Breaking Universality in the Lower Order Terms in the 1-level and 2-level Density of Holomorphic Cusp Newforms
Lawrence Dillon, Xiaoyao Huang, Say-Yeon Kwon, Meiling Laurence, Steven J. Miller, Vishal Muthuvel, Luke Rowen, Pramana Saldin, and Steven Zanetti
TL;DR
This paper investigates the lower-order terms in the 1-level and 2-level density of holomorphic cusp newforms, revealing that these terms depend on the arithmetic of the family and break the expected universality predicted by the Katz-Sarnak conjecture.
Contribution
It extends Miller's work by explicitly computing family-dependent lower-order correction terms in the densities of holomorphic cusp newforms up to a refined error bound.
Findings
Lower-order terms depend on the arithmetic of the family.
Different prime factorizations of the level lead to different lower-order terms.
The results break the universality of the main behavior predicted by the Katz-Sarnak conjecture.
Abstract
The Katz-Sarnak density conjecture states that, as the analytic conductor , the distribution of the normalized low-lying zeros (those near the central point ) converges to the scaling limits of eigenvalues clustered near 1 of subgroups of . There is extensive evidence supporting this conjecture for many families, including the family of holomorphic cusp newforms. Interestingly, there are very few choices for the main term of the limiting behavior. In 2009, S. J. Miller computed lower-order terms for the 1-level density of families of elliptic curve -functions and compared to cuspidal newforms of prime level; while the main terms agreed, the lower order terms depended on the arithmetic of the family. We extend his work by identifying family-dependent lower-order correction terms in the weighted 1-level and 2-level densities of holomorphic cusp newforms up…
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Taxonomy
TopicsAnalytic Number Theory Research · Algebraic Geometry and Number Theory · Advanced Algebra and Geometry
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@tocpagenum#7 .@tocpagenum#7
Breaking Universality in the Lower Order Terms in the 1-level and 2-level Density of Holomorphic Cusp Newforms
Lawrence Dillon
Department of Mathematics, University of Washington, Seattle, WA 98105
,
Xiaoyao Huang
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109
,
Say-Yeon Kwon
Department of Mathematics, Princeton University, Princeton, NJ 08544
,
Meiling Laurence
Department of Mathematics, Yale University, New Haven, CT 06520
,
Steven J. Miller
Department of Mathematics & Statistics, Williams College, Williamstown, MA 01267
,
Vishal Muthuvel
Department of Mathematics, Columbia University, New York, NY 10027
,
Luke Rowen
Department of Mathematics, Carleton College, Northfield, MN 55057
,
Pramana Saldin
Department of Mathematics, University of Wisconsin, Madison, WI 53706
and
Steven Zanetti
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109
Abstract.
The Katz-Sarnak density conjecture states that, as the analytic conductor , the distribution of the normalized low-lying zeros (those near the central point ) converges to the scaling limits of eigenvalues clustered near 1 of subgroups of . There is extensive evidence supporting this conjecture for many families, including the family of holomorphic cusp newforms. Interestingly, there are very few choices for the main term of the limiting behavior. In 2009, S. J. Miller computed lower-order terms for the 1-level density of families of elliptic curve -functions and compared to cuspidal newforms of prime level; while the main terms agreed, the lower order terms depended on the arithmetic of the family. We extend his work by identifying family-dependent lower-order correction terms in the weighted 1-level and 2-level densities of holomorphic cusp newforms up to error, sharpening Miller’s error. We consider cases where the level is prime or when the level is a product of two, not necessarily distinct, primes. We show that the rates at which the prime factors of the level tend to infinity lead to different lower-order terms, breaking the universality of the main behavior.
Date: August 29, 2025.
Acknowledgments: This research was supported by the National Science Foundation under Grant No. DMS-2241623, as well as Columbia University, Princeton University, the University of Michigan, the University of Washington, Williams College, and Yale University.
Contents
1. Introduction
The Katz-Sarnak conjecture states that as we take the limit as the analytic conductor goes to infinity for families of -functions, the zeros lying near the central point agree with the scaling limits of eigenvalues near 1 of a classical compact group ([KS99], [KS99a], [Hej94], [Mon73], [Odl87], [Odl01], [RS96]). Supporting evidence for the conjecture has been gathered for various families, by studying moments of -functions ([CF00], [Con+05], [KS00], [KS00a], [KS03]) as well as by investigating the -level density for suitable test functions ([DM06], [FI03], [Gül05], [HR03], [HM07], [Mil04], [ÖS99], [RR07], [Roy01], [Rub01], [You06]).
Specifically, given an -function associated with a holomorphic cusp newform of level and weight , we assume the Generalized Riemann Hypothesis (GRH).111We do not need the GRH to define any of the statistics described below. However, assuming the GRH allows us to relate the -level density with the spacing between the zeros. Thus, all non-trivial zeros of are of the form . We then enumerate zeros by their imaginary part with by symmetry. We define the -level density by
[TABLE]
where222In general, we do not need the test function to be a product of coordinate-wise test functions. is a Schwartz test function and is the analytic conductor of . We rescale the zeros near the central point by (in all our families of interest, ). The Katz-Sarnak conjecture states that for a family of -functions ordered by their conductors, we have
[TABLE]
where represents the limiting distribution of a similar statistic for the eigenvalues of random matrices in some classical compact group as their rank goes to infinity. While the main terms in the expansions of the -level density have been shown to agree with random matrix theory, the lower order terms break this universality and provide insight in the arithmetic properties of the family and can serve to differntiate families. For example, [You05] analyzed lower order terms arising from families of elliptic curves, revealing that the number of low lying zeros is highly family-dependent (see also [GZ23], [FM15], [FI03a] for investigations of the lower order terms for other families of -functions).
