms [10].The innovation of our work is to achieve sharper estimates for the class-I zeros than those obtained by Ivić in [22].
We seek to obtain an upper bound for the class-I zeros by applying the Halász-Montgomery inequality.Ivić's argument is then to combine Huxley's subdivision technique with the direct insertion of mixed moment bounds for the zeta function into this inequality (cf.Remark 3.2).Our improvement is based on two aspects.The first one is an optimization of the parameters in the mixed moment estimates.Here we rely on the newly established exponent pair (13/84 + ε, 55/84 + ε) due to Bourgain [5].The second aspect that leads to an additional improvement is the incorporation of a dichotomy technique developed by Bourgain in [3] to achieve the current record of the range of validity of the density hypothesis for the Riemann zeta function.The crucial point of the dichotomy is that it allows one to apply Heath-Brown's estimate on double zeta sums [11] which is more efficient than the mixed moment bounds in certain ranges.In Bourgain's original paper the dichotomy approach is a bit difficult to follow; one of the goals of this paper is to explain the underlying ideas and its advantages more clearly.
In addition to improving the range of validity of the density hypothesis for the zeros of the L-functions associated with holomorphic cusp forms, our argument can, with only mild adjustments, also be applied to obtain a zero-density estimate for the Riemann zeta function.In order to further demonstrate the strength of the dichotomy method we prove: Theorem 1.2.There holds This improves on the condition 155/174 ≤ σ ≤ 17/18 obtained by Ivić ([20]; [21,Thm. 11.2]) in 1980.Observe that 155/174 ≈ 0.8908 and 279/314 ≈ 0.8885.
The paper is organized as follows.In Section 2 we recall the classical zero-detection method to divide the zeros into class-I zeros and class-II zeros and explain how to handle the class-II zeros.In Section 3 we revisit Ivić's original argument involving mixed moment estimates.In Section 4 we study Bourgain's dichotomy technique in this context; we derive a large value estimate for Dirichlet polynomials from which Theorem 1.1 follows.Finally, the proof of Theorem 1.2 will be completed in Appendix A.
We adopt the convention that ε stands for a small positive quantity.Throughout the paper, we allow ε to change by at most a constant factor on places that we do not always specify.We let χ E denote the indicator function of a set E. We use ≪ and ≫ to denote Vinogradov's notation, while implied constants depend at most on ε and the cusp form f .
The authors wish to thank Harold Diamond and János Pintz for providing access to the articles [4] and [22], respectively.Furthermore, we would also like to thank Olivier Robert for his swift reply clarifying that his result [31,Thm. 1] does not imply that (1/12, 3/4) is an exponent pair.

The zero-detection method
Our starting point is a zero-detection method which has become standard by now.As our further analysis heavily uses the concepts that are introduced by this method, we opt, for the convenience of the reader, to briefly recall here the main ideas involved in this technique.
Let X, Y, T > 1.We consider an approximate inverse for L(s, f ) given by where µ f (n) is the multiplicative function for which This gives where Introducing the weight e −n/Y and exploiting the Mellin inversion formula for e −x , one finds, for 1/2 < Re s < 1, where we picked up the residue at z = 0 while shifting the line of integration Thus, if ρ = β + it is a zero of L(s, f ) with 1/2 < β < 1 and |t| ≤ T , then either (2.1) The zeros ρ with β ≥ σ and |t| ≤ T for which (2.1) holds are referred to as class-I zeros while those for which (2.2) holds are called class-II zeros.As a zero must inevitably belong to one (or both) of these classes we obtain, for 1/2 < σ < 1, where is the set of class-I, resp.class-II zeros, and |R j | denotes their cardinality.For both of these classes we now consider a (saturated) subset Rj of well-spaced zeros; those are subsets of R j for which the imaginary parts of the zeros are well-spaced in the sense that (2.4) where t r are the imaginary parts of the class-II zeros, then we find where we have set X = T ε .We square the above inequality and as the γ r are wellspaced because the class-II zeros are, we may apply 3 Good's second moment estimate [10] to obtain 3 Good's mean value theorem yields The sum version that we use here can be derived from the integral form along the same lines as it is done for the Riemann zeta function, cf.argument in [21, p. 200].Alternatively, one may derive a second moment estimate on L ′ (s, f ) and apply Gallagher's lemma, as is e.g.done in [33].
Therefore, upon choosing Y = T , we obtain | R2 | ≪ T 2(1−σ)+ε and this already concludes the analysis of the class-II zeros.

