# Band-Mott mixing hybridizes the gap in Fe2Mo3O8

K. Park  These authors contributed equally to this work  Department of
Chemistry, University of Tennessee, Knoxville, Tennessee 37996, USA  G. L.
Pascut  These authors contributed equally to this work  MANSiD Research Center
and Faculty of Forestry, Applied Ecology Laboratory, Stefan Cel Mare
University (USV), 13 University Rd, Suceava 720229, Romania  G. Khanal
Department of Physics and Astronomy, Rutgers University, Piscataway, New
Jersey 08854, USA  M. O. Yokosuk  Department of Chemistry, University of
Tennessee, Knoxville, Tennessee 37996, USA  Xianghan Xu  Department of Physics
and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA  Bin Gao
Department of Physics and Astronomy, Rice University, Houston, Texas 77005,
USA  M. J. Gutmann  ISIS Facility, STFC-Rutherford Appleton Laboratory, Didcot
OX11 OQX, United Kingdom  A. P. Litvinchuk  Texas Center for Superconductivity
and Department of Physics, University of Houston, Houston, Texas 77204, USA
V. Kiryukhin  Department of Physics and Astronomy, Rutgers University,
Piscataway, New Jersey 08854, USA  Rutgers Center for Emergent Materials,
Rutgers University, Piscataway, New Jersey 08854, USA  S. -W. Cheong
Department of Physics and Astronomy, Rutgers University, Piscataway, New
Jersey 08854, USA  Rutgers Center for Emergent Materials, Rutgers University,
Piscataway, New Jersey 08854, USA  Laboratory for Pohang Emergent Materials
and Max Planck POSTECH Center for Complex Phase Materials, Pohang University
of Science and Technology, Pohang 790-784, Korea  D. Vanderbilt  Department of
Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
K. Haule  Department of Physics and Astronomy, Rutgers University, Piscataway,
New Jersey 08854, USA  J. L. Musfeldt<EMAIL_ADDRESS>Department of
Chemistry, University of Tennessee, Knoxville, Tennessee 37996, USA
Department of Physics and Astronomy, University of Tennessee, Knoxville,
Tennessee 37996, USA

###### Abstract

We combined optical spectroscopy and first principles electronic structure
calculations to reveal the charge gap in the polar magnet Fe2Mo3O8. Iron
occupation on the octahedral site draws the gap strongly downward compared to
the Zn parent compound, and subsequent occupation of the tetrahedral site
creates a narrow resonance near the Fermi energy that draws the gap downward
even further. This resonance is a many-body effect that emanates from a flat
valence band in a Mott-like state due to screening of the local moment -
similar to expectations for a Zhang-Rice singlet, except that here, it appears
in a semi-conductor. We discuss the unusual hybridization in terms of orbital
occupation and character as well as the structure-property relationships that
can be unveiled in various metal-substituted systems (Ni, Mn, Co, Zn).

## Introduction

3$d$-containing materials are well known for strong electron correlations,
narrow band widths, site- and orbital-selective states, and robust magnetism
whereas 4- and 5$d$ systems are recognized for strong spin–orbit coupling,
increased hybridization, extended orbitals, and a tendency toward dimerization
Streltsov and Khomskii (2017). Combining these qualities in mixed metal
materials leads to a variety of unexpected properties. Examples include
interpenetrating sublattices with independent spin dynamics and ground states
in Sr2CoOsO6 Morrow  _et al._ (2013); Yan  _et al._ (2014), self-healing
photoelectrode materials like CuRhO2 Gu  _et al._ (2014), covalency-driven
collapse of spin-orbit coupling in Ba5CuIr3O12 Ye  _et al._ (2018), an ultra-
high coercive field in Sr3NiIrO6 Singleton  _et al._ (2016); O’Neal  _et al._
(2019), magnetoelectric coupling in Co4Nb2O9 Khanh  _et al._ (2016),
surprising spin entropy effects across the magnetic quantum phase transition
in CoNb2O6 Liang  _et al._ (2015), and nonreciprocal directional dichroism in
Ni3TeO6 Yokosuk  _et al._ (2020). Another mixed metal system with exciting
properties and curious hybridization is Fe2Mo3O8 \- also known as the mineral
Kamiokite Inosov (2018). While magnetism and magnetoelectric coupling have
been widely studied Sheckelton  _et al._ (2012); Mourigal  _et al._ (2014);
Wang  _et al._ (2015); Kurumaji  _et al._ (2015); Li  _et al._ (2017); Chen
(2018); Solovyev and Streltsov (2019); Nikolaev  _et al._ (2021), the charge
excitations are highly under-explored.

Fe2Mo3O8 is a polar magnet with giant magnetoelectric coupling, strong
Dzyaloshinski-Moriya interactions, valence bond condensation (creating a
cluster magnet), and the possibility of orbitally-selective transitions
Sheckelton  _et al._ (2012); Mourigal  _et al._ (2014); Wang  _et al._ (2015);
Kurumaji  _et al._ (2015); Li  _et al._ (2017); Chen (2018); Solovyev and
Streltsov (2019); Nikolaev  _et al._ (2021). Zinc substitution, first on the
tetrahedral Fe site and then on the octahedral Fe site Kurumaji  _et al._
(2015); Streltsov  _et al._ (2019), is of interest for magnetic properties as
well Nakayama  _et al._ (2011); Mourigal  _et al._ (2014); Inosov (2018);
Streltsov  _et al._ (2019). The structure of Fe2Mo3O8 consists of corner-
shared tetrahedral and octahedral sites separated by layers of Mo trimers
[Fig. 1(a,b)] McCarroll  _et al._ (1957); Ansell and Katz (1966); Sheckelton
_et al._ (2012). The FeO4 tetrahedron is significantly elongated and
distorted, and the FeO6 octahedron is trigonally distorted as well, leading to
a C3v point group on both tetrahedral and octahedral Fe sites. As a result,
Fe2Mo3O8 has no inversion symmetry. The system has a 61 K magnetic ordering
transition to a collinear antiferromagnetic state with a concomitant
structural distortion Varret  _et al._ (1972); Czeskleba  _et al._ (1972);
Wang  _et al._ (2015); Stanislavchuk  _et al._ (2020); Reschke  _et al._
(2020). Antiferromagnetic antiphase domain boundaries have been imaged in this
state Kim  _et al._ (2018). Fe2Mo3O8 also displays a 5 T transition to the
ferrimagnetic state with an extremely large magnetoelectric coefficient Wang
_et al._ (2015); Kurumaji  _et al._ (2015). Ni2Mo3O8 also hosts robust
magnetoelectric coupling with a field-tunable coupling mechanism Tang  _et
al._ (2021). Spectroscopic highlights in Fe2Mo3O8 include (i) nonreciprocal
directional dichroism Yu  _et al._ (2018), phonon trends across $T_{\rm N}$
Stanislavchuk  _et al._ (2020); Reschke  _et al._ (2020), and a variety of
magnetic excitations in the terahertz range Csizi  _et al._ (2020), (ii)
Mössbauer to confirm the 2+ charge on the iron site Varret  _et al._ (1972);
Czeskleba  _et al._ (1972), and (iii) studies of charge transfer via time-
dependent optical Kerr effects Sheu  _et al._ (2019) complemented by first
principles electronic structure calculations Biswas  _et al._ (2017);
Streltsov  _et al._ (2019); Reschke  _et al._ (2020).

In order to place the charge excitations on a firm foundation, we measured the
optical properties of the $A_{2}$Mo3O8 family of materials (where $A$ = Fe,
Ni, Mn, Zn) and compared our findings with complementary electronic structure
calculations. We show that the 1.7 eV gap in Zn2Mo3O8 is determined by the
charge excitations of the Mo trimer. Replacing Zn on the octahedral site with
Fe yields FeZnMo3O8. This system has a substantially reduced and renormalized
gap determined by Fe-O hybridized bands that appear due to the periodic
lattice potential. Further substitution yields Fe2Mo3O8 which has both
trigonally-distorted octahedral and tetrahedral sites occupied by Fe atoms,
and the gap is further reduced to 1.0 eV. Here, the charge gap is more complex
to describe because it has mixed band and Mott features. In other words, some
orbitals hybridize strongly with oxygen and form very narrow bands whereas
other orbitals exhibit real space localization and are Mott insulating. Mixed
band and Mott gaps are commonly called orbitally- or site-selective Mott
states Chen  _et al._ (2020); Pascut and Haule (2020); Lichtenstein  _et al._
(2001); Anisimov  _et al._ (2002); de’Medici  _et al._ (2005); Haule and
Pascut (2017). What distinguishes Fe2Mo3O8 from other orbitally-selective Mott
systems is the narrow many-body resonance emanating from the edge of the flat
valence band. The Kondo effect is, of course, normally studied in metals. In
this work, we show that the Kondo effect can also appear in mixed metal
semiconductors. In addition, the gap in Fe2Mo3O8 is sensitive to magnetic
ordering at 61 K due to the heavily mixed character of the charge excitations.
Moreover, the $d$-to-$d$ excitations on the distorted octahedral Fe site are
vibronically activated, and spin-orbit related features ride on top of the
distorted tetrahedral on-site excitations below the magnetic ordering
transition. We discuss these findings in terms of band-Mott mixing in 3- and
4$d$-containing quantum cluster magnets.

Figure 1:  (a) Crystal structure of the $A_{2}$Mo3O8 compounds where $A$ = Mn,
Fe, Co, Ni, Zn. $A$(T) and $A$(O) represent ions in trigonally-distorted
tetrahedral and octahedral environments, respectively. (b) Schematic view of
the Mo trimer. (c-e) Absorption spectra of Zn2Mo3O8, FeZnMo3O8, and Fe2Mo3O8
measured at room temperature. (f) A Tauc plot reveals the direct band gap of
Zn2Mo3O8 and the Fe substituted analogs. (g) Band gap schematic showing the
impact of two different types of T and O-site substitution. The upper and
lower trend lines correspond to the (Fe,Zn)2Mo3O8 series and the $A_{2}$Mo3O8
($A$ = Fe, Mn, Ni) materials, respectively. (h,i) Calculated optical
conductivity of the $A_{2}$Mo3O8 materials.

## Methods

High quality single crystals of Fe2Mo3O8, the Zn-substituted analogs
FeZnMo3O8, and Zn2Mo3O8, as well as Mn2Mo3O8 and Ni2Mo3O8 were grown by
chemical vapor transport as discussed previously Wang  _et al._ (2015).
Special care was taken to assure the stoichiometry of FeZnMo3O8. Crystals were
polished to control optical density and expose the hexagonal face. A Bruker 55
Fourier transform infrared spectrometer equipped with a microscope attachment
was used to measure transmittance over the 0.41 - 2.0 eV energy range.
Absorption was calculated as $\alpha(E)=-\frac{1}{d}{\rm
ln}({\mathcal{T}}(E$)), where ${\mathcal{T}}$($E$) is the transmittance and
_d_ is the thickness. Performing these measurements in transmittance rather
than reflectance avoids light leakage problems. Temperature was controlled by
an open-flow cryostat.

For theoretical calculations we used the density functional theory (DFT) as
implemented in WIEN2k Blaha  _et al._ (2019) and a charge-self-consistent
dynamical mean field theory (DMFT) as implemented in the eDMFT code Haule
(2020, 2018). At the DFT level, we used the generalized gradient approximation
Perdew-Burke-Ernzerhof (GGA-PBE) functional Perdew  _et al._ (1996), with
RKmax = 7.0 and 312 k-points in the irreducible part of the 1st Brillouin
zone. At the eDMFT level, we used the fully rotationally invariant Coulomb
interaction, a nominal double counting scheme Haule (2015), with the
$d$-orbital occupations for double counting corrections for Mn, Fe, Co and Ni
set to be 5, 6, 7 and 8, respectively. The temperature is fixed at $500\,$K.
To define the DMFT projector, we used quasi-atomic orbitals by projecting
bands in a large hybridization window (-10 to +10 eV) with respect to the
Fermi level, in which partially screened Coulomb interactions have values of U
= 10 eV and JH = 1 eV in Mn, Fe, Co and Ni ions. In order to solve the
auxiliary quantum impurity problem, a continuous-time quantum Monte Carlo
method in the hybridization-expansion (CT-HYB) was used Haule (2007), where
the five $d$ orbitals for the Mn, Fe, Co and Ni ions (grouped according to the
local C3v point group symmetry) were chosen as our correlated subspace in a
single-site DMFT approximation. For the CT-HYB calculations, up to 108 Monte
Carlo steps were employed for each Monte Carlo run. The self-energy on the
real axis was obtained using the analytical continuation maximum entropy
method for the local cumulant as explained in Haule . During the calculation,
the position of the chemical potential was kept fixed within the gap. The
experimental crystal structures used for our computations Stanislavchuk  _et
al._ (2020) as well as details of our calculations are given in the
Supplementary information Sup .