Of particular importance to us, [Mil09] demonstrated how the lower-order terms in the 1-level density can provide insight into the arithmetic properties of the family being studied by looking specificaly at one-parameter families of elliptic curves. Moreover, in his thesis [Mil04] and some subsequent papers ([Mil05], [Mil09]), S. J. Miller noticed that different one parameter families of elliptic curve -functions have different lower order terms for the moments of their Satake parameters, and this has implications for the distribution of zeros near the central point, explaining some of the observed excess rank. The calculations are complicated by the fact that the lower-order terms for the -level densities can be greatly impacted by the small primes. This led him to develop averaging formulas to numerically approximate all the contributions. He proved that while the main terms agree, different families had different lower-order terms. He then compared that to the lower order terms for the family of all cuspidal newforms, for simplicity of prime level N tending to infinity.
We investigate how the lower-order terms of the weighted first-level and second-level density over the space of holomorphic cusp newforms, which are respectively defined by
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
vary depending on how the prime factors of the level of the family approach infinity. By doing this, we break the universality of the main term behavior suggested by the Katz-Sarnak Conjecture.
We consider four different scenarios in which the level approaches infinity.
- •
The first case is going to infinity through primes.
- •
The second case is when for two primes with fixed and
- •
The third case is when for two primes with and where
- •
Lastly, we consider the case when .
Here, we weight each by non-negative weights and define
[TABLE]
Moreover, both of our test functions and are even Schwartz functions whose Fourier transform has compact support.
We write in terms of using inclusion-exclusion.
[TABLE]
where depending on whether the functional equation associated with the completed -function is even or odd. Then, we get
[TABLE]
Whenever there is no ambiguity, we write the family as . We assume that the reader is familiar with standard properties of -functions (see for example [IK04]), and in particular with Satake parameters. We denote the Satake parameters of at by and . The Satake parameters at are related to to the corresponding Hecke eigenvalue, . We have,
[TABLE]
[TABLE]
Our main tool is the explicit formula (see [ILS00] (4.11), for example),
[TABLE]
where
[TABLE]
We define
[TABLE]
so that using (1.12), we can write:
[TABLE]
Expanding (1.13), we have
[TABLE]
In addition, define
[TABLE]
and
[TABLE]
Then we can rewrite and respectively by
[TABLE]
[TABLE]
Therefore, by computing and up to error for arbitrary and even Schwartz functions with compactly supported fourier transforms, we can calculate and up to error.
By doing so, we greatly extend Miller’s work [Mil09], allowing the level to be a product of two, not necessarily distinct, primes. The assumption of prime level greatly simplifies the inclusion-exclusion analysis and resulting formulas, as well as provides explicit formulas for the sign of the functional equation, which allows us to split by sign. When is not prime, we perform more delicate analysis with main ingredients coming from ILS [ILS00] when is square-free, while we use tools from [Bar+16] for general .
While all the families we study share the same main term, they have different lower-order terms, depending on the factorization of the level. We can compute up to errors of size . In all the cases we study, so long as the largest factor of is at least a fixed power smaller than , the lower-order terms agree to this degree of precision with those from the case when the level is prime. On the other hand, the lower-order terms differ when the smallest prime is at most a given size. Therefore, whether or not there is a space of oldforms of comparable size to the newforms turns out to be the reason for the breaking of the universality of behavior.
Through computing the lower order terms for each of the four scenarios of how goes to infinity, we prove that for the case in which is fixed has different lower order terms compared to other cases where both factors go to infinity at rate for some which agree with the prime case. In other words, as long as both factors of the level go to infinity fast enough, the factorization of the level does not affect the lower order terms of first nor second level density up to error. We conjecture that the same phenomenon holds in general.
Conjecture 1.1**.**
Given , as long as the prime factors of go to infinity fast enough relative to the error we are computing up to, the lower order terms of the level density are the same as the prime case. Interesting new factors emerge when one of the factors goes to infinity slower than the reciprocal of the error term, say at the rate of , , or .
In Section 3, we compute up to error. In Section 4, we compute explicitly and where
[TABLE]
extending [Mil09]’s result to improve the error term. See Theorem 4.1 for definitions of and . In Section 5, we compute , and to write
[TABLE]
See Theorem 5.1 for definitions of , and . Then, we have an explicit formula for in terms of and up to error and an explicit formula for in terms of and up to error. Once we have a formula for both and , we start investigating how the level approaches infinity affects the lower-order terms of and . In Section 6, we compute explicitly the terms A^{\prime}_{r,\mathcal{F}}(p),\ A_{r,\mathcal{F}}(p),B^{\prime\prime}_{r_{1},r_{2},\mathcal{F}}(p_{1},p_{2}),\ B^{\prime}_{r_{1},r_{2},\mathcal{F}}(p_{1},p_{2}),\ and which are defined repectively in (4.1),(4.2), (5.3), (5.4), (5.5). If the family we are averaging over is clear, we omit specifying it in a subscript. These quantities determine the value of and for each of the four families we are considering. To this end, we use the Peterson trace formula and [Bar+16]’s generalized formula. Lastly, in Section 7, we substitute the results from Section 6 into the formula of and we obtained in Sections 4 and 5 in order to evaluate the lower order terms.