Representative class-I zeros.
The rest of the argument is then to bound the contribution of the class-I zeros.First we shall restrict the well-spaced class-I zeros even further.By a dyadic subdivision of the interval (X, The elements of R1 that additionally satisfy (2.5) are called representative well-spaced zeros and this subset will be denoted as R. We remark that (2.3) remains valid upon replacing |R 1 | by |R|.
Next, we are going to find some very useful estimates allowing us to bound the size of a set of representative well-spaced zeros in terms of the moduli of certain Dirichlet polynomials.It also turns out that the most problematic range is when M is small; the following argument shall allow us to take care of the range M < T 1/2 such that the critical range for Let ν be a fixed integer and let A be a multiset consisting of elements of R. We shall actually set ν = 1 in the proof of Theorem 1.1 and ν = 2 for Theorem 1.2.We consider an integer power k such that M k ≤ Y ν+ε < M k+1 .Hence, as we have set If we let c ′′ n = 0 after the point where the above maximum is reached, but c ′′ n = c ′ n otherwise, we obtain the last inequality being derived from Cauchy-Schwarz.If we now set b(n) = b(n, A) = ǫc ′′ n for a sufficiently small ǫ such that |b(n)| ≤ 1, we have In particular, if one selects A = R, we get If one were to trivially estimate the right-hand side, one would obtain the bound N 1−σ+ε |R| and this delivers no information at all as it is way worse than the trivial bound |R|.Our goal in the next section is therefore to realize there is indeed sufficient cancellation in (2.7) to extract some non-trivial information.
We do emphasize here again that N and b(n) do depend on the set A. Throughout the rest of the paper, we shall write b(n) and N when we refer to the set R. If any other set of representative well-spaced zeros A is considered, we shall explicitly mention the dependence of b(n) and N on A. On the other hand we note that

Ivić's Estimate
In this section we deduce a first non-trivial estimate on the number of class-I zeros.The first step is to apply the Halász-Montgomery inequality to realize there is cancellation in (2.7).The following lemma is a reformulation of the estimate in [21,Eq. 11.40].We closely follow here the proof of [21,Thm. 11.2].Lemma 3.1.Let A ⊆ R be a set of representative well-spaced class-I zeros (where we do not allow repetition).For ℓ ∈ Z, define Proof.As N ≍ N(A), the estimate (3.1) is equivalent upon replacing N with N(A).
Throughout the rest of the proof, however, we shall simply write N for N(A) in order not to overload the notation unnecessarily.By applying the Halász-Montgomery inequality [28, Lemma 1.7, p. 6] to (2.6), we get where t r , t s denote the imaginary parts of elements of A and We switch the contour to the line Re w = 1/2 which is allowed since ζ is polynomially bounded and Γ decays exponentially on vertical strips.We pass over a simple pole at w = 1 − it with residue O(Ne −|t| ) and our equation becomes The first term on the right hand side is o (|A| 2 ) as the members of A are well-spaced.Moreover, by the definition of ∆ A (ℓ) , we have As σ > 1/2 we arrive at (3.1) after inserting all these estimates in (3.2).
As usual if we do not mention the subscript A for ∆ A when we are referring to A = R.
It thus remains to find adequate estimates for the integral in (3.1).For this we shall appeal to moment estimates on the zeta function.Let B 0 , B 1 , q 0 , q 1 be positive numbers for which q 0 , q 1 ≥ 2, and where χ denotes the characteristic function of a set.In what follows, we rely on an assumption of the form involving mixed moment estimates for the zeta function.