## Results and Discussion

### Optical response of Fe2Mo3O8 and the $A$-substituted analogs ($A$ = Zn,
Mn, Ni)

Figure 1(c-e) summarizes the optical properties of the (Fe,Zn)Mo3O8 family of
materials. The absorption spectrum of the parent compound, Zn2Mo3O8, is low
and flat in the near infrared, rising on approach to the O 2$p$ $\rightarrow$
Mo 3$d$ charge transfer excitation. The direct band gap is 1.75 eV. Because
Zn2+ has a $d^{10}$ configuration, there are no $d$-to-$d$ on-site
excitations. Zn2Mo3O8 therefore provides an opportunity to study how the Mo
trimer interacts with oxygen in isolation. At the same time, it is an
important scaffold upon which additional complexity can be built.

Sequential $A$-site substitution of Fe, first into the distorted octahedral
site in FeZnMo3O8, here denoted as Fe(O), and then into the distorted
tetrahedral site in Fe2Mo3O8, henceforth Fe(T), lowers the charge gap
significantly [Fig. 1(f)]. We find direct gaps of 1.2 and 1.0 eV for FeZnMo3O8
and Fe2Mo3O8, respectively. The gap values were determined from Tauc plots of
(${\alpha}{\cdot}E$)2 vs. energy Pankove (2010).

Figure 2: (a, d) Tauc plot of (${\alpha}{\cdot}E$)2 vs. energy for Fe2Mo3O8
and a plot of band gap vs. temperature with a fit to the Varshni model. (b, e)
Close-up view of the Fe2+ $d$-to-$d$ on-site excitation on the tetrahedral
site showing the fine structure that develops due to spin-orbit coupling below
the 61 K magnetic ordering transition. (c, f) Close-up view of the Fe2+
$d$-to-$d$ on-site excitation on the octahedral site and oscillator strength
vs. temperature along with a fit to a modified vibronic coupling model
Ballhausen (1962); Stoneham (2001); O’Neal  _et al._ (2017).

Figure 1(d) displays the absorption of FeZnMo3O8. The charge excitations
across the gap consist of mixed O 2$p$ \+ Mo 3$d$ \+ Fe(O) 3$d$ transitions.
The lowest energy excitation across this gap comes from Fe(O) hybridizing with
Mo-O trimers. The 1.2 eV gap is substantially lower in energy than the
fundamental Mo-O band gap, which theoretically remains roughly equal to that
in Zn2Mo3O8. As Fe2+ populates the octahedral site in FeZnMo3O8, an on-site
$d$-to-$d$ transition arises near 0.95 eV. It overlaps strongly with the
leading edge of the charge transfer band and is activated by vibronic coupling
[Fig. 2(c,f)] Ballhausen (1962); Stoneham (2001); O’Neal  _et al._ (2017).
Notice that absorption is low and flat near 0.5 eV - a sign of crystal quality
and stoichiometry. Once the Fe(T) site is populated as well (as in Fe2Mo3O8),
the gap is reduced further, and two different types of $d$-to-$d$ on-site
excitations are identified inside the charge gap. As shown in Fig. 1(e), Fe on
the trigonally-distorted tetrahedral site contributes additional atomic-like
excitations centered at 0.5 eV. While oscillator strength is fully conserved
as a function of temperature [Fig. 2(e)], a great deal of fine structure due
to spin-orbit coupling rides on top of the band below the 61 K magnetic
ordering transition [Fig. 2(b)]. A similar response develops in Fe2+:ZnSe
Evans  _et al._ (2017). The behavior of the Fe2+ on-site $d$-to-$d$
excitations in Fe2Mo3O8 is summarized in Fig. 2(b,c,e,f) and further discussed
in Supplementary information Sup . The full sequence of gap values is shown
schematically in Fig. 1(g).

To test the influence of $A$-site substitution on the band gap and strength of
the metal to MoO-trimer hybridization, we measured the optical properties of
the Mn and Ni analogs of Fe2Mo3O8 [Fig. S2, Supplementary information] Sup .
Mn2Mo3O8 and Ni2Mo3O8 have charge gaps of 1.65 and 1.7 eV, respectively - very
similar to that of the Zn end member. This result is attributable to 3$d$
orbital filling and character. The Mn system has a half-filled $d$-manifold,
which corresponds to a high spin Mott state with a large gap. The $d^{8}$
configuration in the Ni analog also has two holes and is thus in the large gap
Mott insulating state. As a result, there is little mixing between the metal
center and Mo-O trimer, hence these transition metals do not play an active
role in determining the low-energy excitations across the gap. We mention in
passing that the Mn and Ni compounds have on-site $d$-to-$d$ excitations as
well [Fig. S2, Supplementary information] Sup .

On the other hand, Fe2Mo3O8 has both Mott-type and band-insulating orbitals,
which strongly hybridize with the Mo trimer. These interactions enable the Fe
centers to control the low energy physics as discussed below. Moreover, we
find that the band gap of Fe2Mo3O8 is sensitive to the 61 K magnetic ordering
transition and the associated structural distortion [Fig. 2(a,d)]. This is
different from the other $A_{2}$Mo3O8 compounds, where the temperature
dependence of the band gaps are in excellent overall agreement with the
Varshni model Sarswat and Free (2012); O’Donnell and Chen (1991) implying no
(or extremely subtle) structural aspects to the magnetic ordering transitions.
That the band gap decreases across $T_{\rm N}$ is due to coupling of charge,
structure, and magnetism and the flat bands emanating from the trigonally-
distorted tetrahedral Fe site.

Figure 3:  Density of states (DOS) for $A_{2}$Mo3O8 ($A$ = Fe and Zn): (a)
total DOS; (b-g) atom- and orbital-projected DOS. The (T) and (O) symbols
refer to trigonally-distorted tetrahedral and octahedral environments. The
vertical solid lines are placed at zero chemical potential. The schematic
insets of gray tetrahedra$/$octahedra are guides to the eye pointing the
reader to the electronic states of the transition metal ions in the
corresponding environment. (h, i) Orbital projected spectral functions for
Fe2Mo3O8 and FeZnMo3O8 (blue - $e$(1) tetrahedra; green - $a_{1}$ and $e$(2)
octahedra; red - $a_{1}$, $e$(2) tetrahedra and $e$(1) octahedra)

.

Figure 4:  Atom- and orbital-projected density of states (DOS) together with a
schematic view of the orbital occupation and character for $A_{2}$Mo3O8 ($A$ =
Mn, Fe, Co and Zn). Each panel shows a schematic view of the orbital
occupation (left side) and DOS (right side). Panels (a, c, e, g) refer to ions
in trigonally-distorted tetrahedral environments ($A$(T)), whereas panels (b,
d, f, h) refer to ions in trigonally-distorted octahedral environments
($A$(O)).

###  .1 Strong hybridization, resonance, and interaction with the Mo trimer

Figure 1(h) displays the theoretical optical conductivity of Zn2Mo3O8,
FeZnMo3O8 and Fe2Mo3O8 computed using a combination of Density Functional
Theory and embedded Dynamical Mean Field Theory (DFT + eDFMT) methods Haule
(2020, 2018). Here, we find the same trend of decreasing charge gap with Fe
substitution. In Zn2Mo3O8, the size of the optical gap is $\approx$1.7 eV
which decreases to approximately 1.5 and 1.4 eV in FeZnMo3O8 and Fe2Mo3O8,
respectively. Figure 1(i) compares the theoretical optical conductivity of the
Mn, Ni and Co analogs. We find that the predicted gap is larger in all of
these compounds (near $1.55\,$eV) as compared to Fe2Mo3O8. The edge of the gap
is very smooth and temperature smeared. This is quite different from Fe2Mo3O8
where the gap is smaller with an additional peak at the onset.

To better understand what determines the low energy excitations and character
of the gap in this class of compounds, we calculated the local density of
states for the full set of $A_{2}$Mo3O8 materials [Fig. 3]. While the optical
gap in general is different than the gap of the single-particle excitations
measured by the local density of states, the two are very similar in these
compounds. This is because the band gap is direct in Zn2Mo3O8, and the
hybridized bands in the Fe-containing compounds are extremely narrow. Hence
momentum-conserving excitations have essentially the same gap size as the
finite momentum single particle excitations. The insets show that the position
of the conduction band in FeZnMo3O8 and Fe2Mo3O8 decreases slightly as
compared to Zn2Mo3O8, although the change is small. Most of the action is in
the valence bands, where the FeZn band edge moves considerably upward. In
Fe2Mo3O8, a very narrow many-body excitation forms at the onset of the gap.
This is the origin of the first peak in the optical conductivity [Fig. 1(h,i)]
and the reason that the gap is drawn strongly downward in this system.

Figure 3(b-g) displays the projected density of states (DOS) per transition
metal center and per orbital in the distorted tetrahedral (T) and octahedral
(O) environments. The local point group symmetry around each iron center is
C3v. Therefore, the $e$ and $t_{2}$ levels at the distorted tetrahedral site
break into $e$(1), $a_{1}$, and $e$(2) orbitals. Similarly symmetry at the
trigonally-distorted Fe octahedral site breaks $t_{2g}$ and $e_{g}$ into
$e$(2), $a_{1}$, and $e$(1) states. These energy levels are shown in Fig. 4,
and the symmetry breaking is diagrammed in Fig. S1, Supplementary Information
Sup .

Getting back to Figure 3(b-g), we notice that the Mo states (dotted grey line)
are very similar across this entire family of materials. The low-energy
excitations are, however, not on the Mo site when Fe is present. Figure 3(b)
reveals that the sharp many-body resonance near $-0.4\,$eV emanates primarily
from the Fe center in the trigonally-distorted tetrahedral environment,
$e$(1). Panel (c) shows that a broader, but still reasonably sharp excitation
around $-0.8\,$eV arises from the doubly-degenerate $e$(2) state on the
distorted octahedral site. Since both of these fairly sharp excitations come
from band formation via hybridization, the Mo partial density of states also
has a small peak at the same energy (Figs. S19 and S20). This demonstrates the
quasi-particle nature of these peaks, which are Kondo-like and come from local
spin screening on the aforementioned $e$(1) and $e$(2) distorted tetrahedral
and octahedral orbitals, respectively.

What is exciting about this finding is that screening and many body Kondo peak
formation Yee  _et al._ (2010); Hewson (1993) are normally expected in a metal
- not an insulator. The possibility of a Kondo resonance in a semiconductor
like Fe2Mo3O8 is potentially quite interesting, opening the door to deeper
exploration of the Kondo effect in a significantly wider variety of materials.
While broader peaks like that at $-0.8\,$eV are not uncommon in transition
metal compounds and appear for example in monoxides Mandal  _et al._ (2019a,
b), the very narrow resonance emanating from the $e$(1) orbitals on the
distorted tetrahedral site is unique to Fe2Mo3O8. We assign it as an analog of
the Zhang-Rice singlet in cuprates, Zhang and Rice (1988); Eskes and Sawatzky
(1988) arising due to screening of the spin $1/2$ hole on the trigonally-
distorted tetrahedral Fe $e$(1) sites in the Mott insulating state. This
characteristic peak appears with very well defined energy. It also mixes
strongly with the Mo-O trimers (see supplement Fig. S5).