Computing the 2-level density, as opposed to calculating just the 1-level density, comes with various new obstructions. From doing inclusion-exclusion, the error term of is dictated by the sharpness of the error term in the 1-level density; if we continue to use inclusion-exclusion for a general -level density, the error term will continue to be dictated by the 1-level density. This is because terms occurs with just one factor of while occurs with . Therefore, we need to obtain more refined lower-order terms for 1-level density. Furthermore, as the level increases, the number of terms that need to be considered greatly increases; computing requires far more terms than . In addition, looking at our work with the harmonic weights, new behaviors appear in the moments; there, careful case work is required depending on the factorization of the level. Notably, the case where is nearly identical to with both factors going to infinity, while new challenges and behaviors emerge when one of the factors is fixed.
Remark 1.2**.**
One helpful analogy to our results due to Miller [Mil09] is demonstrated by the proof of the Central Limit Theorem (CLT). Suppose are ‘nice’ independent, identically distributed random variables with mean and variance . Then, the running average converges to the standard normal as . The universality suggested by the CLT is that, modulo normalization, the main terms are independent of the initial distribution. However, the higher moments of the distribution determine the rate of convergence to (See the Berry-Esseen Theorem [Ber41], [Ess42]). We observe a similar phenomenon with the 2-level density. Universally, as suggested by Katz and Sarnak, we know what the main terms should be in the limit as . However, the higher moments of the Fourier coefficients determine the lower-order terms of the 2-level density.
Remark 1.3**.**
Erik Samuelsson [Sam] of the University of Gothenburg is working on a similar problem, but for a fixed level () with the weight going to infinity.
2. Preliminaries
We define the congruence subgroup of by
[TABLE]
We are interested in how congruence subgroup group acts on certain holomorphic functions , where denotes the upper half complex plane. Specifically, we are interested in functions that admit a symmetry when acted on by by the following relation:
[TABLE]
If these functions satisfy the extra stipulation that they vanish at their cusps, we call these functions a holomorphic cusp form of weight and level . Such functions admit a Fourier expansion
[TABLE]
where the coefficients are normalized so that 333Since we are looking at cusp forms, ; for general forms, this is not necessarily true. We denote the space of all holomorphic cusp forms of weight and level by . This space is a finite-dimensional Hilbert space whose inner product is given by
[TABLE]
where . This inner product is called the Petersson inner product.
In 1970, Atkin and Lehner [AL70] built a theory on newforms of . For every form in , we can induce a form in where and . The induced forms are referred to as “oldforms,” while the rest, which form an orthogonal space to the space spanned by oldforms, are called “newforms.” While contains several oldforms, we filter them out and focus on the remaining set of exclusively newforms. These newforms satisfy many desirable properties like being eigenfunctions of the Hecke operators (Hecke eigenforms for short) for all . The Hecke operators are given by
[TABLE]
Moreover, we can find an orthogonal basis of the newforms. See [ILS00] for construction of this basis. Since is finite-dimensional, we know is finite; in fact, from [ILS00a, (2.73)],
[TABLE]
where is Euler’s totient function. Given Hecke eigenform , we refer to its eigenvalue under as the Hecke eigenvalue of , . Thanks to Deligne [Del74], we know that . Moreover, Hecke eigenvalues of are closely related to its Fourier coefficients, satisfying the following relation
[TABLE]
Furthermore, the Hecke eigenvalues of possess useful multiplicative properties,
[TABLE]
which means that if , then
[TABLE]
Using this multiplicative property, we define the -function associated to as:
[TABLE]
For a cusp form , we define the normalized Fourier coefficients by
[TABLE]
with We wish to consider the sum
[TABLE]
where is an orthonormal basis of This sum is computed using the Petersson trace formula [Pet32][IK04].
Proposition 2.1**.**
We have
[TABLE]
where if and is 0 otherwise, denotes the Bessel function, and is the classical Kloosterman sum.
Effective bounding yields the following estimation from [ILS00].
Proposition 2.2**.**
For ,
[TABLE]
where the implied constant is absolute and
Moreover, we define
[TABLE]
We wish to evaluate the arithmetically weighted sum
[TABLE]
Iwaniec, Luo, and Sarank derived the following expression in the case of squarefree.
Proposition 2.3**.**
Let be squarefree, , and . Then
[TABLE]
In addition, Corollary 2.10 in [ILS00] provides the following estimate:
Proposition 2.4**.**
Let be squarefree, , and . Then,
[TABLE]
As Proposition 2.4 requires to be squarefree, for general , we use proposition 4.1 in [Bar+16] which removes the squarefree restriction on .
Proposition 2.5**.**
Let and . Then
[TABLE]
3. Computing
We start the computation of the 1st and 2nd level density by computing up to error.
Theorem 3.1**.**
Let be an even Schwartz function whose Fourier transform has compact support and let be the digamma function. We have the following estimate.
[TABLE]
Proof of Theorem 3.1. We start by proving a lemma.
Lemma 3.2**.**
Let be an even Schwartz function whose Fourier transform has compact support and let be the digamma function. We have the following estimate.
[TABLE]
- Proof.
Using (8.363.3) of [GR65] gives us that the leading term is . Therefore, we subtract the leading term and compute the error term more accurately. Using the fact that is even
[TABLE]
Suppose where is an open subset of . Suppose further that . Taylor expanding about the yields
[TABLE]
for all such that where is the radius of convergence. Moreover, we know from Cauchy’s Inequality that . We apply the formula in (3.4) with , , , and . We remark that and hence . Then, we get
[TABLE]
for all such that .
