$\operatorname{Map}(S^{1},{\operatorname{Spec}}(\tilde{R}))\times{\operatorname{Spec}}\mathbb{F}_{p}\simeq\operatorname{Map}(S^{1},{\operatorname{Spec}}(R))$

over ${\operatorname{Spec}}\mathbb{F}_{p}$. We show that such a geometric
lifting occurs in many instances in the setting of
$\widehat{\mathbb{G}}$-Hochschild homology.

######  Construction 9.4.

Let $\widehat{\mathbb{G}}$ be a height $n$ formal group over $\mathbb{F}_{p}$
and let $R$ be an commutative $\mathbb{F}_{p}$-algebra. Let
$\widehat{\mathbb{G}}_{un}$ denote the universal deformation of
$\widehat{\mathbb{G}}$, which is a formal group over the
$R_{\widehat{\mathbb{G}}}^{un}$. As in section 8.3, we let
$\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}}$ denote its Cartier dual over this
$E_{\infty}$-ring.

######  Theorem 9.5.

Let $\widehat{\mathbb{G}}$ be a height $n$ formal group over $\mathbb{F}_{p}$
and let $X$ be an $\mathbb{F}_{p}$ scheme. Suppose there exists a lift
$\tilde{X}$ over the spectral deformation ring
$R^{un}_{\widehat{\mathbb{G}}}$. Then there exists a homotopy pullback square
of spectral algebraic stacks

$\textstyle{\operatorname{Map}(B\mathsf{Fix}_{\widehat{\mathbb{G}}},X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\scriptstyle{p^{\prime}}$$\textstyle{\operatorname{Map}(B\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}},\tilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{Spec(\mathbb{F}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{{\operatorname{Spec}}(R^{un}_{\widehat{\mathbb{G}}})}$

displaying
$\operatorname{Map}(B\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}},\tilde{X})$ as a
lift of $\operatorname{Map}(B\mathsf{Fix}_{\widehat{\mathbb{G}}},X)$.

###### Proof.

Given a map $p:X\to Y$ of spectral schemes, there is an induced morphism of
$\infty$-topoi

$p^{*}:\operatorname{Shv}^{\acute{e}t}_{Y}\to\operatorname{Shv}^{\acute{e}t}_{X}$

This pullback functor is symmetric monoidal, and moreover behaves well with
respect to the internal mapping objects. Now let
$X={\operatorname{Spec}}\mathbb{F}_{p}$ and
$Y={\operatorname{Spec}}R_{\widehat{\mathbb{G}}}^{un}$ and let $p$ be the map
induced by the universal property of the spectral deformation ring $R$. In
this particular case, this means there will be an equivalence

$p^{*}\operatorname{Map}(B\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}},\tilde{X})\simeq\operatorname{Map}(p^{*}B\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}},p^{*}\tilde{X})\simeq\operatorname{Map}(B\mathsf{Fix}_{\widehat{\mathbb{G}}},X)$

since $\tilde{X}\times{\operatorname{Spec}}\mathbb{F}_{p}\simeq\ X$ and
$p^{*}B{\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}}\simeq
B\mathsf{Fix}_{\widehat{\mathbb{G}}}}$. ∎

From this we conclude that the $\widehat{\mathbb{G}}$-Hochschild homology has
a lift in the geometric sense, in that there is a spectral mapping stack over
${\operatorname{Spec}}R^{un}_{\widehat{\mathbb{G}}}$ which base changes to
$\operatorname{Map}(B\widehat{\mathbb{G}}^{\vee},X)$. We would like to
conclude this at the level of global section $E_{\infty}$ algebras. This is
not formal unlesss we have a more precise understanding of the regularity
properties of
$\operatorname{Map}(B\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}},X)$ for an
affine spectral scheme $X={\operatorname{Spec}}A$.

Indeed, there is a map

$R\Gamma(\operatorname{Map}(B\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}},\tilde{X}),\mathcal{O})\otimes\mathbb{F}_{p}\to
R\Gamma(\operatorname{Map}(p^{*}B\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}},p^{*}\tilde{X}),\mathcal{O})$
(9.6)

but it is not a priori clear that this is an equivalence. In particular, we
have the following diagram of stable $\infty$-categories

$\textstyle{{\operatorname{Mod}}_{R^{un}_{\widehat{\mathbb{G}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{*}}$$\scriptstyle{\phi^{*}}$$\textstyle{\operatorname{QCoh}(\operatorname{Map}(B\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}},\tilde{X}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime*}}$$\textstyle{{\operatorname{Mod}}_{\mathbb{F}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime*}}$$\textstyle{\operatorname{QCoh}(\operatorname{Map}(B\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}},\tilde{X}))}$

for which we would like to verify the Beck-Chevalley condition holds; i.e.
that the following canonically defined map

$\rho:p^{*}\circ\phi_{*}\to\phi^{\prime}_{*}\circ p^{\prime*}$

is an equivalence. Here $\phi_{*}$ and $\phi^{\prime}_{*}$ are the right
adjoints and may be thought of as global section functors. This construction
applied to the structure sheaf $\mathcal{O}$ recovers the map (9.6).

This would follow from Proposition 2.1 upon knowing either that the spectral
stack
$\operatorname{Map}(B\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}},\tilde{X})$ is
representable by a derived scheme or, more generally if it is of finite
cohomological dimension. In fact it is the former:

######  Theorem 9.7.

Let $\widehat{\mathbb{G}}$ be as above and let $X={\operatorname{Spec}}A$
denote a spectral scheme. Then the mapping stack
$\operatorname{Map}(B\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}},X)$ is
representable by a spectral scheme.