In order to test these predictions, we compare the theoretical optical
conductivity [Fig. 1(h)] to the measured absorption spectrum of Fe2Mo3O8 [Fig.
1(f)]. Overall, the calculated optical conductivity and the measured
absorption spectrum are very consistent - although the features are not as
well-defined as we might prefer. As a reminder, the spectral functions are
very flat on the valence edge due to Fe occupation on the distorted
tetrahedral site. This causes a sharp many-body resonance in the density of
states [Fig 3(b)] which manifests as a small peak on the leading edge of the
theoretical optical conductivity. The contribution from the distorted
octahedral site is similar but less pronounced. Our calculations therefore
predict that the many-body resonance on the valence band edge [Fig 3(b,c)]
should lower the gap. This is exactly what we find. Obviously this structure
is strongly broadened in the experimental result, but the presence of these
states causes the gap to be drawn noticeably downward in this system [Fig.
1(f)]. Similar reasoning applies to FeZnMo3O8, although only the distorted
octahedral site is operative. Because the density of states are momentum-
averaged, we also show the spectral functions [Fig. 3(h,i)].

### Structure-property relations in the metal-substituted analogs

Figure 4 compares the density of states in Fe2Mo3O8 with several other
transition metal analogs which have orbital filling between $d^{5}$ and
$d^{8}$ \- namely the Mn, Co and Ni analogs. We also show a schematic view of
the orbital occupation for both the distorted tetrahedral and octahedral sites
(left and right columns, respectively). The gap in each orbital can originate
from the band structure due to the periodic potential (band gap) or from Mott
localization of electrons on a given $A$ site, which we denote as a Mott gap.
In principle, we can distinguish between the two because the single-particle
spectral function is modified from the DFT bands by self-energy effects. We
expect the spectral function to be either zero or finite in the band gap case;
it should diverge inside the gap for the Mott case Pavarini  _et al._ (2017);
Demchenko  _et al._ (2004); Kotliar  _et al._ (2006). Here, the electron state
can no longer be described within the band picture.

The top panels of Fig. 4 show calculations for Mn2Mo3O8 in which the electrons
are in the high-spin $d^{5}$ configuration. The Mott gap opens in all orbitals
on both tetrahedral and octahedra sites, and it is much larger than the band
gap of Mo-trimer. Consequently the charge gap and low energy excitations in
the Mn and Zn compounds are determined by the same Mo-trimer states. This is
why they appear so similar.

Fe2Mo3O8 is the most interesting of the series showing a complex interplay of
band gaps, Mott gaps, and quasi-particle multiplets. The $a_{1}$ and $e$(2)
states on the distorted tetrahedra are Mott insulating with a large gap. As
discussed above, the doubly degenerate $e$(1) orbitals contain one hole, which
is equally distributed among the two orbitals, and the Mott mechanism opens
the gap - even though the self-energy pole is less strong and the gap smaller
than in the $a_{1}$ and $e$(2) states. Moreover, at the edge of the gap,
strong hybridization, directly computable by the DMFT hybridization function,
shows a very strong and narrow peak due to many body screening effects. With
this mechanism, the spin-$1/2$ emanating from the $e$(1) orbitals on the
tetrahedral site is screened by the Mo-trimer electrons - a mechanism that is
analogous to the Zhang-Rice state in cuprates Zhang and Rice (1988). Note that
this is different from the sharp peak due to the valence band edge in
transition metal monoxides Mandal  _et al._ (2019a, b). The latter appears in
the antiferromagnetic state where all gaps are band-like in nature and spin
states are split by a Zeeman field. As a consequence, many-body screening of
spin is not possible. Here, spin preserves SU(2) symmetry, and the resonance
screens the local spin on Fe. In addition, the narrow peak due to many body
screening effects disappears in the antiferromagnetic ground state [Fig. S19,
Supplementary information] Sup . The distorted octahedral site in the Fe
$d^{6}$ state contains a combination of a Mott gap in the $e$(1) states and a
band gap in the $a_{1}$ and $e$(2) states. This unusual combination is a so-
called orbitally-selective Mott state Anisimov  _et al._ (2002); de’Medici
_et al._ (2005). Note that the sharp peak at the valence band edge appears as
well, even though it is not as sharp as the resonance from the distorted
tetrahedral Fe $e$(1) site. Its nature is different, as it appears in band-
insulating $a_{1}$ and $e$(2) orbitals similar to the valence band edge
discussed in transition metal monoxides Mandal  _et al._ (2019b, a). Note that
hybridization and band formation with oxygen and Mo electrons are needed to
open the band gap in the octahedral Fe $a_{1}$ and $e$(2) states. This is
because the latter contains only four electrons in three nearly degenerate
$a_{1}$ and $e$(2) orbitals.

Next we discuss the Co analog, which is in the $d^{7}$ configuration. In this
case, the $e$(1) orbitals on the tetrahedral site are fully filled, and the
$a_{1}$ and $e$(2) orbitals are in the high-spin Mott insulating state with a
large gap - larger than the Mo-trimer gap. On the octahedral site, the $e$(1)
orbitals are Mott-insulating with a large gap, and the $a_{1}$ and $e$(2)
states are band-insulating in which Mo and oxygen provide one electron to form
a covalent band with the $a_{1}$ electrons. We note that no sharp low-energy
peak is found at the valence band edge, although in principle such a peak is
possible.

Figure 4(g,h) displays the partial density of states for the Ni analog with
its $d^{8}$ configuration. In this case, the $e$(1) and $a_{1}$ orbitals on
the distored tetrahedral site are fully filled, and the doubly-degenerate
$e$(2) state shows a Mott gap which is comparable to that of the Mo trimer. On
the distorted octahedral site, the $e$(2) and $a_{1}$ states are fully filled,
and the $e$(1) states are in the half-filled Mott insulating state. This Mott
gap is again comparable to the Mo-trimer gap. Hence the reduction of the gap
as compared to the Zn analog is minimal.

## Summary and outlook

To summarize, we measured the optical properties of Fe2Mo3O8 and compared our
findings with first principles electronic structure calculations. We find a
1.1 eV direct gap composed of heavily mixed charge-transfer excitations that
is sensitive to magnetic ordering at 61 K, vibronic coupling that activates
on-site $d$-to-$d$ excitations on the distorted octahedral Fe site, and spin-
orbit related features riding on top of the $d$-to-$d$ excitation on the
distorted tetrahedral Fe site below the magnetic ordering temperature. The
Kondo effect is, of course, usually studied in metals. Here, we show that it
can also appear in a semiconductor that has both Mott and band gaps. Similar
to the metallic Kondo effect, the orbitals with Mott-like gaps develop a many-
body excitation near the valence edge. This draws the gap downward in energy
(from 1.7 eV in Zn2Mo3O8 $\rightarrow$ 1 eV in Fe2Mo3O8) and screens the
magnetic moment. This discovery opens the door to deeper exploration of the
Kondo effect in semiconductors. Fe2Mo3O8 is also a superb platform for
unraveling structure-property relationships. What differentiates Fe2Mo3O8 from
the Zn, Mn, Co, and Ni members of this series is the band-Mott mixing, the
Zhang-Rice resonance, and how the gap is hybridized. Taken together, these
findings enhance our understanding of charge transfer in quantum cluster
magnets and advance the use of this powerful scaffold in new types of charge
storage devices.

## Acknowledgements

Research at the University of Tennessee and Rutgers University is supported by
the NSF-DMREF program (DMR-1629079 and DMR-1629059). G.L.P.’s work was
supported by a grant of the Romanian Ministry of Education and Research, CNCS
- UEFISCDI, project number PN-III-P1-1.1-TE-2019-1767, within PNCDI III.
Access to the x-ray facilities at the Research Complex, Rutherford Appleton
Laboratory is gratefully acknowledged.

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#  Bounding the Order of Vanishing of Cuspidal Newforms via the $n$th Centered
Moments

Sohom Dutta<EMAIL_ADDRESS>Lynbrook High School, San Jose, CA
95129  and  Steven J. Miller<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Mathematics and Statistics,
Williams College, Williamstown, MA 01267

###### Abstract.

Building on the work of Iwaniec, Luo and Sarnak, we use the $n$-level density
to bound the probability of vanishing to order at least $r$ at the central
point for families of cuspidal newforms of prime level $N\to\infty$, split by
sign. There are three methods to improve bounds on the order of vanishing:
optimizing the test functions, increasing the support, and increasing the
$n$-level density studied. Previous work has determined the optimal test
functions for the $1$ and $2$-level densities in certain support ranges, with
the effectiveness of the bounds only marginally increasing by the optimized
test functions over simpler ones, and thus this is not expected to be a
productive avenue for further research. Similarly the support has been
increased as far as possible, and further progress is shown to be related to
delicate and difficult conjectures in number theory. Thus we concentrate on
the third method, and study the higher centered moments (which are similar to
the $n$-level densities but combinatorially easier). We find the level at each
rank for which the bounds on the order of vanishing is the best, thus
producing world-record bounds on the order of vanishing to rank at least $r$
for every $r>2$ (for example, our bounds for vanishing to order at least 5 or
at least 6 are less than half the previous bounds, a significant improvement).
Additionally, we explicitly calculate the optimal test function for the
$1$-level density from previous work and compare it to the naive test
functions for higher levels. In doing so, we find that the optimal test
function for certain levels are not the optimal for other levels, and some
test functions may outperform others for some levels but not in others.
Finally, we explicitly calculate the integrals needed to determine the bounds,
doing so by transforming an $n$-dimensional integral to a $1$-dimensional
integral and greatly reducing the computation cost in the process.

###### Key words and phrases:

Cuspidal Newforms, $L$-functions, order of vanishing, Random Matrix Theory

###### 2010 Mathematics Subject Classification:

11Mxx (primary); 45Bxx (secondary)

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Random Matrix Theory and Statistics of $L$-functions
    3. 1.3 Test Functions
    4. 1.4 $n$-th Centered Moments
    5. 1.5 Main Results
  2. 2 Preliminaries
  3. 3 Bounds on the Order of Vanishing From the 1-Level Density
  4. 4 Bounds from the $n$-Level Densities ($n$ even) with support $v=2/n$
  5. 5 Plots of Bounds for support $v=2/n$

##  1\. Introduction

###  1.1. Background

Building on the work of Iwaniec, Luo and Sarnak [ILS] we determine new world
records for bounding the probability of $L$-functions of cuspidal newforms of
prime level $N\to\infty$, split by sign, vanishing to order at least $r$ at
the central point. We recall some standard definitions (see [IK] for more
details), review connections between $L$-functions and Random Matrix Theory,
and then state our improved results.

######  Definition 1.1 (Cuspidal Newforms).

Let $H^{\star}_{k}(N)$ denote the set of holomorphic cusp forms of weight $k$
that are newforms of level $N$. For every $f\in H^{\star}_{k}(N)$, we have a
Fourier expansion

$f(z)\ =\ \sum_{n=1}^{\infty}a_{f}(n)e(nz).$ (1.1)

We set $\lambda_{f}(n)=a_{f}(n)n^{-(k-1)/2}$, and obtain the $L$-function
associated to $f$

$L(s,f)\ =\ \sum_{n=1}^{\infty}\lambda_{f}(n)n^{-s}.$ (1.2)

The completed $L$-function is

$\Lambda(s,f)\ :=\
\left(\frac{\sqrt{N}}{2\pi}\right)^{s}\Gamma\left(s+\frac{k-1}{2}\right)L(s,f).$
(1.3)

Since $\Lambda(s,f)$ satisfies the functional equation $\Lambda(s,f)\ =\
\epsilon_{f}\Lambda(1-s,f)$ with $\epsilon_{f}=\pm 1$, $H^{\star}_{k}(N)$
splits into two disjoint subsets, $H^{+}_{k}(N)=\\{f\in
H^{\star}_{k}(N):\epsilon_{f}=+1\\}$ and $H^{-}_{k}(N)=\\{f\in
H^{\star}_{k}(N):\epsilon_{f}=-1\\}$. The associated symmetry group of
$H^{\star}_{k}(N)$ is Orthogonal _(O)_ , $H^{+}_{k}(N)$ is Special Orthogonal
even _SO(even)_ , and $H^{-}_{k}(N)$ is Special Orthogonal odd _SO(odd)_.