Bourgain's dichotomy
In this section, inspired by the work of Bourgain [3,4], we will use the dichotomy method to improve the estimates for the integral terms appearing in (3.1).This allows us to obtain a new estimate for the class-I zeros.

Lemmas on Dirichlet polynomials.
In applying Bourgain's method, we shall require some preliminary lemmas on estimations for Dirichlet polynomials.The first lemma gives an upper estimate for pointwise values of a Dirichlet polynomial in terms of an average of the values near the point.It is a slight modification of [3,Lemma 4.48].
where the coefficients b n satisfy |b n | ≤ 1.Then, Proof.Let ψ be a smooth function on R such that ψ, the Fourier transform of ψ, is identically 1 on the interval [1,2] and satisfies The existence of such a function ψ is guaranteed by the Denjoy-Carleman theorem.
Let ψ λ (x) = (1/λ)ψ(x/λ).We have, for N ≥ 2, The result now follows upon realizing that |ψ (log The next one is a simple estimate due to Bourgain [3,Lemma 3.4] for Dirichlet polynomials over difference sets where the index sets are different. . The final lemma is Heath-Brown's estimate on double zeta sums [11, Thm.1] (cf.[3, Lemma 3.7]).It is much deeper and is a crucial ingredient for our argument.Lemma 4.3.Let R be a finite set of well-spaced, cf.(2.4), points such that |t| ≤ T for each t ∈ R. Then

The dichotomy.
Let T 0 and R α be as in Section 3, see (3.4).We recall that we have set Let 0 < δ 1 < 1 be a parameter to be optimized later.We set and let B 0 , B 1 , q 0 , q 1 be the parameters that were introduced in the mixed moment estimates (3.3).Note that the definition for ζ 1 is slightly different than in the previous section because of the lower bound |ζ| ≥ 1.For each fixed α, we distinguish between the following alternatives.Case 1.We have Case 2. Either (4.1) or (4.2) fails.
We consider a collection of ⌈2/δ 2 ⌉ sets R α that cover R. We let I 0 be the index set of α for which (4.1) fails, I 1 be the index set for which (4.2) fails, and I 2 be the index set for which both inequalities hold.Clearly |R| ≪ α∈I 0 R α + α∈I 1 R α + α∈I 2 R α .An additional constraint on the parameter δ 1 will arise below in the analysis of Case 2.