###### Proof.

This will be an application of the Artin-Lurie representability theorem, cf.
[Lur16, Theorem 18.1.0.1]. Given spectral stacks $X,Y$, the derived spectral
mapping stack $\operatorname{Map}(Y,X)$ is representable by a spectral scheme
if and only if it is nilcomplete, infinitesimally cohesive and admits a
cotangent complex and if the truncation $t_{0}(\operatorname{Map}(Y,X))$ is
representable by a classical scheme. By Proposition 5.10 of [HLP14] if $Y$ is
of finite tor-amplitude and $X$ admits a cotangent complex, then so does the
mapping stack $\operatorname{Map}(Y,X)$; in our case $X$ is an honest spectral
scheme which has a cotangent complex. Note that the condition of being finite
tor-amplitude is local on the source with respect to the flat topology (cf.
[Lur16, Proposition 6.1.2.1]. Thus if there exists a flat cover $U\to Y$ such
that the composition $U\to Y\to{\operatorname{Spec}}R$ is of finite tor
amplitude, then $Y\to{\operatorname{Spec}}R$ itself has this property.
Infinitesimal cohesion follows from [TV08, Lemma 2.2.6.13]. The following
lemma takes care of nilcompleteness:

######  Lemma 9.8.

Let $Y$ be a spectral stack over ${\operatorname{Spec}}(R)$ which may be
written as a colimit of affine spectral schemes

$Y\simeq\operatorname{colim}{\operatorname{Spec}}A_{i}$

where each $A_{i}$ is flat over $R$ and let $X$ be a nilcomplete spectral
stack. Then $\operatorname{Map}_{Stk_{R}}(Y,X)$ is nilcomplete.

###### Proof.

The argument is similar to that of an analogous claim appearing in the proof
of Theorem 2.2.6.11 in [TV08]. Let $Y$ be as above. Then

$\operatorname{Map}(Y,X)\simeq\lim_{i}\operatorname{Map}({\operatorname{Spec}}A_{i},X)$

and so it amounts to verify this for when $Y={\operatorname{Spec}}A$ for
$A_{i}$ flat. In this case we see that for
$B\in\operatorname{CAlg}^{\operatorname{cn}}$,

$\operatorname{Map}({\operatorname{Spec}}A,X)(B)\simeq X(A\otimes_{R}B).$

The map

$\operatorname{Map}({\operatorname{Spec}}A,X)(B)\to\lim\operatorname{Map}({\operatorname{Spec}}A,X)(\tau_{\leq
n}B_{n})$

which we need to check is an equivalence now translates to a map

$X(A\otimes_{R}B)\to X(\tau_{\leq n}B\otimes_{R}A)$ (9.9)

We now use the flatness assumption on $A$. Using the general formula (cf.
[Lur, Proposition 7.2.2.13])in this case

$\pi_{n}(A\otimes
B)\simeq\operatorname{Tor}^{0}_{\pi_{0}(R)}(\pi_{0}A,\pi_{n}B)$

we conclude that $\tau_{\leq n}(A\otimes B)\simeq A\otimes\tau_{\leq n}B$.
Thus, 9.9 above becomes a map

$X(A\otimes_{R}B)\to X(\tau_{\leq n}(B\otimes_{R}A))$

which is an equivalence because $X$ was itself assumed to be nilcomplete. ∎

Finally we show that the truncation is an ordinary scheme. Note first of all
that the truncation functor

$t_{0}:SStk\to Stk$

preserves limits and colimits. It is induced from the Eilenberg Maclane
functor

$H:\operatorname{CAlg}^{0}\to\operatorname{CAlg},\,\,\,A\mapsto HA$

which is itself adjoint to the truncation functor on $E_{\infty}$ rings. One
sees that the truncation functor $t_{0}=H^{*}:SStk\to Stk$ will have as a
right adjoint the functor

$\pi_{0}^{*}:Stk\to SStk,$

induced by the $\pi_{0}$ functor

$R\mapsto\pi_{0}R$

Thus it is right exact and preserves colimits. Hence if $Y=BG$ for some
spectral group scheme $G$, then $t_{0}BG\simeq Bt_{0}G$. Now, one has the
identification

$t_{0}\operatorname{Map}(Y,X)\simeq\operatorname{Map}(t_{0}Y,t_{0}X),$

which in our situation becomes

$t_{0}\operatorname{Map}(B\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}},X)\simeq\operatorname{Map}(BG,t_{0}X)$

for some (classical) affine group scheme $G$. Recall that the only classical
maps between $f:BG\to t_{0}X$ between a classifying stack and a scheme
$t_{0}X$ are the constant ones. Hence we conclude that the truncation of this
spectral mapping stack is equivalent to the scheme $t_{0}X$, the truncation of
$X$. ∎

###  9.1 Topological Hochschild homology

As we saw, for a height $n$ formal group $\widehat{\mathbb{G}}$ over a finite
field $k$, there exists a lift $\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}}$ of
the Cartier dual of $\widehat{\mathbb{G}}$; this allows one to define a lift
of $\widehat{\mathbb{G}}$-Hochschild homology. We show that when the formal
group is $\widehat{\mathbb{G}_{m}}$ this lift is precisely topological
Hochschild homology, at least after $p$-completion, as one would expect. For
the remainder of this section we let
$\widehat{\mathbb{G}}=\widehat{\mathbb{G}_{m}}$, the formal multiplicative
group.