###  1.2. Random Matrix Theory and Statistics of $L$-functions

The Riemann Hypothesis states that all zeroes of the Riemann zeta function are
either at the negative even integers (the trivial zeros, as these numbers are
well understood!) or complex numbers with real part $1/2$ (the nontrivial
zeros); the Generalized Riemann Hypothesis (GRH) asserts this is true for all
$L$-functions, in particular for the cuspidal newforms we study.

It turns out that the behavior of many different objects in mathematics and
physics are the same, and this has led to fruitful conversations where one
subject suggests problems and predicts answers in another. In the early 1900s,
random matrix theory was used for applications in statistics and harmonic
analysis. However, a major advance in the subject was made with the seminal
work of Eugene Wigner, who noticed a remarkable connection between
fluctuations in the position of compound nuclei resonances and statistics for
the eigenvalues of random matrices. As more researchers began exploring random
matrix theory, they discovered connections between random matrix theory
(specifically the distribution of eigenvalues of matrices) and number theory
(the distribution of the non-trivial zeros). See [BFMT-B, Ha] and the
references therein for the history and many of the results.

The first statistics studied were the $n$-level correlations and the spacings
between adjacent zeros; see [Hej, Mon, Od1, Od2, RS], where it was observed
that the behavior of zeros far from the central point converged to a universal
behavior, independent of the arithmetic of the $L$-function. This led to the
quest to find a new statistic that was sensitive to the behavior near the
central point, which by the Birch and Swinnerton-Dyer Conjecture was known to
be an important point to study.

Katz and Sarnak [KS1, KS2] introduced a new statistic, the $n$-level density,
which has different values for different families of $L$-functions, and
essentially only depends on the zeros near the central point. They found that
as the level of the forms approach infinity, the statistics for the zeroes of
families of $L$-functions can be modelled by eigenvalue statistics for one of
the classical compact groups (unitary, orthogonal, symplectic).

######  Definition 1.2 ($n$-level Density).

The $n$-level density of an $L$-function $L(s,f)$ is defined as

$D_{n}(f;\phi)\ :=\ \sum_{\begin{subarray}{c}j_{1},\cdots,j_{n}\\\
j_{i}\neq\pm j_{k}\end{subarray}}\phi\left(\frac{\log
c_{f}}{2\pi}\gamma_{f}^{(j_{1})},\dots,\frac{\log
c_{f}}{2\pi}\gamma_{f}^{(j_{n})}\right)$ (1.4)

for a _test function_ $\phi:\mathbb{R}^{n}\to\mathbb{R}$ where $c_{f}$ is the
analytic conductor of $f$. For many applications we assume $\phi$ is a non-
negative even Schwartz function (see 2.5) with compactly supported111A
function $f$ is supported in $(\sigma,\sigma)$ if $f(x)=0$ for all $x$ with
$|x|>\sigma$. Fourier transform and $\phi(0,\ldots,0)>0$.

There is now an extensive series of papers showing that the $n$-level
densities of various families of $L$-functions match the random matrix theory
predictions; see for example the introduction in [C–] for a review of the
literature. Our main result is to use the $n$-level densities to bound how
often forms in the family $\mathcal{F}_{N}$ vanish to a given order (or more),
with $\mathcal{F}_{N}$ being cuspidal newforms of level $N$ and some fixed
weight $k$, with $N\to\infty$ through the primes222This latter condition is
for technical reasons; with additional work (see [BBDDM]) one may take $N$
tending to infinity through all integers. We use the results on the $n$-level
densities of cuspidal newform families as inputs, and thus with standard but
tedious and technical work, we could remove the prime restriction.

###  1.3. Test Functions

To get the best bounds, we must choose a good test function to use in the
$n$-level density. As shown later (see Theorem 3.1), we require the test
function to be even, non-negative, Schwartz, and have a Fourier transform333We
define the Fourier transform of $g$ by $\mathaccent
866{g}(y)=\int_{-\infty}^{\infty}g(x)e^{-2\pi ixy}$. with finite support; we
will later see how to pass from such functions to bounds on vanishing.

To obtain the most information about what is happening at the central point,
we want our test function to be concentrated there and rapidly decay, ideally
a delta spike at the origin. Unfortunately, the closer our test function is to
a delta spike, the larger the support is of its Fourier transform; this is the
mathematical instance of the Heisenberg Uncertainty Principle: we cannot
localize both a function and its Fourier transform. The reason this is an
obstruction is that the $n$-level density requires us to compute certain
weighted sums of the $L$-function coefficients times the Fourier transform of
the test function evaluated at the logarithm of the primes, and we can only
compute these sums if the support of the Fourier transform is suitably
restricted. Thus the goal is to find test functions as close to the delta
spike as we can, subject to being able to compute the resulting sums on the
Fourier transform side.

Throughout this paper, the main test function we use is the naive test
function (so named as it is easy to use and easy to guess is worth using).

######  Definition 1.3 (Naive Test Function).

The naive test function is the Fourier test function pair

$\phi_{\rm naive}(x)\ =\ \left(\frac{\sin(\pi v_{n}x)}{(\pi
v_{n}x)}\right)^{2}\ \ \ ,\ \ \ \mathaccent 866{\phi}_{\rm
naive}(y)=\frac{1}{v_{n}}\left(y-\frac{|y|}{v_{n}}\right)$ (1.5)

for $|y|\ <\ v_{n}$ where $v_{n}$ is the support.

Previous work has improved on the naive test function by finding the optimal
test function for some of the $n$-level densities for certain ranges of
support (see, for example, [BCDMZ, FM, ILS]). This is one of three ways to
improve results; however, the optimal function for the 1-level density leads
to such a small improvement (and there are similarly small gains in the higher
levels) that this avenue is not pursued in this paper. The second approach is
to try to increase the support for the $n$-level density, but doing so
requires resolving difficult combinatorics and technical sums. There has been
a recent breakthrough here, though, in work by Cohen et. al. [C–], who
increased the support for the $n$-level density from $1/(n-1)$ to $2/n$.

We turn to the third method, high level densities. There has been some
progress here; Li and Miller [LiM] used the 4-level density to obtain better
bounds than those from the first and second level densities, for sufficiently
large vanishing at the central point. Unfortunately the higher level densities
have a disadvantage – while they give better bounds for much larger than
expected vanishing at the central point, they give worse bounds for small
vanishing.

This trade-off has never been quantified and analyzed till now, and is the
main goal of this project. In particular, we prove for each $r$ what is the
optimal level density to use (for a fixed test function, usually the naive
one), to bound the probability of vanishing to order at least $r$ at the
central point.

###  1.4. $n$-th Centered Moments

The first results are due to Iwaniec-Luo-Sarnak [ILS], who computed the
1-level density for families of cuspidal newforms split by sign with support
up to $2$. This was extended by Hughes-Miller [HM] to the $n$-level; while it
was expected their results should hold up to $2/n$ for the support, there were
combinatorial obstructions and their results are only valid if the support is
at most $1/(n-1)$; these complications were recently resolved in [C–], and now
the $n$-level is known up to $2/n$.

The following theorem from [C–] is used to generate bounds on the order of
vanishing.

######  Theorem 1.4.

[Theorem 1.1 from [C–].] Let $n\geq 2$ and ${\rm
supp}(\phi)\subset(-\frac{2}{n},\frac{2}{n})$. Define

$\displaystyle\sigma_{\phi}^{2}$ $\displaystyle\ :=\ $ $\displaystyle
2\int_{-\infty}^{\infty}|y|\mathaccent 866{\phi}(y)^{2}dy$ $\displaystyle
R(m,i;\phi)$ $\displaystyle\ :=\ $ $\displaystyle
2^{m-1}(-1)^{m+1}\sum_{l=0}^{i-1}(-1)^{l}\binom{m}{l}$
$\displaystyle\left(-\frac{1}{2}\phi^{m}(0)+\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\mathaccent
866{\phi}(x_{2})\cdots\mathaccent 866{\phi}(x_{l+1})\right.$
$\displaystyle\left.\int_{-\infty}^{\infty}\phi^{m-l}(x_{1})\frac{\sin(2\pi
x_{1}(1+|x_{2}|+\cdots+|x_{l+1}|))}{2\pi x_{1}}dx_{1}\cdots dx_{l+1}\right)$
$\displaystyle S(n,a,\phi)$ $\displaystyle\ :=\ $
$\displaystyle\sum_{l=0}^{\lfloor\frac{a-1}{2}\rfloor}\frac{n!}{(n-2l)!l!}R(n-2l,a-2l,\phi)\left(\frac{\sigma_{\phi}^{2}}{2}\right)^{l}$
(1.6)

and

$\lim_{\begin{subarray}{c}N\to\infty\\\ N\text{\rm
prime}\end{subarray}}\left\langle\left(D(f;\phi)-\left\langle
D(f;\phi)\right\rangle_{\pm}\right)^{n}\right\rangle_{\pm}\ =\
(n-1)!!\sigma_{\phi}^{n}1_{n\ \rm even}\pm S(n,a;\phi).$ (1.7)

444With extra work, we can move to $N$ through the square-free integers in the
limit.

###  1.5. Main Results

We use the $n$-level densities to bound how often forms in the family
$\mathcal{F}_{N}$ vanish with $\mathcal{F}_{N}$ being cuspidal newforms of
level $N$ and some fixed weight $k$, with $N\to\infty$ through the primes.

In order to generate better bounds on the order of vanishing, as remarked
there are three main methods we can turn to: optimizing the test function,
increasing the support, and using higher levels. We concentrate on the last.

Because work done to find the optimal test function produces marginal
improvements, we turn to using higher levels using the recently improved
results with support $2/n$, thus producing record bounds for the order of
vanishing as shown in the following tables. For example, the previous record
for vanishing to order 5 or more was .06580440 and for order 6 it was
.00853841 from [LiM], while we show for vanishing to order 5 or more it is at
most .020408300 while 6 or more .003346510; these are not slight improvements
but a decrease by more than a factor of two! The tables below show our results
for the various symmetry groups; cuspidal newforms with even functional
equations are SO(even), while those with odd signs are SO(odd).

Lowest Bounds for Each Rank for G=SO(even)
---
Rank |  Level Used |  Bound
2 |  1 |  $0.43231300$
4 |  2 |  $0.066666667$
6 |  6 |  $0.003346510$
8 |  8 |  $0.000579210$
10 |  10 |  $1.14380\times 10^{-6}$
12 |  12 |  $1.85901\times 10^{-8}$
14 |  14 |  $2.59310\times 10^{-10}$
16 |  16 |  $3.09185\times 10^{-12}$
18 |  18 |  $3.26332\times 10^{-14}$
20 |  20 |  $3.08920\times 10^{-16}$

Lowest Bounds for Each Rank for G=SO(odd)
---
Rank |  Level Used |  Bound
1 |  N/A |  $1.0000000$
3 |  2 |  $0.111111111$
5 |  2 |  $0.020408300$
7 |  6 |  $0.000292790$
9 |  8 |  $7.65596\times 10^{-6}$
11 |  10 |  $1.53302\times 10^{-7}$
13 |  12 |  $2.50956\times 10^{-9}$
15 |  16 |  $3.03362\times 10^{-11}$
17 |  18 |  $3.10549\times 10^{-13}$
19 |  20 |  $4.18402\times 10^{-17}$

We find that although using higher levels can create better bounds, increasing
the levels to an arbitrarily large value does not lead to better bounds for
small rank. Increasing the level eventually produces trivial bounds (such as
the percent that vanish to a given level is at most a number greater than
100%!). Thus there are only a finite number of levels that need to be checked
to find which level creates the optimal bound for each order of vanishing.
Additionally, we find that the optimal test functions for certain levels will
not necessarily outperform other test functions for higher levels, as shown by
the worse results produced by the $1$-level optimal test function for higher
levels as opposed to the naive test function; this was a very surprising
result, namely that a function which is superior for one $n$ can become
inferior to a test function it beat for larger $n$.

In addition to calculating bounds, we show explicit calculations for the
integrals needed to generate bounds. For support $2/n$, we are able to reduce
an $n$-dimensional integral to a $1$-dimensional integral, thus significantly
reducing the computation time needed to evaluate the integral (it is
unfortunately not solvable in closed form, and requires numerical
approximation, which we do via Simpson’s method).