The Case 1 contribution.
We suppose here that |R| ≪ where the last term is coming from the contribution of |ζ| < 1. Inserting this estimate in (3.1) and rearranging |R α | gives6 , for σ > 3/4, Replacing T 0 by δ 2 T and summing over the index set I 2 then yields In this section we consider the case when The analysis of the case when |R| ≪ α∈I 1 |R α | is analogous.We have incorporated the extra factor T ε in (4.4) as in some places of the analysis we shall add extra restrictions on the set I 0 and the extra factor T ε shall guarantee that (4.4) remains valid under these restrictions.We write for simplicity q and B instead of q 0 and B 0 .First, we translate the failure of (4.1) and the dominance of the index set I 0 into a lower bound for the size of a specific multiset of representative well-spaced class-I zeros.In this part we shall perform numerous dyadic decompositions and exploit the mixed moment estimate (3.3).Afterwards we apply the analysis of Section 2 to find an upper estimate for this multiset in terms of a Dirichlet polynomial which will subsequently be estimated with the technology provided by Lemma 4.3.The compatibility of this upper and lower estimate shall then result in an improved estimate on |R|.
For 0 < δ ′ < 1 we define the set is an integer and |R α | ≤ T 0 because the points of R α are well-spaced and I α has length at most T 0 , we may through a dyadic argument find A priori δ ′ does depend on α, but as there are only O(log T ) possibilities for δ ′ , the pigeonhole principle asserts that one may select a subset of I 0 , which we shall continue to write as I 0 , for which the above expression holds for a single δ ′ and where (4.4) remains valid, possibly with a different value for ε.In conclusion, the parameter δ ′ can be chosen independent of α.
Exploiting now that (4.1) fails, we obtain Next we proceed to narrow the range for the modulus of ζ 0 .For H > 0, we consider level sets As one can cover the support of We emphasize here that |(D α (δ ′ ) + u) ∩ S H,T 0 | denotes the cardinality of the set (depending on the variable u).Again H a priori depends on α, but as there are only O(log T 0 ) valid choices for H, one may select as before a subset of I 0 , which we keep denoting as I 0 , for which (4.4) remains true.Therefore, we may assume without loss of generality that H is independent of α.Consider follows from the trivial convexity bound for ζ.Of course there are better estimates available.Here is also why we invoked the additional bound |ζ 1 | ≥ 1 in the definition of ζ 1 .This additional restriction guarantees that also the support of ζ 1 can be covered by O(log T 0 ) level sets of the form S H,T0 , which is unclear otherwise.
where m stands for the Lebesgue measure and where we have used the trivial estimate Again δ ′′ can be taken independent of α by an appropriate restriction of the index set I 0 and a constraint on the parameter δ 1 ; in fact, we shall require 8 from now on that δ 1 ≫ T −c for some c > 0. Combining all the above inequalities gives We can derive another lower bound on |D α (δ ′ )|.Namely, the trivial bound Together with (4.5) and (4.4) this implies )) 1/q ≪ T c2 for some c 2 > 0. Going back through the inequalities we also have the lower bound ≫ T −c1 for some c 1 > 0 as log H ≍ log T 0 and because δ 1 shall later be picked in such a way that δ 1 ≫ T −c .Therefore, a dyadic covering for W α only requires O(log T ) intervals and this enables one to pick a restriction of I 0 such that (4.4) remains intact.as Hölder's inequality implies |R| (2q−1)/q ≪ δ −(q−1)/q 2 α∈I 0 |R α | (2q−1)/q and Recalling the definition of W α , we may therefore find |u| < T ε such that the set of integers We keep in mind that we have to multiply by δ ′ ≍ ∆ α (ℓ)/|R α | with ℓ ∈ D α (δ ′ ) to eliminate δ ′ and δ ′′ from the right-hand side by virtue of (4.6).Now that we have established a lower bound for a multiset of class-I zeros, we shift our attention to an upper bound.
We now select A to be the multiset where the multiplicity of a zero ρ is according to how many triples (α, ℓ, t) produce ρ.Therefore, |A| = α∈I 0 ℓ∈S ∆ α (ℓ).We now apply the machinery from Section 2, in particular (2.6), to find a Dirichlet polynomial F A (t) = N (A)<n≤2N (A) b n n −it with bounded coefficients b n such that In the penultimate transition we applied a Cauchy-Schwarz estimate on Lemma 4.1 and in the last step we used that there can only be one t ′ with a given t as the zeros in R α are well-spaced and that the term with 1 may be dropped as it only delivers a contribution of at most T ε α∈I 0 ℓ∈S ∆ α (ℓ) which can never be dominant in view of the estimate on the first line.
The remaining sums and integral are estimated via Lemma 4.2 and Lemma 4.3.This gives .
If we now assume that |R| ≤ N, and use the condition N ≥ T 1/2 , then we find Combining this with the lower bound (4.7) and eliminating δ ′ , δ ′′ through (4.6), we arrive at 2 N 3/2 T 1/4 |R| 5/8 |S| 5/8 )T ε .One of the three terms on the right is dominant.We now wish to eliminate |S|.This can be done as the exponent of |S| for each term on the right-hand side lies between 1/q and 1; note that q ≥ 2. Therefore, for each term on the right-hand side, |S| can be eliminated by an appropriate interpolation of the two left side terms.After a few calculations one obtains Collecting the contributions (4.3) and (4.8) from Cases 1 and 2, we get the following bound for |R|: Lemma 4.4.Let N ≥ T 1/2 and R be a set of representative well-spaced class-I zeros.If B 0 , B 1 > 0 and q 0 , q 1 ≥ 2 are parameters for which (3.3) holds and |R| ≤ N, then for any T −c ≪ δ 1 < 1 (for some c > 0) and T −1 ≤ δ 2 ≤ 1, we have From the analysis of Section 2 it only remains to find an estimate for the representative well-spaced class-I zeros and from Section 3 we may suppose that (3.6) holds and that σ < (3q * − 4)/(4q * − 4B * − 4).We also recall the restriction σ ≥ 1 − q j /(4B j + 4q j − 4), j = 0, 1, we encountered in Section 3. We consider first the case that |R| ≤ N. Applying Lemma 4.4 with admissible parameters q 0 , q 1 , B 0 , B 1 gives Let us choose for T 1/2 < N ≤ T the parameters9 δ 1 and δ 2 in such a way that where (q, B) is the couple (q j , B j ), j = 0, 1, for which δ B−1 2 T B N (3−4σ)q/2 is maximal.This is equivalent to , whence T −c < δ 1 < 1 (with e.g.c = max j=0,1 {7q j /4 + 3B j /2} + 2) and T −1 ≤ δ 2 ≤ 1 in view10 of (3.6).Inserting this choice in the estimate for |R| gives, as N ≥ T 1/2 , and further imposing the restriction σ ≥ 1 − q/(4B + 4q), where the second, third and fourth summand give new restrictions on σ.Summarizing, for j = 0, 1, we obtain the set of constraints provided that also q j > B j + 8/5 which ensures that the denominators in the above fractions are all positive, as otherwise we would not obtain any range for σ.The density hypothesis holds under these restrictions 11 for σ and if |R| ≤ N. Now, suppose that |R| > N. As N ≥ T 1/2 , this implies that |R| > T 1/2 .Select now a subset of representative well-spaced class-I zeros R ′ such that |R ′ | = ⌊T 1/2 ⌋.Now |R ′ | ≤ N and the entire analysis above can be performed for R ′ to give |R ′ | ≪ T 2(1−σ)+ε ≪ T 1/2−ε , if σ > 3/4 say, which is impossible (for large enough T ).Therefore |R| must have been smaller than N to begin with.
As the proof method is very similar to the proof of the density hypothesis in the range σ ≥ 1407/1601 for L(s, f ) discussed in detail in the paper, we only point out the differences.Moreover, since Ivić had already shown Theorem 1.2 when 155/174 ≤ σ ≤ 17/18, we shall only work under the hypothesis 279/314 ≤ σ < 155/174.