Let $X$ be a fixed spectral stack. We remark that there exists an adjunction
of $\infty$-topoi:

$\mathcal{S}\rightleftarrows SStk_{X}$

where one has on the right hand side the $\infty$-category of spectral stacks
over $X$.

First, one has the following proposition; here we think of $S^{1}$ as a
“constant stack induced by the adjunction

$\pi^{*}:\mathcal{S}\rightleftarrows SStk_{R}:\pi_{*}$

######  Proposition 9.10.

There exists a canonical map

$S^{1}\to B\mathsf{Fix}^{un}_{\widehat{\mathbb{G}}}$

of group objects in the $\infty$-category of spectal stacks over
$\mathbb{S}_{p}$.

###### Proof.

By [MRT19, Construction 3.3.1], there is a canonical map

$\mathbb{Z}\to\mathsf{Fix}$ (9.11)

in the category of fpqc abelian sheaves over
${\operatorname{Spec}}\mathbb{Z}_{(p)}$. We claim that the (discrete) group
scheme $\mathsf{Fix}$ is none other than the truncation of the spectral group
scheme

$\widehat{\mathbb{G}}_{un}^{\vee}\to{\operatorname{Spec}}\mathbb{S}_{p}$

This follows from the fact that $\widehat{\mathbb{G}}_{un}^{\vee}$ is flat
over $\mathbb{S}_{p}$, as the corresponding Hopf algebra is flat. As a result,
the base change of this spectral group scheme along the map

${\operatorname{Spec}}\mathbb{Z}_{p}\to{\operatorname{Spec}}\mathbb{S}_{p}$

is itself flat over $\mathbb{Z}_{p}$ and in particular is $0$-truncated. By
definition, this is $\mathsf{Fix}$. Now, there is an adjunction

$i^{*}:SStk_{\mathbb{S}_{p}}\rightleftarrows Stk_{\mathbb{Z}_{p}}:t_{0}$

against which the map (9.11) is lifted to a map

$\mathbb{Z}\to\widehat{\mathbb{G}}_{un}^{\vee}.$

in $SStk_{\mathbb{S}_{p}}$. This will be a map of group objects, since the
adjoint pair preserves the group structure. Delooping this, we obtain the
desired map

$S^{1}\simeq B\mathbb{Z}\to B\widehat{\mathbb{G}}_{un}^{\vee}.$

∎

Let $X={\operatorname{Spec}}A$ be an affine spectral scheme. By taking mapping
spaces, the above proposition furnishes a map

$\operatorname{Map}(B\widehat{\mathbb{G}}_{un}^{\vee},X)\to\operatorname{Map}(S^{1},X);$

applying global sections further begets a map

$f:\operatorname{THH}(A)\to
R\Gamma(\operatorname{Map}(B\widehat{\mathbb{G}}_{un}^{\vee},X),\mathcal{O})$

of $E_{\infty}$ $\mathbb{S}_{p}$-algebras.

######  Theorem 9.12.

Let

$f_{p}:\operatorname{THH}(A;\mathbb{Z}_{p})\to
R\Gamma(\operatorname{Map}(B\widehat{\mathbb{G}}_{un}^{\vee},X),\mathcal{O})^{\widehat{}}_{p}$

denote the $p$-completion of the above map. Then $f$ is an equivalence.

###### Proof.

Since this is a map of $p$-complete spectra, it is enough to verify it is an
equivalence upon tensoring with the Moore spectrum $\mathbb{S}_{p}/p$. In
fact, since these are both connective spectra, one can go further and test
this simply by tensoring with $\mathbb{F}_{p}$ (eg. by [Mao20, Corollary
A.33]) Hence, we are reduced to showing that

$\operatorname{THH}(A;\mathbb{Z}_{p})\otimes_{\mathbb{S}_{p}}\mathbb{F}_{p}\to
R\Gamma(\operatorname{Map}(B\widehat{\mathbb{G}}_{un}^{\vee},X),\mathcal{O})^{\widehat{}}_{p}\otimes\mathbb{F}_{p}$

is an equivalence of $E_{\infty}$ $\mathbb{F}_{p}$-algebras. By generalities
on topological Hochschild homology, we have the following identification of
the left hand side:

$\operatorname{THH}(A;\mathbb{Z}_{p})\otimes_{\mathbb{S}_{p}}\mathbb{F}_{p}\simeq\operatorname{HH}(A\otimes_{\mathbb{S}_{p}}\mathbb{F}_{p}/\mathbb{F}_{p}).$

Now we can use Theorem 9.7 to identify the right hand side with the global
sections of the following mapping stack

$\operatorname{Map}(B\widehat{\mathbb{G}}_{un}^{\vee},X)\times{\operatorname{Spec}}\mathbb{F}_{p}\simeq\operatorname{Map}(B\widehat{\mathbb{G}}_{un}^{\vee}\times{\operatorname{Spec}}\mathbb{F}_{p},X\times{\operatorname{Spec}}\mathbb{F}_{p})$

By Proposition 9.3, this is precisely
$\operatorname{HH}(A\otimes_{\mathbb{S}_{p}}\mathbb{F}_{p}/\mathbb{F}_{p})$,
whence the equivalence. ∎

##  10 Filtrations in the spectral setting

In Section 6 an interpretation of the HKR filtration on Hochshild homology was
given in terms of a degeneration of $\widehat{\mathbb{G}_{m}}$ to
$\widehat{\mathbb{G}_{a}}$. Moreover, this was expressed as an example of the
deformation to the normal cone construction of section 5.