After reviewing some number theory and complex analysis preliminaries we
continue to a review of work done by [ILS] to generate bounds for the
$1$-level density. We show the bounds obtained by [ILS] by both the naive and
optimal test function while explicitly enumerating the optimal Fourier pair
for the $1$-level density. We then continue to results for the bounds with
higher levels for support $1/(n-1)$ and extend this to support $2/n$. Lastly,
we display plots and tables of the bounds of the order of vanishing based on
the rank and level.

##  2\. Preliminaries

We record some needed definitions and standard results; see [SS] for more
details and proofs.

######  Definition 2.1 (Fourier Transform).

The Fourier transform of a function $\phi(x)$ is

$F(\phi(x))\ :=\ \mathaccent 866{\phi}(y)\ =\
\int_{-\infty}^{\infty}\phi(x)e^{-2\pi ixy}dx.$ (2.1)

######  Theorem 2.2 (Plancherel Theorem).

For a Fourier pair $f(x)$ and $\mathaccent 866{f}(y)$ such that $E(t)$ is
square-integrable,

$\int_{-\infty}^{\infty}|f(x)|^{2}dt\ =\ \int_{-\infty}^{\infty}|\mathaccent
866{f}(y)|^{2}dy.$ (2.2)

######  Remark 2.3.

As a result of the polarization identity 555The polarization identity relates
the inner product of two vectors with the norm of the two vectors., for
square-integrable functions $f$ and $g$, the Plancherel Theorem can be re-
expressed as

$\int_{-\infty}^{\infty}f(x)\overline{g(x)}dx\ =\
\int_{-\infty}^{\infty}\mathaccent 866{f}(y)\overline{\mathaccent
866{g}(y)}dy.$ (2.3)

Recall the convolution of two functions $f$ and $g$ is defined by

$(f\ast g)(x)\ =\ \int_{-\infty}^{\infty}f(t)g(x-t)dx.$ (2.4)

######  Theorem 2.4 (Convolution Theorem).

For functions $f(t)$ and $g(t)$,

$\mathcal{F}[f\ast g]\ =\ \mathcal{F}[f]\mathcal{F}[g].$ (2.5)

######  Definition 2.5 (Schwartz space).

The Schwartz space is the set of all functions such that $f$ is infinitely
differentiable and the derivatives of $f$ decay faster than any polynomial. A
function is Schwartz if it is a function in the Schwartz space.

######  Definition 2.6 (Even Indicator Function).

The Even Indicator function is defined to be

$1_{\rm even}(n)\ :=\ \begin{cases}1&\text{{\rm if} }x\text{{\rm is\ even}}\\\
0&\text{{\rm if} }x\text{{\rm is\ odd}}.\end{cases}$ (2.6)

Since we encounter integrals later which we cannot evaluate in closed form, we
use Simpson’s Rule to approximate the integral using only function values at
certain points while having a low error term.

######  Theorem 2.7 (Simpson’s Rule).

Let $[a,b]$ be an interval that is split into $s$ equal sub-intervals with $s$
even and $h\ =\ (b-a)/s$. Then, the integral $\int_{a}^{b}f(x)dx$ is
approximated by

$\int_{a}^{b}f(x)dx\ \approx\
\frac{h}{3}\left[f(x_{0})+4\sum_{j=1}^{\frac{n}{2}}f(x_{2j-1})+2\sum_{j=1}^{\frac{n}{2}-1}f(x_{2j})+f(x_{n})\right]$
(2.7)

where $x_{j}=a+jh$ for $j=0,1,2,...,n$ and the difference between the integral
and the sum, ${\rm Err}(f)$, is bounded by

$|{\rm Err}(f)|\ \leq\
\frac{h^{4}}{180}(b-a)\max_{\epsilon\in[a,b]}|f^{(4)}(\mathcal{E})|.$ (2.8)

######  Theorem 2.8 (Fubini’s Theorem).

If $\iint_{X\times Y}|f(x,y)|d(x,y)<\infty$,

$\iint_{X\times Y}|f(x,y|d(x,y)\ =\ \int_{X}\left(\int_{Y}f(x,y)dy\right)dx\
=\ \int_{Y}\left(\int_{X}f(x,y)dx\right)dy.$ (2.9)

######  Definition 2.9 (Double Factorial).

For positive integer $n$,

$n!!\ :=\ \begin{cases}n(n-2)(n-4)\cdots 4\cdot 2&\text{{\rm for\ $n$\
even}}\\\ n(n-2)(n-4)\cdots 3\cdot 1&\text{{\rm for\ $n$\ odd}}.\end{cases}$
(2.10)

######  Definition 2.10.

Define the family of forms in $\mathcal{F}_{N}$ that vanish to exact order $r$
to be $\mathcal{F}_{N,r}$.

######  Definition 2.11.

Define the percent of $f\in\mathcal{F}_{N}$ that vanish to exact order $r$ to
be

$p_{r}(\mathcal{F}_{N})\ :=\ \frac{|\mathcal{F}_{N,r}|}{|\mathcal{F}_{N}|}.$
(2.11)

######  Definition 2.12.

For a fixed $r$, define $p_{r}(\mathcal{F})$ to be the limit of the percent of
forms in $\mathcal{F_{N}}$ whose order of vanishing is $r$ as $N$ tends to
infinity through the primes:

$p_{r}(\mathcal{F})\ :=\
\lim_{N\to\infty}\frac{|\mathcal{F}_{N,r}|}{|\mathcal{F}_{N}|}.$ (2.12)

######  Definition 2.13.

For a fixed $r$, define $q_{r}(\mathcal{F})$ to be the limit of the percent of
forms in $\mathcal{F_{N}}$ that vanish to order at least $r$ as $N$ tends to
infinity through the primes:

$q_{r}(\mathcal{F})\ =\ \sum_{i\geq r}p_{i}(\mathcal{F}).$ (2.13)

######  Definition 2.14.

The one-level density of an $L$-function $L(s,f)$ is

$D(f,\phi)\ :=\
\sum_{\gamma_{f}}\phi\left(\frac{\gamma_{f}}{2\pi}\log{c_{f}}\right),$ (2.14)

where $c_{f}$ is the analytic conductor of $f$. The average or expectation of
$D(f,\phi)$ over a family $\mathcal{F}_{N}$ is

$E(\mathcal{F}_{N},\phi)\ :=\
\frac{1}{|\mathcal{F}_{N}|}\sum_{f\in\mathcal{F}_{N}}D(f,\phi).$ (2.15)

######  Definition 2.15.

The mean of the $1$-level density of a family is denoted $\mathcal{F}_{N}$ is
$\mu(\phi,\mathcal{F}_{N})$.

##  3\. Bounds on the Order of Vanishing From the 1-Level Density

As we use the same functions and similar techniques as [ILS], We quickly
summarize their work to obtain bounds on the order of vanishing using the
$1$-level density, and then discuss the complications that arise in extending
these arguments to the $n$-level densities.

######  Theorem 3.1.

[Bounds on the $1$-level density from [ILS]] Let $\phi$ be a non-negative,
even Schwartz function with ${\rm supp}(\mathaccent
866{\phi})\subset(-\sigma,\sigma)$ for some finite $\sigma$. Let $\mathcal{G}$
be the group associated to the family $\\{\mathcal{F}_{N}\\}$ (i.e., Unitary,
Symplectic, Orthogonal, SO(even), SO(odd)). _Set_

$g_{\mathcal{F}}(\phi)\ :=\ \int_{-\infty}^{\infty}\mathaccent
866{\phi}(y)\mathaccent 866{W}_{\mathcal{G}(\mathcal{F})}(y)dy.$ (3.1)

_For a given $r$, as $N\to\infty$ the percent of forms in the family
$\mathcal{F}_{N}$ that vanish to order exactly $r$ is bounded by_

$p_{r}\ \leq\ \frac{1}{r}(g_{\mathcal{F}}(\phi)).$ (3.2)

######  Corollary 3.2.

Let $\phi$ be the naive test function with ${\rm supp}(\mathaccent
866{\phi})\subset(-v_{n},v_{n})$, Then

$\displaystyle p_{r}(\mathcal{F})$ $\displaystyle\ \leq\ $
$\displaystyle\frac{1}{r}\left(\frac{1}{v_{n}}+\frac{1}{2}\right)\ \ \textrm{
for G = O}$ $\displaystyle p_{r}(\mathcal{F})$ $\displaystyle\ \leq\ $
$\displaystyle\left\\{\begin{array}[]{lr}\frac{1}{r}\left(\frac{1}{v_{n}}+\frac{1}{2}\right)&\text{if
}v_{n}\ \leq\ 1\\\
\frac{1}{r}\left(\frac{2}{v_{n}}-\frac{1}{2v_{n}^{2}}+\epsilon\right)&\text{if
}v_{n}\geq 1\end{array}\right\\}\ \ \textrm{for G = SO(even)}$ (3.5)
$\displaystyle p_{r}(\mathcal{F})$ $\displaystyle\ \leq\ $
$\displaystyle\left\\{\begin{array}[]{lr}\frac{1}{r}\left(\frac{1}{v_{n}}+\frac{1}{2}+\epsilon\right)&\text{if
}v_{n}\ \leq\ 1\\\ \frac{1}{r}\left(1+\frac{1}{2v_{n}^{2}}\right)&\text{if
}v_{n}\ \geq\ 1\end{array}\right\\}\ \ \textrm{for G = SO(odd)}.$ (3.8)

###### Proof of Corollary 3.2.

To use Theorem 3.1 we need to choose a good pair of functions, $\phi$ and
$\mathaccent 866{\phi}$. In [ILS] the authors remark that a particularly good
choice is the following (what we call the naive test function):

$\phi_{v}(x)\ =\ \left(\frac{\sin(\pi vx)}{\pi vx}\right)^{2},\ \ \ \
\mathaccent 866{\phi}_{v}(y)\ =\
\frac{1}{v_{n}}\left(1-\frac{|y|}{v_{n}}\right)$ (3.9)

for $|y|<v$.

To find the bounds on the order of vanishing, we first must find $g$ for
specific cases. From (3.1), we get

$\displaystyle g_{\mathcal{F}}(v_{n})$ $\displaystyle\ =\ $
$\displaystyle\frac{1}{v_{n}}+\frac{1}{2}\ \ \textrm{ for G = O}$
$\displaystyle g_{\mathcal{F}}(v_{n})$ $\displaystyle\ =\ $
$\displaystyle\left\\{\begin{array}[]{lr}\frac{1}{v_{n}}+\frac{1}{2}&\text{if
}v_{n}\leq 1\\\ \frac{2}{v_{n}}-\frac{1}{2v_{n}^{2}}&\text{if }v_{n}\geq
1\end{array}\right\\}\ \ \textrm{for G = SO(even)}$ (3.12) $\displaystyle
g_{\mathcal{F}}(v_{n})$ $\displaystyle\ =\ $
$\displaystyle\left\\{\begin{array}[]{lr}\frac{1}{v_{n}}+\frac{1}{2},&\text{if
}v_{n}\leq 1\\\ 1+\frac{1}{2v_{n}^{2}}&\text{if }v_{n}\geq
1\end{array}\right\\}\ \ \textrm{for G = SO(odd)}.$ (3.15)

We now want to find the bounds for the order of vanishing as a function of the
support $v$. Substituting in our values for $g_{\mathcal{F}}(v)$ from (3) into
(3.2) we get

$\displaystyle p_{r}(\mathcal{F})$ $\displaystyle\ \leq\ $
$\displaystyle\frac{1}{r}\left(\frac{1}{v_{n}}+\frac{1}{2}\right)\ \ \textrm{
for G = O}$ $\displaystyle p_{r}(\mathcal{F})$ $\displaystyle\ \leq\ $
$\displaystyle\left\\{\begin{array}[]{lr}\frac{1}{r}\left(\frac{1}{v_{n}}+\frac{1}{2}\right)&\text{if
}v\ \leq\ 1\\\
\frac{1}{r}\left(\frac{2}{v_{n}}-\frac{1}{2v_{n}^{2}}\right)&\text{if }v\geq
1\end{array}\right\\}\ \ \textrm{for G = SO(even)}$ (3.18) $\displaystyle
p_{r}(\mathcal{F})$ $\displaystyle\ \leq\ $
$\displaystyle\left\\{\begin{array}[]{lr}\frac{1}{r}\left(\frac{1}{v_{n}}+\frac{1}{2}\right)&\text{if
}v\ \leq\ 1\\\ \frac{1}{r}\left(1+\frac{1}{2v_{n}^{2}}\right)&\text{if }v_{n}\
\geq\ 1\end{array}\right\\}\ \ \textrm{for G = SO(odd)}.$ (3.21)