Now we integrate (3.5) against the test function for and (3.3). Then
[TABLE]
We now show the tail part of the integral is negligible. Consider the series expansion of from [AS72], which holds for all :
[TABLE]
where is the Euler–Mascheroni constant. For we then have that for all :
[TABLE]
We also know that because is Schwartz, for any polynomial . Therefore, using this fact along with (3), we find
[TABLE]
Similarly, we know that
[TABLE]
In addition, we know that
[TABLE]
is because is Schwartz. Therefore, combining (3), (3), and (3.10), and noticing that
[TABLE]
we have that
[TABLE]
as desired. ∎
By the same argument, but for , we get the following lemma.
Lemma 3.3**.**
Let be an even Schwartz function whose Fourier transform has compact support and let be the digamma function. Then
[TABLE]
Using the two above lemmas, we have Theorem 3.1. ∎
4. Computing up to error
Similar to Miller [Mil09], we define
[TABLE]
where . This brings us to the following theorem.
Theorem 4.1**.**
Let
[TABLE]
where is the average log conductor, then
[TABLE]
where
[TABLE]
Proof. Identical to Miller’s analysis in [Mil09], we break the explicit formula from [ILS00] into the case when and and use properties of the Satake parameters to obtain
[TABLE]
The first three sums are already in the desired form, as they can easily be expressed as weighted averages of Hecke eigenvalues. We now look to evaluate the last sum. Through Taylor expansion, we see that
[TABLE]
Thus we have
[TABLE]
Substituting (4.9) into the last sum in (4.7), we get
[TABLE]
Define
[TABLE]
We now write equation (4) as
[TABLE]
Substituting gives
[TABLE]
In Miller’s proof of Theorem 1.1, he shows
[TABLE]
We now compute See Appendix A for the proof.
Lemma 4.2**.**
For prime , we have
[TABLE]
Substituting equation (4.2) into equation (4) yields the main term in (4.1). Now, we look at the error term
[TABLE]
As for all , we have
[TABLE]
Thus we have
[TABLE]
As the sum converges by the comparison test, our error term is sufficient. ∎
5. Computing up to error
We are interested in finding a formula for where
[TABLE]
up to error term.
For fixed we break this sum given by (5) into four cases depending on whether each and divide or not. We freely change the order of summation because convergence is guaranteed by and being Schwartz. Using the fact that if , then , we obtain:
[TABLE]
where Further, we introduce the notations:
[TABLE]
Whenever there is no confusion, we drop the indication of the family , and write , , or . We now compute each of the sums explicitly, using the notations above.
Theorem 5.1**.**
Define , , . Let We have that
[TABLE]
where
[TABLE]
[TABLE]
with the convention that , and for every and prime.
To prove the theorem, we show that the first line of (5) is the sum of the second and third line of (5) equals , and the fourth line of (5) equals . Below, we break the proof into three parts, where each part shows one of the equalities mentioned above.
5.1. First line of (5): ,
This case is simple. Using the notation established above, the first line of (5) is equal to
[TABLE]
5.2. Sum of Second and Third Lines of (5): , and ,
We begin by noting that the sums for and are essential symmetric with the exception of the test function. Therefore, for the second and third line of (5) sum to
[TABLE]
For fixed, we denote by the expression inside of the brackets in (5.11). By symmetry, it suffices to compute () for .
We start by breaking the into three parts, depending on whether or .
[TABLE]
We write where, for example, is the first line in (5.2). Using the definition of in (5.4) and in (4.2), as well as the convention that (for any prime ), we can easily deal with the case when and . We have,
[TABLE]
and
[TABLE]
We now aim to simplify , which is the case when . The purpose of the following lemma is to remove the dependency of in the argument of by arguing that we can replace with at the cost of .
Lemma 5.2**.**
Suppose and are two even Schwartz test functions with Fourier transform supported in . Then,
[TABLE]
- Proof.
By assumption, is even and therefore . Thus, using the Taylor expansion around the origin, we notice that
[TABLE]
Moreover, we know that for every and prime
[TABLE]
Thus, because for , we have that if ,
[TABLE]
Therefore, we get the following estimate:
[TABLE]
where the last line follows using the fact that and the geometric series formula. In fact, the proof is complete because and are convergent sums. ∎
With Lemma 5.2 and using the notation of from (4.11), we have
[TABLE]
We know from [Mil09] that
[TABLE]
Moreover, by observing that , we have that
[TABLE]
Now substituting (5.13), (5.14), and (5.21) for , and respectively, we get that the second line of (5) equals .
5.3. Fourth line of (5): ,
Lastly, we show that up to error, the fourth line of (5) equals .
For fixed primes , let
[TABLE]
The fourth line of (5) is equal to
[TABLE]
We show that the finite part, line (5.23), equals while the infinite part, line (5.24), equals
The former immediately follows from the multiplicativity of Hecke eigenvalues. We know that for all prime , and . For the latter, we need a lemma. See Appendix B for the proof of it.
Lemma 5.3**.**
Suppose and are even Schwartz functions with and having support in . We then have the following estimate:
[TABLE]
Lemma 5.4**.**
Suppose and are even Schwartz functions with and having compact support in . We can estimate every term of (5.24) up to the desired error by replacing by and summing over primes . For example, for the first term, we have
[TABLE]
The proof of Lemma 5.4 follows from the proof of Lemma 5.3. ∎
Now using lemmas 5.3 and 5.4, as well as using formulas for from [Mil09], we get
[TABLE]
Performing a similar substitution, we have
[TABLE]
With this, we can see that is equivalent to (5.24), finishing the proof of Theorem 5.1.