Lemma 4 . 2 .
Let a n , b n (1 ≤ n ≤ N) be complex numbers such that |a n | ≤ b n .Let R, S ⊆ R be two finite sets.Then t∈R s∈S N n=1

5 .
A large value estimate.

11 . 8 30σ− 11 ≤ N ≤ T 6 65−84σ • 35−54σ 30σ− 11 .
In view of (A.1) the above estimate is always valid if σ ≥ 155/174 and this concludes Ivić's argument.For the remaining range, we may thus assume(A.2)TThe analysis of Section 4 is mostly analogous, except at the end in the treatment of case 2 where instead of the bound N ≥ T 1/2 we shall use N ≥ T 8 30σ−11 and instead of |R| ≤ N we use the modified T 24(1−σ)/(30σ−11)−ε ≤ |R| ≤ N. (A.3)This results in the bound N 2σ δ ′ α∈I 0 |R α ||S ∩ D α (δ ′ )| 2. The tail of the sum n>Y log2 Y c n n −s e −n/Y is o(1) as Y → ∞.If | Im s| ≤ T , then also the tails | Im z| ≥ log 2 T of the final integral become o(1) as T → ∞ if X is polynomially bounded in T ,say, in view of the exponential decay on vertical lines of the Γ function and the trivial estimate M X (1/2 + iu) ≪ X 1/2+ε .Therefore, for Y, T sufficiently large, we have by as many as O(log T 0 ) level sets of the form S H,T 0 in view 7 of 1 ≤ |ζ 0 (1/2 + it)| ≤ T ), one can find through another dyadic argument a number H such that