∎

The best results to date give the support for the $1$-level density to be
${\rm supp}(\mathaccent 866{\phi})\subset(-2,2)$. As a result, we get

$\displaystyle p_{r}(\mathcal{F})$ $\displaystyle\ \leq\ $
$\displaystyle\frac{1}{r}\ \ \textrm{ for G = O}$ $\displaystyle
p_{r}(\mathcal{F})$ $\displaystyle\ \leq\ $ $\displaystyle\frac{7}{8r}\ \
\textrm{for G = SO(even)}$ $\displaystyle p_{r}(\mathcal{F})$ $\displaystyle\
\leq\ $ $\displaystyle\frac{9}{8r}\ \ \textrm{for G = SO(odd)}.$ (3.22)

The test function $\phi_{v}(x)$ is not the optimal test function that we can
use. The function that satisfies

$\mathaccent 866{\phi}_{\rm optimal}(y)\ :=\ (f_{0}\ast\overline{f_{0}})(y)$
(3.23)

for

$f_{0}(x)\ :=\
\frac{\cos\left(\frac{|x|}{2}-\frac{\pi+1}{4}\right)}{\sqrt{2}\sin\left(\frac{1}{4}\right)+\sin\left(\frac{\pi+1}{4}\right)}\
\ ,\ \ 0\leq|x|\leq 1$ (3.24)

for $G={\rm SO(even)}$ and

$f_{0}(x)\ :=\
\frac{\cos\left(\frac{|x|}{2}+\frac{\pi-1}{4}\right)}{3\sin\left(\frac{\pi+1}{4}\right)-2\sin\left(\frac{\pi-1}{4}\right)}\
\ ,\ \ 0<|x|<1$ (3.25)

for $G={\rm SO(odd)}$ from [ILS] has been proven to be the optimal test
function to use for the $1$-level density and can create better bounds. We now
want to calculate the test function using the values provided by [ILS] and
compare the bounds from the optimal test function to that of the naive test
function.

######  Lemma 3.3.

Let

$f(x)\ :=\ \frac{\cos\left(\frac{x}{2}-a\right)}{b}\ \ ,\ \ 0\leq|x|\leq 1.$
(3.26) $\displaystyle(f\ast\overline{f})(y)$ $\displaystyle\ =\ $
$\displaystyle\frac{1}{b^{2}}\left[\frac{1}{2}(|y|-1)\cos\left(\frac{|y|}{2}\right)+\cos\left(\frac{1}{2}-2a\right)\sin\left(\frac{1-|y|}{2}\right)\right.$
(3.27) $\displaystyle\
+\left.\frac{1}{2}|y|\cos\left(2a-\frac{|y|}{2}\right)+\sin\left(\frac{|y|}{2}\right)\right]$

for $0\leq|y|\leq 1$ and

$(f\ast\overline{f})(y)\ =\
\frac{1}{b^{2}}\left[\frac{1}{2}(|y|-2)\sin\left(\frac{1-|y|}{2}\right)+\sin\left(\frac{1-|y|}{2}\right)\right]$
(3.28)

for $1\leq|y|\leq 2$.

###### Proof.

Since $f$ is a real-valued function, $\overline{f}\ =\ f$. Therefore, we want
to find $(f\ast f)(y)$. Since our resulting function is an even function with
support $v_{n}=2$, we consider the values of $y\in[0,2]$ and reflect over the
$y$-axis to find the values of $y\in[-2,0]$. Using the definition of a
convolution, we get

$(f\ast f)(y)\ =\
\frac{1}{b^{2}}\int_{y-1}^{1}\cos\left(\frac{|x|}{2}-A\right)\cos\left(\frac{|y-x|}{2}-A\right)dx.$
(3.29)

We first consider the case of $y\in[0,1]$. We proceed to split the integral
into 3 cases: $x<0$, $x>0$ and $y\geq x$, $x>0$ and $y<x$. The contribution to
the integral from each case is $V_{1}$, $V_{2}$, and $V_{3}$ respectively.

For $x<0$, we get

$\displaystyle V_{1}\ $ $\displaystyle=\
\frac{1}{b^{2}}\int_{y-1}^{0}\cos\left(\frac{y}{2}\right)\cos\left(x+2a-\frac{y}{2}\right)dx$
$\displaystyle=\
\frac{1}{b^{2}}\left[\frac{1}{2}(y-1)\cos\left(\frac{y}{2}\right)+\cos\left(\frac{1}{2}-2a\right)\sin\left(\frac{1-y}{2}\right)\right].$
(3.30)

From $x>0$ and $y\geq x$, we get

$\displaystyle V_{2}\ $ $\displaystyle=\
\frac{1}{b^{2}}\int_{0}^{y}\cos\left(\frac{2x-y}{2}\right)\cos\left(\frac{y}{2}-2a\right)dx$
$\displaystyle=\
\frac{1}{b^{2}}\left[\frac{1}{2}y\cos\left(2a-\frac{y}{2}\right)+\sin\left(\frac{y}{2}\right)\right].$
(3.31)

From $x>0$ and $y<x$, we get

$\displaystyle V_{3}\ $ $\displaystyle=\
\frac{1}{b^{2}}\int_{y}^{1}\cos\left(\frac{x}{2}-a\right)\cos\left(\frac{y-x}{2}-r\right)dx$
$\displaystyle=\
\frac{1}{b^{2}}\left[\frac{1}{2}(y-1)\cos\left(\frac{y}{2}\right)+\cos\left(\frac{1}{2}-2a\right)\sin\left(\frac{1-y}{2}\right)\right].$
(3.32)

Since $V_{1}=V_{2}$, we get

$\displaystyle(f\ast f)(y)\ $ $\displaystyle=\ 2V_{1}+V_{3}$ $\displaystyle=\
\frac{1}{b^{2}}\left[\frac{1}{2}(y-1)\cos\left(\frac{y}{2}\right)+\cos\left(\frac{1}{2}-2a\right)\sin\left(\frac{1-y}{2}\right)\right.$
$\displaystyle\
+\left.\frac{1}{2}y\cos\left(2a-\frac{y}{2}\right)+\sin\left(\frac{y}{2}\right)\right]$
(3.33)

for $0\leq|y|\leq 1$. We now consider the case of $y\in[0,2]$. Since
$x\in[0,1]$, $x>0$ and $y-x>0$. Therefore, for $y\in[1,2]$, we get

$\displaystyle(f\ast f)(y)\ $ $\displaystyle=\
\frac{1}{b^{2}}\int_{y-1}^{1}\cos\left(\frac{x}{2}-a\right)\cos\left(\frac{y-x}{2}-a\right)dx$
$\displaystyle=\
\frac{1}{b^{2}}\left[\frac{1}{2}(y-2)\sin\left(\frac{1-y}{2}\right)+\sin\left(\frac{1-y}{2}\right)\right].$
(3.34)

∎

We now calculate $\phi$ using $\mathaccent 866{\phi}$. We first show that the
Fourier transform of $\mathaccent 866{\phi}$ is the same as the inverse
Fourier transform of $\mathaccent 866{\phi}$. Let $\mathcal{F}(g)$ denote the
Fourier transform of $g$. Since $\mathaccent 866{\phi}$ is even, we get

$\mathcal{F}(\mathaccent 866{\phi})\ =\ \int_{-\infty}^{\infty}\mathaccent
866{\phi}(y)e^{-2\pi ixy}dy\ =\ \int_{\phi}(y)e^{2\pi ixy}dy\ =\
\mathcal{F}^{-1}(\mathaccent 866{\phi}).$ (3.35)

Therefore, using Theorem (2.4), we get

$\phi(x)\ =\ \mathcal{F}^{-1}(\mathaccent 866{\phi(x)})\ =\ \mathcal{F}((f\ast
f)(y)).$ (3.36)

From the Convolution Theorem ((2.4)), we get

$\mathcal{F}(f\ast f(y))\ =\ \mathcal{F}(f)F(f)\ =\ \mathcal{F}(f)^{2}.$
(3.37)

Therefore, we need to compute the Fourier transform of

$f(x)\ =\ \frac{\cos\left(\frac{|x|}{2}-a\right)}{b}$ (3.38)

for $|x|\in[0,1]$. From the definition of Fourier transforms, we get

$\displaystyle\mathaccent 866{f}(y)\ $ $\displaystyle=\
\int_{-1}^{1}\frac{\cos\left(\frac{|x|}{2}-a\right)}{b}e^{-2\pi ixy}dx$
$\displaystyle=\
\int_{-1}^{1}\frac{\cos\left(\frac{|x|}{2}-a\right)}{b}\cos(2\pi
xy)dx+i\int_{-1}^{1}\frac{\cos\left(\frac{|x|}{2}-a\right)}{b}\sin(2\pi
xy)dx.$ (3.39)

Since the right integral is the integral from $-1$ to $1$ of an odd function,
it evaluates to $0$. Therefore, we get

$\displaystyle\mathaccent 866{f}(y)\ $ $\displaystyle=\
\int_{-1}^{1}\frac{\cos\left(\frac{|x|}{2}-a\right)}{b}\cos(2\pi xy)dy$
$\displaystyle=\ \frac{16\left(\cos(2\pi
x)\sin\left(\frac{1}{2}-a\right)+\sin(a)-4\pi
x\cos\left(\frac{1}{2}-a\right)\sin(2\pi
x)\right)^{2}}{(b-16b\pi^{2}x^{2})^{2}}.$ (3.40)

Substituting in our values of $a$ and $b$ for the cases of SO(even) and
SO(odd), we get

$\displaystyle p_{r}(\mathcal{F})\ \leq\ \frac{0.8645...}{r}\ \ \ {\rm for\
G=SO(even)}$ (3.41) $\displaystyle p_{r}(\mathcal{F})\ \leq\
\frac{1.1145...}{r}\ \ \ {\rm for\ G=SO(odd)}.$ (3.42)

We now present the graphs of the optimal and naive test functions and their
respective Fourier transforms. As shown in the plots, the two functions have
values that are close to each other and thus produce similar bounds.

Comparison of Bounds for the $1$-level density for $G={\rm SO(even)}$
---
Rank |  Bound From naive test function |  Bound From optimal test function
2 |  0.43750000 |  0.43231300
4 |  0.21875000 |  0.21615700
6 |  0.14583333 |  0.14410400
8 |  0.10937500 |  0.10807800
10 |  0.08750000 |  0.08646260
12 |  0.07291670 |  0.07205220
14 |  0.06250000 |  0.06175900
16 |  0.05468750 |  0.05403910
18 |  0.04861110 |  0.04803848
20 |  0.04375000 |  0.04323130

Comparison of Bounds for the $1$-level density for $G=\rm{SO(odd)}$
---
Rank |  Bound From naive test function |  Bound From optimal test function
1 |  1.12500000 |  1.11454000
3 |  0.37500000 |  0.37151300
5 |  0.22500000 |  0.22908000
7 |  0.16071400 |  0.15922000
9 |  0.12500000 |  0.12383838
11 |  0.10227300 |  0.10132200
13 |  0.08653850 |  0.08573380
15 |  0.07500000 |  0.07430270
17 |  0.06617650 |  0.06556120
19 |  0.05921050 |  0.05866000
21 |  0.05357140 |  0.05307333

As shown in the tables, the improvements in the bounds by the optimal function
are marginal, with a difference in the bounds only showing up in the
hundredths digit for the non-trivial bounds. Previous work has been done by
[BCDMZ, FM, ILS] to generate the optimal test functions for the $1$\- and
$2$-level densities. However, since the gain is so small, we choose to pursue
alternate methods rather than optimizing the test functions for higher levels;
we choose to look at higher levels to generate better bounds.