6. Formulas for Family Specific Terms
We use the harmonic weights444The harmonic weights are nearly constant across the family: by [Iwa90], [HL94],[Mil09] we have uniformly for in the family. If we allow ineffective constants, the factor can be replaced by for sufficiently large .There are other ways of normalizations. [Mil09] and our paper averages with . Rouymi [Rou11] uses the Petersson-harmonic weight . Rankin-Selberg shows , so . In addition, using this weight, Rouymi derived formulas for level for fixed prime and . We also note that different weights exhibit different behaviors as seen in [KR19] and [Dil+25]. to facilitate computing explicitly the asymptotic. Recall we are working with the weights
[TABLE]
where and are defined in (2.14) and (2.13) respectively. We consider the family of holomorphic cusp newforms with weight and level . We consider four different scenarios in which the level approaches infinity.
- •
is prime.
- •
for two primes , with and fixed.
- •
for two primes with and where
- •
where is prime
The goal of this section is to compute explicitly the terms and , which determine the value of and . We will find we only see significant differences in the moment when is fixed and
Recall that using the definition of the above terms involve the sum of the weights over the family Therefore, we start by computing for each scenario.
6.1. Sum of the Weights
We compute the sum of the weights for each scenario below. In brief, we do so by first appealing the Peterson trace formula and resulting bounds in [ILS00] in the square-free case and to proposition 4.1 in [Bar+16] for the case. Below, Lemma 6.1 computes for the first three square-free cases while Lemma 6.2 computes it for case.
Lemma 6.1**.**
Let be square-free with being distinct primes with
[TABLE]
Here, while can be both prime constants if , or or prime and prime if . Then,
[TABLE]
where
[TABLE]
Proof. We begin by noticing that , which by Proposition 2.3 ([ILS00] Lemma 2.7) becomes
[TABLE]
Notice that if , then is a prime and we get that
[TABLE]
On the other hand, if , is product of 2 distinct primes; so, we have:
[TABLE]
Using Proposition 2.2 ([ILS00] Cor 2.2), we have that
[TABLE]
where the implied constant is absolute in We first evaluate the two sums that appear in both the prime and being the product of two distinct primes cases, then move to the sums specific to the two prime case.
Case 1:
We see that since
[TABLE]
we have
[TABLE]
for the first case.
Case 2:
We have that
[TABLE]
As if and only if and is 0 otherwise
[TABLE]
Since the implied constant is absolute and is constant, we can combine the sum of the big O terms into one error, and obtain
[TABLE]
We now see
[TABLE]
Thus we have
[TABLE]
We have now exhausted the cases for prime, giving us an error term of We now operate under the assumption that if , then is prime.
Case 3:
Similarly,
[TABLE]
for the same reasons as in Case 2. We now turn our attention to the error term
[TABLE]
Thus we have
[TABLE]
Case 4:
Latly, we have
[TABLE]
We note that this error term requires additional care as when , the method used in Case 3 achieves an error. Thus we write
[TABLE]
For the first sum, since , we have
[TABLE]
The second sum requires more care. Define . We can rewrite the sum as
[TABLE]
Combining these together yields
[TABLE]
Here we note that if , left term dominates, otherwise the right term dominates, thus
[TABLE]
[TABLE]
Combining this with the other error terms and observing that yields the lemma. ∎
While the Peterson trace formula applies for square-free level, we wish to investigate beyond this condition. Thus we appeal to proposition 4.1 in [Bar+16] to obtain the following lemma.
Lemma 6.2**.**
Let for some prime , then
[TABLE]
We have
[TABLE]
By applying Proposition 2.5, we have
[TABLE]
We now use [ILS00] Proposition 2.2 to compute and :
[TABLE]
Similarly,
[TABLE]
Thus, we have
[TABLE]
∎
6.2. Computing Terms
Recall that our extended explicit formula is expressed in terms of weighted average moments of Hecke eigenvalues, namely
[TABLE]
wherein restrictions are placed on relative to the level . We wish to compute these average weighted moments for our four cases: prime, with fixed and distinct from , with both distinct and neither fixed, and lastly . We summarize our results in the following four lemmas.
Remark 6.3**.**
Note that unless otherwise stated, the value of the moments in the following lemmas is 0.
Lemma 6.4**.**
For where is a prime, the following holds:
[TABLE]
Proof of Lemma 6.4. First, we show asymptotics for and We know that if then Therefore, we first determine how behaves. We know that Therefore, we have
[TABLE]
Moreover, we compute for To do this, we use (6.13) and (6.12):
[TABLE]
where is the Catalan number (see sequences in A000108, A000984 in [Slo]). For the last equality, we use the fact that We compute and for various and Because is prime, notice that if prime divides , then . First, notice that
[TABLE]
because is prime, thus square-free, and in this case, for all prime
Secondly, for , we have
[TABLE]
Recall for , by equation 2.4 we have
[TABLE]
Therefore using (6.1), we find
[TABLE]
In addition, note that
[TABLE]
We can compute recursively using the fact that .