In Section 9, we further saw that these $\widehat{\mathbb{G}}$-Hochshchild
homology theories may be lifted beyond the integral setting. A natural
question then arises: do the filtrations come along for the ride as well?
Namely, does there exist a filtration on
$\operatorname{THH}^{\widehat{\mathbb{G}}}(-)$ which recovers upon base-
changing along $R^{un}_{\widehat{\mathbb{G}}}\to k$, the filtered object
corresponding to $HH^{\widehat{\mathbb{G}}}(-)$?

We will not seek to answer this question here. However we do give a reason why
some negative results might be expected. As mentioned in the introduction,
many of the constructions do work integrally. For example, one can talk about
the deformation to the normal cone
$Def_{\mathbb{A}^{1}/\mathbb{G}_{m}}(\widehat{\mathbb{G}})$ of an arbitrary
formal group over ${\operatorname{Spec}}\mathbb{Z}$. If we apply this to
$\widehat{\mathbb{G}_{m}}$ we obtain a degeneration of the formal
multiplicative group to the formal additive group. We let
$Def_{\mathbb{A}^{1}/\mathbb{G}_{m}}(\widehat{\mathbb{G}_{m}})^{\vee}$ the
Cartier dual, as in section 4. In [Toë19] the Cartier dual to
$\widehat{\mathbb{G}_{m}}$ is described to be
${\operatorname{Spec}}(Int(\mathbb{Z}))$, the spectrum of the ring of integer
valued polynomials on $\mathbb{Z}$. Moreover it is shown that
$B{\operatorname{Spec}}(Int(\mathbb{Z}))$ is the affinization of $S^{1}$,
hence one can recover (integral) Hochshild homology mapping out of this.

Let us suppose there exists a lift of $Def(\widehat{\mathbb{G}_{m}})^{\vee}$
to the sphere spectrum, which we shall denote by
$Def^{\mathbb{S}}(\widehat{\mathbb{G}_{m}})^{\vee}$. This would allow us to
define a mapping stack in the $\infty$-category
$sStk_{\mathbb{A}^{1}/\mathbb{G}_{m}}$ of spectral stacks over the spectral
variant of $\mathbb{A}^{1}/\mathbb{G}_{m}$. By the results of [Mou19], this
comes equipped with a filtration on its cohomology, which we would like to
think of as recovering topological Hochschild homology.

However, over the special fiber
$B\mathbb{G}_{m}\to\mathbb{A}^{1}/\mathbb{G}_{m}$, we would expect that such a
lift $Def^{\mathbb{S}}(\widehat{\mathbb{G}_{m}})^{\vee}$ recovers the formal
additive group $\widehat{\mathbb{G}_{a}}$. More precisely, we would get a
formal group over the sphere spectrum
$\widehat{\mathbb{G}}\to{\operatorname{Spec}}\mathbb{S}$ which pulls back to
the formal additive group $\mathbb{G}_{a}$ along the map
$\mathbb{S}\to\mathbb{Z}$. However, by [Lur18, Proposition 1.6.20], this can
not happen. Indeed there it is shown that $\widehat{\mathbb{G}_{a}}$ does not
belong to the essential image of
$\operatorname{FGroup}(\mathbb{S})\to\operatorname{FGroup}(\mathbb{Z})$.

We summarize this discussion into the following proposition.

######  Proposition 10.1.

There exists no lift of
$Def_{\mathbb{A}^{1}/\mathbb{G}_{m}}(\widehat{\mathbb{G}}_{m})$ over to the
sphere spectrum. In particular, there exists no formal group
$\widetilde{\widehat{\mathbb{G}}}$ over $\mathbb{A}^{1}/\mathbb{G}_{m}$,
relative to $\mathbb{S}$ such that
$\widetilde{\widehat{\mathbb{G}}}\times{\operatorname{Spec}}\mathbb{Z}\simeq
Def_{\mathbb{A}^{1}/\mathbb{G}_{m}}(\widehat{\mathbb{G}_{m}})$.

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# Jacob’s ladders and infinite set of transmutations of asymptotic complete
hybrid formula on level curves in Gauss’ plane

Jan Moser  Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
<EMAIL_ADDRESS>
###### Abstract.

In this paper we have obtained new phenomenon lying in the following: every
fixed asymptotic complete hybrid formula (we call it as mother formula)
generates infinite set of new formulas (transmutations) such that every new
formula expresses a close binding between some subset of $\\{|\zeta(s)|\\}$
and subset of moduli of certain integral and meromorphic functions.