##  4\. Bounds from the $n$-Level Densities ($n$ even) with support $v=2/n$

Previously, the largest support found was from [HM] which was $v=1/(n-1)$.
This has been extended to $v=2/n$ for the $n$-level density by [C–]. The
increased support results in a great increase in the complexity of the
integrals. For support $v=1/(n-1)$, we only have a one-dimensional integral,
which can be easily integrated with methods like Mathematica’s NIntegrate,
Simpson’s Method, or Riemann approximations (we can also do directly through a
contour integral, though some work is required as there is a pole of high
order on the line of integration). However, the extended support introduces
$n$-dimensional integrals and sums over $n$-dimensional integrals which
renders these methods unusable or far too computationally intensive.

In this section, we describe how to obtain bounds using the $n$-level density
and support $v_{n}=2/n$. Additionally, we detail methods to lower the
complexity of the calculations, including converting our $n$-dimensional
integral to a $1$-dimensional integral, which allows us to use the
aforementioned methods.

######  Theorem 4.1.

Theorem 1.1 from [C–] Let $n\geq 2$ and ${\rm
supp}(\phi)\subset(-\frac{2}{n},\frac{2}{n})$. Define

$\sigma_{\phi}^{2}\ :=\ 2\int_{-\infty}^{\infty}|y|\mathaccent
866{\phi}(y)^{2}dy,$ (4.1) $R(m,i;\phi)\ :=\
2^{m-1}(-1)^{m+1}\sum_{l=0}^{i-1}(-1)^{l}\binom{m}{l}\\\
\left(-\frac{1}{2}\phi^{m}(0)+\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\mathaccent
866{\phi}(x_{2})\cdots\mathaccent 866{\phi}(x_{l+1})\right.\\\
\left.\int_{-\infty}^{\infty}\phi^{m-l}(x_{1})\frac{\sin(2\pi
x_{1}(1+|x_{2}|+\cdots+|x_{l+1}|))}{2\pi x_{1}}dx_{1}\cdots dx_{l+1}\right),$
(4.2)

,

$S(n,a,\phi)\ :=\
\sum_{l=0}^{\lfloor\frac{a-1}{2}\rfloor}\frac{n!}{(n-2l)!l!}R(n-2l,a-2l,\phi)\left(\frac{\sigma_{\phi}^{2}}{2}\right)^{l}.$
(4.3)

Then

$\lim_{\begin{subarray}{c}N\to\infty\\\ N\text{\rm
prime}\end{subarray}}\left\langle\left(D(f;\phi)-\left\langle
D(f;\phi)\right\rangle_{\pm}\right)^{n}\right\rangle_{\pm}\ =\
(n-1)!!\sigma_{\phi}^{n}1_{\rm even}(n)\pm S(n,a;\phi).$ (4.4)

######  Remark 4.2.

Since the integrand is an even function in $x_{1},x_{2},\dots,x_{l+1}$, we can
re-write $R$ as

$R(m,i;\phi)\ :=\ 2^{m-1}(-1)^{m+1}\sum_{l=0}^{i-1}(-1)^{l}\binom{m}{l}\\\
\left(-\frac{1}{2}\phi^{m}(0)+2^{l+1}\int_{0}^{\infty}\cdots\int_{0}^{\infty}\mathaccent
866{\phi}(x_{2})\cdots\mathaccent 866{\phi}(x_{l+1})\right.\\\
\left.\int_{0}^{\infty}\phi^{m-l}(x_{1})\frac{\sin(2\pi
x_{1}(1+|x_{2}|+\cdots+|x_{l}+1|))}{2\pi x_{1}}dx_{1}\cdots dx_{l+1}\right).$
(4.5)

This decreases the computation by a factor of $1/2^{l+1}$ when computing the
integral.

######  Remark 4.3.

We can rewrite (4.4) as

$\lim_{\begin{subarray}{c}N\to\infty\\\ N\text{\rm
prime}\end{subarray}}\frac{1}{|\mathcal{F}_{N}|}\sum_{f\in\mathcal{F}_{N}}\left(\sum_{j}\phi\left(\gamma_{f,j}c_{n}\right)-\mu(\phi,\mathcal{F}_{N})\right)^{n}\
=\ 1_{n\ \rm even}(n-1)!!\sigma_{\phi}^{n}\pm S(n,a;\phi)$ (4.6)

for $\phi$ an even Schwartz function with ${\rm supp}(\mathaccent
866{\phi})\subset(-\frac{2}{n},\frac{2}{n})$ and even $n=2m$.

######  Corollary 4.4.

For $\phi$ an even Schwartz function with ${\rm supp}(\mathaccent
866{\phi})\subset(-\frac{1}{n-1},\frac{1}{n-1})$ and even $n=2m$, we have

$\lim_{\begin{subarray}{c}N\to\infty\\\ N\text{\rm
prime}\end{subarray}}\frac{1}{|\mathcal{F}_{N}|}\sum_{f\in\mathcal{F}_{N}}\left(\sum_{j}\phi\left(\gamma_{f,j}c_{n}\right)-\mu(\phi,\mathcal{F})\right)^{n}\
=\ 1_{n\ \rm even}(n-1)!!\sigma_{\phi}^{n}\pm S(n,a;\phi).$ (4.7)

###### Proof.

From (4.7), we get

$\lim_{\begin{subarray}{c}N\to\infty\\\ N\text{\rm
prime}\end{subarray}}\frac{1}{|\mathcal{F}_{N}|}\sum_{f\in\mathcal{F}_{N}}\left(\sum_{j}\phi\left(\gamma_{f,j}c_{n}\right)-\mu(\phi,\mathcal{F}_{N})\right)^{n}\
=\ 1_{n\ \rm even}(n-1)!!\sigma_{\phi}^{n}\pm S(n,a;\phi).$ (4.8)

We now consider the following term, $T$, and show that it equals the LHS of
(4.8)

$T\ =\ \lim_{\begin{subarray}{c}N\to\infty\\\ N\text{\rm
prime}\end{subarray}}\frac{1}{|\mathcal{F}_{N}|}\sum_{f\in\mathcal{F}_{N}}\left(\sum_{j}\phi\left(\gamma_{f,j}c_{n}\right)-\mu(\phi,\mathcal{F}_{N})+\mu(\phi,\mathcal{F}_{N})-\mu(\phi,\mathcal{F})\right)^{n}.$
(4.9)

Applying the Binomial Theorem, we get

$T\ =\ \lim_{\begin{subarray}{c}N\to\infty\\\ N\text{\rm
prime}\end{subarray}}\frac{1}{|\mathcal{F}_{N}|}\sum_{f\in\mathcal{F}_{N}}\sum_{k=0}^{n}\binom{n}{k}\left(\sum_{j}\phi(\gamma_{f,j}c_{n})-\mu(\phi,\mathcal{F}_{N})\right)^{k}\left(\mu(\phi,\mathcal{F}_{N})-\mu(\phi,\mathcal{F})\right)^{n-k}.$
(4.10)

Since $\lim_{\begin{subarray}{c}N\to\infty\\\ N\text{\rm
prime}\end{subarray}}\left(\mu(\phi,\mathcal{F})-\mu(\phi,\mathcal{F}_{N})\right)\
=\ 0$, we would like to say each term vanishes in the limit except the $k=0$
term. Unfortunately more care is needed, as it is possible that the two terms
in each product could reinforce each other. A standard application of the
Cauchy-Schwarz inequality however suffices, We now have the square root of two
sums. The first involves $n$-level densities to even arguments, and is $O(1)$,
while the second involves our differences and thus tends to zero. Thus the
only term that contributes in the limit is the first, and

$\lim_{\begin{subarray}{c}N\to\infty\\\ N\text{\rm
prime}\end{subarray}}\frac{1}{|\mathcal{F}_{N}|}\sum_{f\in\mathcal{F}_{N}}\left(\sum_{j}\phi\left(\gamma_{f,j}c_{n}\right)-\mu(\phi,\mathcal{F}_{N})\right)^{n}\\\
=\lim_{\begin{subarray}{c}N\to\infty\\\ N\text{\rm
prime}\end{subarray}}\frac{1}{|\mathcal{F}_{N}|}\sum_{f\in\mathcal{F}_{N}}\left(\sum_{j}\phi\left(\gamma_{f,j}c_{n}\right)-\mu(\phi,\mathcal{F})\right)^{n}.\
$ (4.11)

Substituting this back into our original equation, we get

$\lim_{\begin{subarray}{c}N\to\infty\\\ N\text{\rm
prime}\end{subarray}}\frac{1}{|\mathcal{F}_{N}|}\sum_{f\in\mathcal{F}_{N}}\left(\sum_{j}\phi\left(\gamma_{f,j}c_{n}\right)-\mu(\phi,\mathcal{F})\right)^{n}\
=\ 1_{n\ \rm even}(n-1)!!\sigma_{\phi}^{n}\pm S(n,a;\phi).$ (4.12)

∎

We use the following from [ILS].

######  Theorem 4.5.

For a given test function $\phi$ where ${\rm supp}(\mathaccent
866{\phi})\subset(-1,1)$, the main term of the mean of the 1-level density of
$\mathcal{F}_{N}$ is

$\mu(\phi,\mathcal{F})\ :=\ \mathaccent
866{\phi}(0)+\frac{1}{2}\int_{-1}^{1}\mathaccent 866{\phi}(y)dy.$ (4.13)

######  Theorem 4.6.

For an even $n$ with $r\geq\mu(\phi,\mathcal{F})/\phi(0)$,

$p_{r}(\mathcal{F})\ \leq\ \frac{(n-1)!!\sigma_{\phi}^{n}\pm
S(n,\frac{n}{2};\phi)}{(r\phi(0)-\mu(\phi,\mathcal{F}))^{n}}.$ (4.14)

###### Proof.

For even-level densities, the contribution to the sum in Corollary (4.4) by
forms in which there are not $r$ zeroes at the central point is positive (as
$n$ is even), so removing them cannot increase the sum. Therefore, from
Corollary (4.4), we have

$\lim_{\begin{subarray}{c}N\to\infty\\\ N\text{\rm
prime}\end{subarray}}\frac{1}{|\mathcal{F}_{N}|}\sum_{f\in\mathcal{F}_{N,r}}\left(r\phi(0)+B_{f}(\phi)-\mu(\phi,\mathcal{F})\right)^{n}\
\leq\ 1_{n\ \rm even}(n-1)!!\sigma_{\phi}^{n}\pm S(n,a;\phi).$ (4.15)

We have $(r\phi(0)+B_{f}(\phi)-\mu(\phi,\mathcal{F}_{N}))^{n}$; we would like
to say dropping $B_{f}(\phi)$ cannot increase the sum, but if the first two
terms are less than the third, this is not the case. By our assumption on $r$,
however, we see the sum with and without $B_{f}(\phi)$ is positive and thus
dropping it leads to an upper bound:

$\lim_{\begin{subarray}{c}N\to\infty\\\ N\text{\rm
prime}\end{subarray}}\frac{1}{|\mathcal{F}_{N}|}\sum_{f\in\mathcal{F}_{N,r}}\left(r\phi(0)-\mu(\phi,\mathcal{F})\right)^{n}\
\leq\ 1_{n\ \rm even}(n-1)!!\sigma_{\phi}^{n}\pm S(n,a;\phi).$ (4.16)

Since the quantity $r\phi(0)-\mu(\phi,\mathcal{F}_{N})$ is not dependent on
$f$, we can separate it from the summation:

$\left(r\phi(0)-\mu(\phi,\mathcal{F})\right)^{n}\lim_{\begin{subarray}{c}N\to\infty\\\
N\text{\rm
prime}\end{subarray}}\frac{1}{|\mathcal{F}_{N}|}\sum_{f\in\mathcal{F}_{N,r}}1\
\leq\ 1_{n\ \rm even}(n-1)!!\sigma_{\phi}^{n}\pm S(n,a;\phi).$ (4.17)

Since $\sum_{f\in\mathcal{F}_{N,r}}1=|\mathcal{F}_{N,r}|$, we get

$\left(r\phi(0)-\mu(\phi,\mathcal{F}_{N})\right)^{n}\lim_{\begin{subarray}{c}N\to\infty\\\
N\text{\rm prime}\end{subarray}}\frac{|\mathcal{F}_{N,r}|}{|\mathcal{F}_{N}|}\
\leq\ 1_{n\ \rm even}(n-1)!!\sigma_{\phi}^{n}\pm S(n,a;\phi).$ (4.18)

Using the definition for the percent that vanish to exact order $r$ from
(2.11), we get

$\lim_{\begin{subarray}{c}N\to\infty\\\ N\text{\rm
prime}\end{subarray}}p_{r}(\mathcal{F}_{N})\ \leq\
\frac{(n-1)!!\sigma_{\phi}^{n}\pm
R_{n}(\phi)}{(r\phi(0)-\mu(\phi,\mathcal{F}_{N}))^{n}}.$ (4.19)

Taking the limit as $N\to\infty$, we get

$p_{r}(\mathcal{F})\ \leq\ \frac{1_{n\ \rm even}(n-1)!!\sigma_{\phi}^{n}\pm
S(n,a;\phi)}{(r\phi(0)-\mu(\phi,\mathcal{F}_{N}))^{n}}.$ (4.20)

∎

For quicker computation, we can convert the multi-dimensional $R$ integral to
a one-dimensional integral.