Thus, we have that for and
[TABLE]
where is the Catalan number. The second-to-last equality follows because the sums here are finite. ∎
Next, we compute the terms A^{\prime}_{r,\mathcal{F}}(p),\ A_{r,\mathcal{F}}(p),B^{\prime\prime}_{r_{1},r_{2},\mathcal{F}}(p_{1},p_{2}),\ B^{\prime}_{r_{1},r_{2},\mathcal{F}}(p_{1},p_{2}),\ and for the other three cases we consider. The proof of them is similar to that of Lemma 6.4 and can be found in Appendix C.
Lemma 6.5**.**
Suppose where for fixed prime and Then
[TABLE]
Lemma 6.6**.**
Let and with prime and , where Then,
[TABLE]
Lemma 6.7**.**
Let and with Then,
[TABLE]
Remark 6.8**.**
Notice that for the three cases where prime, product of two primes with both factors going to infinity, and square, the terms and do not admit a main term. On the other hand, for where is fixed, and have main terms. Moreover, in all four cases depending on , even up to the error terms, the and terms are equal. This is an important remark to keep in mind because it will save us a lot of computations later.
7. Lower Order Terms
Theorem 7.1**.**
Suppose the test function is an even Schwartz function with supported in for 555The restriction for the support arises in the proof. If is prime or where both and goes to infinity, is negligible, i.e.,
[TABLE]
However, when for fixed we have
[TABLE]
Proof. We compute the term
[TABLE]
from up to the desired error.
**Case 1. ** When is prime, by Lemma 6.4
[TABLE]
Note that in this case is the only prime factor of and hence the sum becomes
[TABLE]
**Case 2. **When with and is fixed, by Lemma 6.5
[TABLE]
Notice that in this case, the main term only comes from when and is even. Therefore, the main term is given by
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The equality follows from the power series expansion of the even function Moreover, the error term is bounded by
[TABLE]
Thus, we get that in this case,
[TABLE]
Case 3. When and , by Lemma 6.6 we have
[TABLE]
Then, the term becomes an error term because
[TABLE]
∎
Theorem 7.2**.**
Suppose the test function is an even Schwartz function with supported in for . For all our cases, we have
[TABLE]
where
[TABLE]
and where we let and define be the error.
Proof of Theorem 7.2, We note that, regardless of factorization, we have
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Thus we have
[TABLE]
Since
[TABLE]
the error terms for all the finite sums in Theorem (4.1) that are multiplied by a Schwartz function are sufficiently small and we have that the sums equal
[TABLE]
We now deal with the terms
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In the appendix D, we estimate many sums up to our required error, which we now employ to compute these expressions. In many cases, the error terms are left with convergent but non-elementary integrals that must be estimated numerically. Since the main terms for and agree and is an error, using Lemma D.1, we have
[TABLE]
as the main term for these two sums. We now show the error terms in these cases are sufficiently small. Using compact support, we have our error terms as
[TABLE]
which for restricted support are all sufficient. We now evaluate the last two sums. Using (7.7) and (7.8), we have
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We now aim to evaluate
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We first show the tail is negligible. We see
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Further,
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Thus the tail is negligible. We now show the error term that arises from in the truncated sum is also negligible. Using the same asymptotic for the rational function in the sum, we have that the error is
[TABLE]
which is negligible for our support. Thus, we have
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as our main term. This sum can be extended to infinity at the cost of , which is negligible. Thus, we have shown
[TABLE]
Substituting our constants in yields the theorem. ∎
Theorem 7.3**.**
Suppose the test functions are even Schwartz functions with supported in for . If is prime, where both and goes to infinity, or then is negligible, i.e.,
[TABLE]
However, when for fixed we find that
[TABLE]
Theorem 7.4**.**
Suppose the test functions are even Schwartz function with supported in for . If is prime, where both and goes to infinity, or then is negligible, i.e.
[TABLE]
However, when for fixed we have
[TABLE]
where
[TABLE]
For , we start by noticing that the expressions for and are the same up to error terms for the families with prime level or families with level being two distinct primes. The error term only differs by an exponent on , making the computations more or less the same.
First, we compute the main terms; the computation is exactly the same for all three families.
Theorem 7.5**.**
Suppose the test functions are even Schwartz function with supported in for . Define , with the second derivative of . Then for all of our cases
[TABLE]
where
[TABLE]
Theorem 7.6**.**
Suppose the test functions are even Schwartz functions with supported in for , and with the second derivative of . Then for all of our cases,
[TABLE]
where
[TABLE]
Proof Sketch of Theorems 7.3, 7.4,7.5, and 7.6.
The above theorems follow from substituting the terms and obtained from Lemmas 6.4, 6.5, 6.6, and 6.7 for each case into the formulas for and given in Theorem 5.1.
There are two parts to the proof of the theorem. The first is evaluating the main terms from merely plugging in what we’ve already proved. For example, one of the main terms that come up in is, after plugging in our results,
[TABLE]
Notice that the substitution still has a sum over of evaluated at . The general procedure is as follows: to move the dependence on outside of the sum, we substitute its power series expansion and apply the Prime Number Theorem to compute the sum associated to the constant term. The other sums are generally error terms, since they involve larger powers of in the denominator. Often, we use techniques involving generating functions to simplify our expressions.
In the example (7.20), we find that the expression is equal to, up to error,
[TABLE]
As mentioned above, the remaining lemmas we use to get rid of the dependency on the test function as well as to simplify evaluating the main terms can be found in Appendix D.
The second part of the proof is to argue that the error terms of terms and obtained from Lemmas 6.4, 6.5, 6.6, and 6.7 become error terms in computing and as well. Below are the two main lemmas we use to bound the errors.