Dedicated to old alchemists

###### Key words and phrases:

Riemann zeta-function

##  1\. Introduction

### 1.1.

Let us remind that the following sets of values

(1.1)
$\begin{split}&\left\\{|\zeta\left(\frac{1}{2}+it\right)|^{2}\right\\}\\\
&\\{f_{1}(t)\\}=\\{\sin^{2}t\\},\ \\{f_{2}(t)\\}=\\{\cos^{2}t\\},\
\\{f_{3}(t)\\}=\\{\cos 2t\\},\\\ &t\in[\pi L,\pi L+U],\
U\in(0,\pi/4),L\in\mathbb{N}\end{split}$

generate the following secondary asymptotic complete hybrid formula (see [8],
(3.7), $k_{1}=k_{2}=k_{3}=1$)

(1.2)
$\begin{split}&\left|\zeta(\frac{1}{2}+i\alpha_{1}^{1,1})\right|^{2}\sin^{2}(\alpha_{0}^{1,1})-\\\
&-\left\\{1+\mathcal{O}(\frac{\ln\ln L}{\ln
L})\right\\}\left|\zeta(\frac{1}{2}+i\alpha_{1}^{2,1})\right|^{2}\cos^{2}(\alpha_{0}^{2,1})+\\\
&+\left|\zeta(\frac{1}{2}+i\alpha_{1}^{3,1})\right|^{2}\cos(2\alpha_{0}^{3,1})=0,\\\
&\forall\\-L\geq L_{0}>0,\end{split}$

($L_{0}$ is sufficiently big), where

(1.3) $\begin{split}&\alpha_{r}^{l,1}=\alpha_{r}(U,\pi L,f_{l}),\ r=0,1,\
l=1,2,3,\\\ &\alpha_{0}^{l,1}\in(\pi L,\pi L+U),\
\alpha_{1}^{l,1}\in(\overset{1}{\wideparen{\pi L}},\overset{1}{\wideparen{\pi
L+U}}),\end{split}$

and

$[\overset{1}{\wideparen{\pi L}},\overset{1}{\wideparen{\pi L+U}}]$

is the first reverse iteration (by means of the Jacob’s ladder, see [4]) of
the basic segment

$[\pi L,\pi L+U]=[\overset{0}{\wideparen{\pi L}},\overset{0}{\wideparen{\pi
L+U}}].$

######  Remark 1.

The components of the main $\zeta$-disconnected set (for our case)

$\Delta(\pi L,U,1)=[\pi L,\pi L+U]\cup[\overset{1}{\wideparen{\pi
L}},\overset{1}{\wideparen{\pi L+U}}]$

are separated each from other by gigantic distance $\rho$, (see [4], (5.12),
comp. [7], (2.2) – (2.9))):

(1.4) $\rho\\{[\pi L,\pi L+U];[\overset{1}{\wideparen{\pi
L}},\overset{1}{\wideparen{\pi L+U}}]\\}\sim\pi(1-c)\frac{L}{\ln L},\
L\to\infty,$

($c$ stands for the Euler’s constant).

### 1.2.

In this paper we obtain, for example, the following transmutations of the
mother formula (1.2). For every fixed and admissible $U$ and $L$ (see (1.1),
(1.2)), the formula (1.2) generates:

  * (a)

continuum sets

(1.5)
$\overset{(1)}{\Omega}_{l}(U,L),\overset{(2)}{\Omega}_{l}(U,L),\subset\mathbb{C},\
l=1,2,3$

such that for any elements

$s_{l}^{1}\in\overset{(1)}{\Omega}_{l},s_{l}^{2}\in\overset{(2)}{\Omega}_{l}$

we have the following formula (transmutation of (1.2))

(1.6)
$\begin{split}&|\zeta(s_{1}^{1})|^{2}|\zeta(s_{1}^{2})|^{2}-\left\\{1+\mathcal{O}\left(\frac{\ln\ln
L}{\ln
L}\right)\right\\}|\zeta(s_{2}^{1})|^{2}|\zeta(s_{2}^{2})|^{2}+|\zeta(s_{3}^{1})|^{2}|\zeta(s_{3}^{2})|=0,\end{split}$

  * (b)

the continuum set

(1.7) $\overset{(5)}{\Omega}_{l}(U,L)\subset\mathbb{C},l=1,2,3$

such that for any elements

$s_{l}^{1}\in\overset{(1)}{\Omega}_{l},s_{l}^{5}\in\overset{(5)}{\Omega}_{l}$

we have the following formula (transmutation of (1.2))

(1.8)
$\frac{|\zeta(s_{1}^{1})|^{2}}{|\Gamma(s_{1}^{5})|^{2}}-\left\\{1+\mathcal{O}\left(\frac{\ln\ln
L}{\ln
L}\right)\right\\}\frac{|\zeta(s_{2}^{1})|^{2}}{|\Gamma(s_{2}^{5})|^{2}}+\frac{|\zeta(s_{3}^{1})|^{2}}{|\Gamma(s_{3}^{5})|}=0,$

  * (c)

the continuum set

(1.9) $\overset{(7)}{\Omega}_{l}(U,L,k_{l})\subset\mathbb{C},\ l=1,2,3,\
[(k_{1})^{2},(k_{2})^{2},(k_{3})^{2}]\in(0,1)^{3}$

(for every fixed and admissible $k_{1},k_{2},k_{3}$) such that for any fixed
elements

$s_{l}^{1}\in\overset{(1)}{\Omega}_{l},s_{l}^{7}\in\overset{(5)}{\Omega}_{l}$

we have the following transmutation of (1.2)

(1.10)
$\begin{split}&|\zeta(s_{1}^{1})|^{2}|\operatorname{sn}(s_{1}^{7},k_{1})|^{2}-\\\
&-\left\\{1+\mathcal{O}\left(\frac{\ln\ln L}{\ln
L}\right)\right\\}|\zeta(s_{2}^{1})|^{2}|\operatorname{cn}(s_{2}^{7},k_{2})|^{2}+|\zeta(s_{3}^{1})|^{2}|\operatorname{dn}(s_{3}^{7},k_{3})|=0,\end{split}$

(for Jacobi’s elliptic functions of moduli $k_{1},k_{2},k_{3}$).