######  Lemma 4.7.

Let

$I_{1}(x,v)\ :=\ \frac{2i\pi vx-e^{2i\pi vx}+1}{2\pi^{2}v^{2}x^{2}}\ \ \ ,\ \
\ I_{2}(x,v)\ :=\ \frac{1-e^{-2i\pi vx}-2i\pi
vx}{2\pi^{2}v^{2}x^{2}}=I_{1}(x,v).$ (4.21) $R(m,i;\phi)\ =\
2^{m-1}(-1)^{m+1}\sum_{l=0}^{i-1}(-1)^{l}\binom{m}{l}\\\
\left(-\frac{1}{2}\phi^{m}(0)+\int_{-\infty}^{\infty}\phi^{m-l}(x)\frac{I_{1}(x)^{l}e^{2\pi
ix}-I_{2}(x)^{l}e^{-2\pi ix}}{4\pi ix}\right).$ (4.22)

###### Proof.

Let

$T\ :=\ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\mathaccent
866{\phi}(x_{2})\cdots\mathaccent 866{\phi}(x_{l+1})\\\
\int_{-\infty}^{\infty}\phi^{m-l}(x_{1})\frac{\sin(2\pi
x_{1}(1+|x_{2}|+\cdots+|x_{l+1}|))}{2\pi x_{1}}dx_{1}.\cdots dx_{l+1}.$ (4.23)

Then, we get

$R(m,i;\phi)\ =\
2^{m-1}(-1)^{m+1}\sum_{l=0}^{i-1}(-1)^{l}\binom{m}{l}\left(-\frac{1}{2}\phi^{m}(0)+T\right).$
(4.24)

We reduce $T$ to a 1-dimensional integral. To do so, we first show that the
conditions for Fubini’s Theorem hold which allows us to switch the order of
integration. Therefore, we show that integral of the absolute value converges.
Let

$T^{\prime}\ :=\
\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\mathaccent
866{\phi}(x_{2})\cdots\mathaccent 866{\phi}(x_{l+1})\\\
\int_{-\infty}^{\infty}\phi^{m-l}(x_{1})\left|\frac{\sin(2\pi
x_{1}(1+|x_{2}|+\cdots+|x_{l+1}|))}{2\pi x_{1}}\right|dx_{1}\cdots dx_{l+1}.$
(4.25)

Since $\phi$ and $\mathaccent 866{\phi}$ are non-negative and the absolute
value is non-negative, the value inside the integrand is non-negative.
Multiplying by $(1+|x_{2}|+\cdots+|x_{l+1})/(1+|x_{2}|+\cdots+|x_{l+1}|)$, we
get

$T^{\prime}\ =\
\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\mathaccent
866{\phi}(x_{2})\cdots\mathaccent
866{\phi}(x_{l+1})(1+|x_{2}|+\cdots+|x_{l+1}|)\\\
\int_{-\infty}^{\infty}\phi^{m-l}(x_{1})\left|\frac{\sin(2\pi
x_{1}(1+|x_{2}|+\cdots+|x_{l+1}|))}{2\pi
x_{1}(1+|x_{2}|+\cdots+|x_{l+1}|)}\right|dx_{1}\cdots dx_{l+1}.$ (4.26)

Since $|\sin(u)/u|\leq 1$ for all $u$, we get

$T^{\prime}\ \leq\
\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\mathaccent
866{\phi}(x_{2})\cdots\mathaccent
866{\phi}(x_{l+1})(1+|x_{2}|+\cdots+|x_{l+1}|)\int_{-\infty}^{\infty}\phi^{m-l}(x_{1})dx_{1}\cdots
dx_{l+1}.$ (4.27)

Since $\mathaccent 866{\phi}$ decays faster than any polynomial, the integral
converges so Fubini’s Theorem holds. Therefore, we can apply Fubini’s Theorem
to get

$T\ =\ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\mathaccent
866{\phi}(x_{2})\cdots\mathaccent 866{\phi}(x_{l+1})\\\
\int_{-\infty}^{\infty}\phi^{m-l}(x_{1})\frac{\sin(2\pi
x_{1}(1+|x_{2}|+\cdots+|x_{l+1}|))}{2\pi x_{1}}dx_{1}\cdots dx_{l+1}.$ (4.28)

Converting sine to exponential form, we get

$T\ =\ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\mathaccent
866{\phi}(x_{2})\cdots\mathaccent 866{\phi}(x_{l+1})\\\
\int_{-\infty}^{\infty}\phi^{m-l}(x+1)\frac{e^{2\pi ix_{1}}\cdots e^{2\pi
ix_{1}|x_{l+1}|}-e^{-2\pi ix_{1}}\cdots e^{-2\pi ix_{1}|x_{l+1}|}}{4\pi
ix_{1}}dx_{2}\cdots dx_{l+1}dx_{1}.$ (4.29)

To evaluate this integral, we find

$\displaystyle\int_{-\infty}^{\infty}\mathaccent 866{\phi}(y)e^{2\pi
ix_{1}|y|}\ $ $\displaystyle=\ 2\int_{0}^{v_{n}}\mathaccent
866{\phi}(y)e^{2\pi ix_{1}y}dy$ $\displaystyle=\ \frac{2i\pi vx-e^{2i\pi
vx}+1}{2\pi^{2}v^{2}x^{2}}=I_{1}(x,v).$ (4.30)

Additionally, we find

$\displaystyle\int_{-\infty}^{\infty}\mathaccent 866{\phi}(y)e^{-2\pi
x_{1}|y|}\ $ $\displaystyle=\ 2\int_{0}^{v_{n}}\mathaccent
866{\phi}(y)e^{-2\pi ix_{1}|y|}dy$ $\displaystyle=\ \frac{1-e^{-2i\pi vx-2i\pi
vx}}{2\pi^{2}v^{2}x^{2}}=\ I_{2}(x,v).$ (4.31)

Plugging these values into (4.28), we get

$\displaystyle T\ $ $\displaystyle=\
\int_{-\infty}^{\infty}I_{1}(x_{1})^{l}\phi^{m-l}(x_{1})\frac{e^{2\pi
ix_{1}}}{4\pi
ix_{1}}-\int_{-\infty}^{\infty}I_{2}(x_{1})^{l}\phi^{m-l}(x_{1})\frac{e^{-2\pi
ix_{1}}}{4\pi ix_{1}}$ $\displaystyle=\
\int_{-\infty}^{\infty}\phi^{m-l}(x)\frac{I_{1}(x)^{l}e^{2\pi
ix}-I_{2}(x)^{l}e^{-2\pi ix}}{4\pi ix}.$ (4.32)

Therefore, plugging in our value of $T$ from (4), we get

$R(m,i;\phi)\ =\ 2^{m-1}(-1)^{m+1}\sum_{l=0}^{i-1}(-1)^{l}\binom{m}{l}\\\
\left(-\frac{1}{2}\phi^{m}(0)+\int_{-\infty}^{\infty}\phi^{m-l}(x)\frac{I_{1}(x)^{l}e^{2\pi
ix}-I_{2}(x)^{l}e^{-2\pi ix}}{4\pi ix}\right).$ (4.33)

∎

######  Remark 4.8.

From the equations of $I_{1}(x,v)$ and $I_{2}(x,v)$, we see that $I_{1}(x,v)$
and $I_{2}(x,v)$ are complex conjugates. Therefore, when calculating the
difference of $I_{1}(x,v)$ and $I_{2}(x,v)$, we only need to calculate
$2\Re(I_{1}(x,v))$ which reduces computation.

We now want to calculate the values of $S$ for small cases to find bounds
using support $2/n$. We first consider the cases of $n=2$ and $n=4$ with the
naive test function.

##  5\. Plots of Bounds for support $v=2/n$

We display bounds found using support $v=2/n$. We first present bounds for the
naive test function.

Figure 1.  Approximate Bounds for the Percent of Vanishing to exact order $r$
for the case G=SO(even) with support $v=2$ for the $1$-level and $v=2/n$ for
the $n$-level with $n$ going from $1$ to $10$ and $r$ from $2$ through $20$
obtained using the naive test function.

Figure 2.  Approximate Bounds for the Percent of Vanishing to exact order $r$
for the case G=SO(even) with support $v=2$ for the $1$-level and $v=2/n$ with
$n$ going from 12 through 20 and $r$ going from $2$ through $20$ obtained
using the naive test function.

Figure 3.  Approximate Bounds for the Percent of Vanishing to exact order $r$
for the case G=SO(odd) with support $v=2$ for the $1$-level and $v=2/n$ with
$n$ going from $1$ through $10$ and $r$ going from $1$ through $21$ obtained
using the naive test function.

Figure 4.  Approximate Bounds for the Percent of Vanishing to exact order $r$
for the case G=SO(odd) with support $v=2$ for the $1$-level and $v=2/n$ with
$n$ going from $12$ through $20$ and $r$ going from $1$ through $21$ obtained
using the naive test function.

We now show our bounds using the optimal test function for the 1-level
density.

Figure 5.  Approximate Bounds for the Percent of Vanishing to exact order $r$
for the case G=SO(even) with support $v=2$ for the $1$-level and $v=2/n$ for
the n-level with $n$ going from $1$ to $10$ and $r$ from $2$ through $20$
obtained using the optimal test function.

Figure 6.  Approximate Bounds for the Percent of Vanishing to exact order $r$
for the case G=SO(even) with support $v=2$ for the $1$-level and $v=2/n$ with
$n$ going from 12 through 20 and $r$ going from $2$ through $20$ obtained
using the optimal test function.

Figure 7.  Approximate Bounds for the Percent of Vanishing to exact order $r$
for the case G=SO(odd) with support $v=2$ for the $1$-level and $v=2/n$ with
$n$ going from $1$ through $10$ and $r$ going from $1$ through $21$ obtained
using the optimal test function.

Figure 8.  Approximate Bounds for the Percent of Vanishing to exact order $r$
for the case G=SO(odd) with support $v=2$ for the $1$-level and $v=2/n$ with
$n$ going from $12$ through $20$ and $r$ going from $1$ through $21$ obtained
using the optimal test function.

As shown in the Figure (1), the 1-level optimal test function does not produce
better results than the naive test function for higher levels. While the
bounds for the 1-level optimal are better for the 1-level, interestingly the
naive test function outperforms it for higher levels and ranks.

See the table in §1.5 for a summary of the bounds attainable using the best
choice of level and the naive test function for each rank from $2$ through
$20$.

## References

  * [BFMT-B] O. Barrett, F. W. K. Firk, S. J. Miller and C. Turnage-butterbaugh, _From Quantum Systems to $L$-Functions: Pair Correlation Statistics and Beyond_, in Open Problems in Mathematics (editors John Nash Jr. and Michael Th. Rassias), Springer-Verlag, 2016, pages 123–171.
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  * [Od2] A. Odlyzko, _The $10^{22}$-nd zero of the Riemann zeta function_. Pages 139–144 in _Proceedings of the Conference on Dynamical, Spectral and Arithmetic Zeta Functions_ , ed. M. van Frankenhuysen and M. L. Lapidus, Contemporary Mathematics Series,AMS, Providence, RI, $2001$.
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