Lemma 7.7**.**
Suppose is a fixed positive integer. Then,
[TABLE]
Proof of Lemma 7.7. Note that
[TABLE]
In the second to last line above, we can bound because
. ∎
Lemma 7.8**.**
Suppose is any fixed positive integer.
[TABLE]
- Proof.
Observe that:
[TABLE]
Again, it is crucial that . The restriction for the support () comes from the fact that and that . ∎
Every error term from those of and obtained from Lemmas 6.4, 6.5, 6.6, and 6.7 is in big O notation where the implied constant is absolute in Therefore, we can bring in all the sum into the big O notation. Moreover, every error term has some infinite sum over or over Whenever we have this sum in , we break the into two parts, into cases when and when In the first case, when , we bound the error using the terms given from Lemmas 6.4, 6.5, 6.6, and 6.7 then apply Lemma 7.7. On the other hand, in the case where we use the bound for all and primes then apply Lemma 7.8.
For example, when we are dealing with the term
[TABLE]
appearing in the computation of , the error term is
[TABLE]
For the first sum, we use bounds from Lemmas 6.4, 6.5, 6.6, and 6.7 then apply Lemma 7.7:
[TABLE]
This error is negligable as long as which we assumed.
For the second sum, we bound then apply Lemma 7.8:
[TABLE]
which is negligible as long as which we assume.
Appendix A Proof of Lemma 4.2
Lemma 4.2. *For prime , we have *
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Proof of Lemma. We recall . We note that since
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We have
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Thus,
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We see that by multiplying by we can write
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and we obtain a similar result for the corresponding term. We now aim to combine
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The denominator becomes
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We note we are leaving the aforementioned expression inside the cube. Thus we have
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Looking at the first term, we express the denominator as and use the power series to obtain
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Thus we get
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which is equal to
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Combining like terms yields
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Plugging this back into equation (A.5) we get
[TABLE]
∎
Appendix B Proof of Lemma 5.3
Lemma 5.3 *Suppose and are even Schwartz function with and having support in We then have the following estimate: *
[TABLE]
Proof of Lemma. Let us write
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Notice that the statement of the lemma is equivalent to saying that . We first show that , then argue that
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The first approximation follows from using Taylor expansion, bounding by 2, and finally observing that and Therefore, we have that .
Now we show that . By a similar approximation,
[TABLE]
With this, the proof of Lemma 5.3 is complete. ∎
Appendix C Proof of Lemmas 6.5, 6.6, and 6.7
In this section, we prove the Lemmas 6.5, 6.6, and 6.7.
*Proof of Lemma 6.5. * The proofs for and are identical to the prime case so we omit it. First we look at . There are two cases to consider: when and when If and is even, then using the fact that , we have that If is odd, using Corollary 2.10 of [ILS00] and substituting for we found in lemma (6.5), we have:
[TABLE]
Moreover, when using triangle inequality and using the fact that for all ,
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Next, we consider . There are four pairs possible for They are: and .
In the first case, where we have
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In the last three cases, with at least one of or being we have:
[TABLE]
Next, we look at By definition, and If we see
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If is even, it follows that
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If is odd, Corollary 2.10 of [ILS00] yields
[TABLE]
Next, suppose we have that Then, trivially bounding by gives:
[TABLE]
completing the proof. ∎
Next, we prove Lemma 6.6.
Proof of Lemma 6.6. The proofs of and is identical to the prime case so we omit it. We can see
[TABLE]
Since , this gives
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We now look at . We see we must have , and . Using the fact that for , we have
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For , we have . We can write
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Using the triangle inequality and the fact that , we find
[TABLE]
∎
Lastly, we look at the proof of Lemma 6.7. Proof of Lemma 6.7. First, we show that , and for are 0. By definition, for , we have that the sum is empty, so . For , by [Bar+16] (2.11), we have for a prime,
[TABLE]
Since the Hecke eigenvalue terms are zero. Thus, for all values of and . Similarly for , for , the sum is empty, so . For , we again have that the Hecke eigenvalue term is zero. Thus, for all values of and . The same reasoning applies to show that for all primes .
Next, we compute . For , which implies , the sum is empty, so . Therefore, assume that . We then have
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Lastly, we compute . By definition,
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For , which implies and equal , the sum is empty, so . Therefore, assume that . We then have
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We now apply [Bar+16] Proposition 4.1, which holds as and are not equal . Using the fact that ,
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By [ILS00] Proposition 2.2,
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and
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After multiplying by and by , the error term associated with absorbs the error terms associated with . As such, we may write
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Dividing by , we have
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Trivial estimation and the fact that gives us
[TABLE]
yielding the lemma. ∎
Appendix D Lemmas Used for Main Terms in Theorems 7.3, 7.4, 7.5, and 7.6.
In this section, we state and prove various lemmas used to compute the main Terms in Theorems 7.3, 7.4, 7.5, and 7.6.
Lemma D.1**.**
Let be a compactly supported even Schwartz test function. As in [You05], define
[TABLE]
Then
[TABLE]
- Proof.
By Abel summation
[TABLE]
Define . Then
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Write as . The first term in (D.2) is
[TABLE]
[TABLE]
The second term in (D.2) is
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Because is even, the Taylor expansions are
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Thus, the second term in (D.2) becomes
[TABLE]
Combining the terms in (D.5) and (D.8) yields the lemma. ∎
Lemma D.2**.**
Let be a compactly supported even Schwartz test function. Then
[TABLE]
- Proof.