### 1.3.

Now we give the following remarks.

######  Remark 2.

The formula (1.1) gives the example of the continuum set of transmutations of
the mother formula (1.2).

######  Remark 3.

Let us remind that the mother formula is written as orthogonality condition
for corresponding vectors. We see that the transmutations (1.6), (1.8) as well
as (1.10) inherit this property.

######  Remark 4.

The method presented in this paper is general one in the following sense:

  * (a)

single secondary asymptotic complete hybrid formula (1.2) generates the
infinite set of its transmutations,

  * (b)

there is an infinite set of asymptotic complete hybrid formulas, (see [5] –
[8] concerning evolution from $\zeta$-factorization formula to complete hybrid
formula).

######  Remark 5.

Let us remind that the general feature of all transmutations is a new
phenomenon that lies in close binding between some subsets of $|\zeta(s)|$ and
subsets of moduli of other integral and meromorphic functions.

######  Remark 6.

This paper is based also on new notions and methods in the theory of the
Riemann zeta-function we have introduced in our series of 48 papers concerning
Jacob’s ladders. These can be found in arXiv [math.CA] starting with the paper
[2].

##  2\. Jacob’s ladder as basis of secondary complete hybrid formula

### 2.1.

Let us remind that the Jacob’s ladder

$\varphi_{1}(t)=\frac{1}{2}\varphi(t)$

was introduced in [2], (see also [3]), where the function $\varphi_{1}(t)$ is
arbitrary solution of the non-linear integral equation (also introduced in
[2])

$\int_{0}^{\mu[x(T)]}Z^{2}(t)e^{-\frac{2}{x(T)}t}{\rm
d}t=\int_{0}^{T}Z^{2}(t){\rm d}t,$

of course,

$\begin{split}&Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right),\\\
&\vartheta(t)=-\frac{t}{2}\ln\pi+\operatorname{Im}\left\\{\ln\Gamma\left(\frac{1}{4}+i\frac{t}{2}\right)\right\\},\end{split}$

where each admissible function $\mu(y)$ generates a solution

$y=\varphi(T;\mu)=\varphi(T);\ \mu(y)>7y\ln y.$

We call the function $\varphi_{1}(T)$ the Jacob’s ladder as an analogue of the
Jacob’s dream in Chumash, Bereishis, 28:12.

######  Remark 7.

By making use of the Jacob’s ladders we have shown (see [2]) that the
classical Hardy-Littlewood integral (1918)

$\int_{0}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|{\rm d}t$

has - in addition to previously known Hardy-Littlewood expression (and other
similar) possessing an unbounded error term at $T\to\infty$ \- the following
infinite set of almost exact representations

$\int_{0}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|{\rm
d}t=\varphi_{1}(T)\ln\\{\varphi_{1}(T)\\}+(c-\ln
2\pi)\varphi_{1}(T)+c_{0}+\mathcal{O}\left(\frac{\ln T}{T}\right)$

as

$T\to\infty,$

where $c$ is the Euler’s constant and $c_{0}$ is the constant from the
Titschmarsh-Kober-Atkinson formula.

### 2.2.

Next, we have obtained (see [2], (6.2)) the following formula

(2.1) $T-\varphi_{1}(T)\sim(1-c)\pi(T);\ \pi(T)\sim\frac{T}{\ln T},\
T\to\infty,$

where $\pi(T)$ is the prime-counting function.

######  Remark 8.

Consequently, the Jacob’s ladder $\varphi_{1}(T)$ can be viewed by our formula
(2.1) as an asymptotic complementary function to the function

$(1-c)\pi(T)$

in the following sense

$\varphi_{1}(T)+(1-c)\pi(T)\sim T,\ T\to\infty.$

Since Jacob’s ladder is exactly increasing function we have the reversely
iterated sequence

$\\{\overset{k}{T}\\}_{k=1}^{k_{0}},\ k_{0}\in\mathbb{N}:\
\varphi_{1}(\overset{k}{T})=\overset{k-1}{T},\ \overset{0}{T}=T>T_{0}>0,\
\overset{k}{T}=\varphi_{1}^{-k}(T),$

where $T_{0}$ is a sufficiently big (we fix $k_{0}$). Next we have (see [4],
(2.5) – (2.7), (5.1) – (5.13)) the following basic property generated by the
Jacob’s ladder.

######  Property.

For every segment

$[T,T+U],\ U=o\left(\frac{T}{\ln T}\right),\ T\to\infty$

there is the following class of disconnected sets

(2.2)
$\Delta(T,U,k)=\bigcup_{r=0}^{k}[\overset{r}{\wideparen{T}},\overset{r}{\wideparen{T+U}}],\
1\leq k\leq k_{0}$

generated by the Jacob’s ladder $\varphi_{1}(t)$.

######  Remark 9.

Disconnected set (2.2) has the following properties (see [4], (2.5) – (2.7)):

  * (a)

lengths of its components are given by

(2.3) $|[\overset{r}{\wideparen{T}},\overset{r}{\wideparen{T+U}}]|\sim
o\left(\frac{T}{\ln T}\right),\ T\to\infty,$

  * (b)

lengths of adjacent segments are given by

(2.4)
$|[\overset{r-1}{\wideparen{T+U}},\overset{r}{\wideparen{T}}]|\sim(1-c)\frac{T}{\ln
T},\ T\to\infty,$

i.e. for distance $\rho$ of two consecutive components we have

(2.5)
$\rho\\{[\overset{r-1}{\wideparen{T}},\overset{r-1}{\wideparen{T+U}}];[\overset{r}{\wideparen{T}},\overset{r}{\wideparen{T+U}}]\\}\sim(1-c)\frac{T}{\ln
T},\ T\to\infty,\ r=1,\dots,k.$

######  Remark 10.

Formula (1.4) follows from (2.5) for $r=1$ and $T=\pi L$.

######  Remark 11.

Asymptotic behavior of the components of the disconnected set (2.2) is as
follows: at $T\to\infty$ these components recede unboundedly each from other
and all together are receding to infinity. Hence, at $T\to\infty$ the set of
the components of (2.2) behaves as one-dimensional Friedman-Hubble universe.

### 2.3.

Finally, let us remind that the functions (see (1.2), (1.3))

$\alpha_{r}^{l,1}=\alpha_{r}(U,\pi L,f_{l}),\ r=0,1,\ l=1,2,3$

are generated by the Jacob’s ladder $\varphi_{1}(T)$ and its iterations as
follows (see [4], (6.3), [5], (1.14), [7], (3.6))

(2.6) $\begin{split}&\alpha_{r}^{l,1}=\varphi_{1}^{1-r}(d_{l}),\ d_{l}=d(U,\pi
L,f_{l}),\ r=0,1,\ l=1,2,3,\\\ &\alpha_{r}^{l,1}\in(\overset{r}{\wideparen{\pi
L}},\overset{r}{\wideparen{\pi L+U}}),\ r=0,1.\end{split}$

##  3\. Transmutation of type $[\zeta,\zeta]$

### 3.1.

Let us remind that the loci

(3.1) $|\zeta(s)|=c,\ c>0,\ s\in\mathbb{C}\setminus\\{1\\}$

is the level curve, (comp. [9], pp. 120 – 122). Of course, the topological
structure of the level curve is (in general) very complicated. For example,
the level curve

$|\zeta(s)|=0.8$

contains three ovals (apart from another) situated in the neighborhood of the
first three roots of the equation

$\zeta\left(\frac{1}{2}+it\right)=0,$

i.e. this one represents the disconnected set, (see [1], p. 371, FIG. 149).

### 3.2.

First, we define for every admissible and fixed $U,L$ (see (1.1), (1.2)) the
following level curves:

(3.2) $\overset{(1)}{\Omega}_{l}(U,L),\ l=1,2,3$

as the loci

(3.3)
$|\zeta(s_{l}^{1})|=\left|\zeta\left(\frac{1}{2}+i\alpha_{1}^{l,1}\right)\right|,$

where

(3.4) $s_{l}^{1}=s_{l}^{1}(U,L)\in\overset{(1)}{\Omega}_{l}\subset\mathbb{C},\
l=1,2,3,$

(of course, the sets $\overset{(1)}{\Omega}_{l}$ are continuum ones).

### 3.3.

Secondly, we use the loci (3.1) for the following values

(3.5) $c_{1}=|\sin\alpha_{0}^{1,1}|,\ c_{2}=|\cos\alpha_{0}^{2,1}|,\
c_{3}=\cos 2\alpha_{0}^{3,1},$

see (1.2) where, of course

$\cos(2\alpha_{0}^{3,1})=\cos[2(\alpha_{0}^{3,1}-\pi L)]>0.$

Consequently, we define for every admissible and fixed $U,L$ the following
three level curves

(3.6) $\overset{(2)}{\Omega}_{l}(U,L),\ l=1,2,3$

as the loci

(3.7) $\begin{split}&|\zeta(s_{1}^{2})|=|\sin\alpha_{0}^{1,1}|,\
s_{1}^{2}=s_{1}^{2}(U,L)\in\overset{(2)}{\Omega}_{1},\\\
&|\zeta(s_{2}^{2})|=|\cos\alpha_{0}^{2,1}|,\
s_{2}^{2}=s_{2}^{2}(U,L)\in\overset{(2)}{\Omega}_{2},\\\
&|\zeta(s_{3}^{2})|=\cos 2\alpha_{0}^{3,1},\
s_{3}^{2}=s_{3}^{2}(U,L)\in\overset{(2)}{\Omega}_{3}.\end{split}$

### 3.4.

Finally, construction of the sets
$\overset{(1)}{\Omega}_{l},\overset{(2)}{\Omega}_{l}$ together with
application of the substitutions (2.3), (3.7) into (1.2) completes the proof
of the following.