By the Convolution Theorem, we have
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Define
[TABLE]
[TABLE]
By Abel summation, we write
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Differentiate with respect to . Then we change the variable and calculate (D.10)
[TABLE]
[TABLE]
Write as , by change of variable, the first term in (D.12) is
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Since is even. The Taylor expansions are
[TABLE]
Substituting (D.14) into yields
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The second term is then equal to
[TABLE]
Combining (D.15) and (D.16) yields the lemma.
∎
Lemma D.3**.**
Let , be even Schwartz functions with and compactly supported. Define
[TABLE]
then we have
[TABLE]
- Proof.
As before, we appeal to the convolution theorem, writing so that
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Recall that we define and let
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Now, by the Abel summation formula and the fact that is bounded,
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To ease notation, substitute and differentiate to get
[TABLE]
[TABLE]
Splitting the error in we first consider the main term. Substituting in again gives
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Now, using the evenness of , we Taylor expand and find
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and note that and Thus, the main term is, after bringing in the factor of ,
[TABLE]
We now compute the error term
[TABLE]
Since we also have from evenness that
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we expand each of and to see that the above is equal to
[TABLE]
∎
Lemma D.4**.**
Let be prime. For and prime, we have
[TABLE]
- Proof.
Define
[TABLE]
By Abel summation
[TABLE]
After calculating the main terms, we have
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By the Prime Number Theorem, E(t)=\theta(t)-t=O\bigl{(}t/\log(t)\bigr{)}, substituting in (D.28) yields
[TABLE]
The constants are absorbed into the big O term, gives
[TABLE]
∎
Lemma D.5**.**
Let be an even Schwartz function, then
[TABLE]
- Proof.
As in [Slo], the generating function for the Catalan numbers can be written in closed-form for :
[TABLE]
Letting , we note that
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so we always have convergence to the above formula. Factoring and subtracting off the first two terms,
[TABLE]
Since
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we have, using ,
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We multiply through by the extra factors to get
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Furthering our effort to simplify the bracket in the sum, note that
[TABLE]
hence, the sum becomes
[TABLE]
which is amenable to our techniques involving the Prime Number Theorem. Let and apply the Abel summation formula with the sequence to get the main term
[TABLE]
arising from Writing and plugging back into the integral, the main term becomes
[TABLE]
since We proceed to compute the error term. We have that the error is
[TABLE]
Now,
[TABLE]
so the error term is
[TABLE]
∎
Lemma D.6**.**
Let be an even Schwartz function, then
[TABLE]
where quantities and are defined explicitly in the proof below.
- Proof.
Let so that by our standard tricks, we have
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Integrating the first term by parts and letting , it suffices to compute
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Expanding using evenness, this gives that the main term is
[TABLE]
where we let
[TABLE]
to later be integrated numerically. We now compute the error term in (D.45). Differentiating, we have
[TABLE]
where
[TABLE]
Using evenness to expand and to compute the integral
[TABLE]
we see that the first term is absorbed into the error. Considering the second term, only the constant term in the expansion of matters for us, yielding an error term of
[TABLE]
∎
Lemma D.7**.**
Let be even Schwartz function. Then
[TABLE]
- Proof.
Rearranging
[TABLE]
We apply Lemma D.1 to the sum over , getting that the above equals
[TABLE]
where we can compute up to numerically, leaving only the consideration of the sum over We apply the standard method of Abel summation, setting so that
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Integrating by parts, the first term gives a contribution of
[TABLE]
Since
[TABLE]
the error term in (D.57)
[TABLE]
which is a constant that we can compute numerically. Hence, after substituting in our expression for from Lemma D.1, we have in total
[TABLE]
Let
[TABLE]
Substituting (D.60) and (D.62) into (D.61) yeilds the lemma. ∎
Lemma D.8**.**
Let be even Schwartz function. Then
[TABLE]
where
- Proof.
We begin the same as in Lemma D.7, obtaining
[TABLE]
We define so that by the Prime Number Theorem it follows that
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The first term is
[TABLE]
Since
[TABLE]
the second term is
[TABLE]
Hence, applying Lemma D.1 to get the value of , we obtain
[TABLE]
where ∎
Lemma D.9**.**
Let be even Schwartz function. Then
[TABLE]
- Proof.
Applying the same argument as in the previous computations and the fact that
[TABLE]
it follows that the sum equals
[TABLE]
By the Prime Nnumber Theorem, letting gives
[TABLE]
The first term is
[TABLE]
For the second term, note that
[TABLE]
so that
[TABLE]
Hence, (D.73) is equal to
[TABLE]
Substituting into (D.72), the sum is
[TABLE]
∎
Lemma D.10**.**
Let be even Schwartz function. Then
[TABLE]
- Proof.
We first simplify the sum over . Let
[TABLE]
and
[TABLE]
where Since and , it follows that
[TABLE]
Plug in , noting that to get that
[TABLE]
and thus
[TABLE]
Multiplying through by what we factored out, we get
[TABLE]
The rest of the argument is analogous to (D.6). We define
[TABLE]
where
[TABLE]
Hence, we can rewrite the sum we want as
[TABLE]
Letting ,
[TABLE]
This is equal to
[TABLE]
We now consider the error term. We have that
[TABLE]
where , , and are polynomials. Expanding and , the main term doesn’t contribute, so the error is
[TABLE]
which is explicitly
[TABLE]
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