######  Theorem 1.

For every fixed and admissible $U,L$ (see (1.1), (1.2)) there are continuum
sets

(3.8) $\overset{(1)}{\Omega}_{l}(U,L),\
\overset{(2)}{\Omega}_{l}(U,L)\subset\mathbb{C},\ l=1,2,3$

such that for every elements

(3.9) $s_{l}^{1}\in\overset{(1)}{\Omega}_{l},\
s_{l}^{2}\in\overset{(2)}{\Omega}_{l}$

we have the following formula (transmutation of (1.2))

(3.10)
$\begin{split}&|\zeta(s_{1}^{1})|^{2}|\zeta(s_{1}^{2})|^{2}-\left\\{1+\mathcal{O}\left(\frac{\ln\ln
L}{\ln
L}\right)\right\\}|\zeta(s_{2}^{1})|^{2}|\zeta(s_{2}^{2})|^{2}+|\zeta(s_{3}^{1})|^{2}|\zeta(s_{3}^{2})|=0.\end{split}$

##  4\. Transmutations of types $[\zeta,\cos]$,
$[\zeta,(s)^{n_{1}},(s)^{n_{2}},(s)^{n_{3}}]$

### 4.1.

We use the loci

(4.1) $|\cos s|=c,\ c>0,\ s\in\mathbb{C}$

for the three values (3.5). Consequently, we define for every admissible and
fixed $U,L$ the following level curves

(4.2) $\overset{(3)}{\Omega}_{l},\ l=1,2,3$

as the loci

(4.3) $\begin{split}&|\cos s_{1}^{3}|=|\sin\alpha_{0}^{1,1}|,\
s_{1}^{3}=s_{1}^{3}(U,L)\in\overset{(3)}{\Omega}_{1},\\\ &|\cos
s_{2}^{3}|=|\cos\alpha_{0}^{2,1}|,\
s_{2}^{3}=s_{2}^{3}(U,L)\in\overset{(3)}{\Omega}_{2},\\\ &|\cos
s_{3}^{3}|=\cos 2\alpha_{0}^{3,1},\
s_{3}^{3}=s_{3}^{3}(U,L)\in\overset{(3)}{\Omega}_{3}.\end{split}$

Now we have the following.

######  Theorem 2.

For every fixed and admissible $U,L$ there are the sets

(4.4) $\overset{(1)}{\Omega}_{l}(U,L),\
\overset{(3)}{\Omega}_{l}(U,L)\subset\mathbb{C},\ l=1,2,3$

such that we have the following formula (transmutation of (1.2))

(4.5) $\begin{split}&|\zeta(s_{1}^{1})|^{2}|\cos
s_{1}^{3}|^{2}-\left\\{1+\mathcal{O}\left(\frac{\ln\ln L}{\ln
L}\right)\right\\}|\zeta(s_{2}^{1})|^{2}|\cos
s_{2}^{3}|^{2}+|\zeta(s_{3}^{1})|^{2}|\cos s_{3}^{3}|=0.\end{split}$

### 4.2.

For the functions

(4.6) $(s)^{n_{1}},\ (s)^{n_{2}},\ (s)^{n_{3}},\
(n_{1},n_{2},n_{3})\in\mathbb{N}^{3}$

we define the level curves

(4.7) $\overset{(4)}{\Omega}_{l}(U,L,n_{l}),\ l=1,2,3$

as the loci

(4.8) $\begin{split}&|s_{1}^{4}|^{n_{1}}=|\sin\alpha_{0}^{1,1}|,\\\
&|s_{2}^{4}|^{n_{2}}=|\cos\alpha_{0}^{2,1}|,\\\ &|s_{3}^{4}|^{n_{3}}=\cos
2\alpha_{0}^{3,1}\end{split}$

for every fixed and admissible $U,L,n_{1},n_{2},n_{3}$. Now we have (see
(1.2), (3.4), (4.8)) the following.

######  Theorem 3.

For every fixed and admissible $U,L,n_{1},n_{2},n_{3}$ there are the sets

(4.9)
$\overset{(1)}{\Omega}_{l}(U,L),\overset{(4)}{\Omega}_{l}(U,L,n_{l})\subset\mathbb{C},\
l=1,2,3$

such that we have the following formula (transmutation of (1.2))

(4.10)
$\begin{split}&|\zeta(s_{1}^{1})|^{2}|s_{1}^{4}|^{2n_{1}}-\left\\{1+\mathcal{O}\left(\frac{\ln\ln
L}{\ln
L}\right)\right\\}|\zeta(s_{2}^{1})|^{2}|s_{2}^{4}|^{2n_{2}}+|\zeta(s_{3}^{1})|^{2}|s_{3}^{4}|^{n_{3}}=0.\end{split}$

######  Remark 12.

The formula (4.10) gives an example of infinite set of transmutations of the
mother formula (1.2).

##  5\. Transmutations of types $[\zeta,\frac{1}{\Gamma}]$,
$[\zeta,J_{p_{1}},J_{p_{2}},J_{p_{3}}]$,
$[\zeta,\operatorname{sn},\operatorname{cn},\operatorname{dn}]$

### 5.1.

For the integral function

(5.1) $\frac{1}{\Gamma(s)},\ s\in\mathbb{C}$

we define the level curves

(5.2) $\overset{(5)}{\Omega}_{l}(U,L),\ l=1,2,3$

as the loci

(5.3) $\begin{split}&\frac{1}{|\Gamma(s_{1}^{5})|}=|\sin\alpha_{0}^{1,1}|,\\\
&\frac{1}{|\Gamma(s_{2}^{5})|}=|\cos\alpha_{0}^{2,1}|,\\\
&\frac{1}{|\Gamma(s_{3}^{5})|}=\cos 2\alpha_{0}^{3,1},\end{split}$

for every fixed and admissible $U,L$. Now we have (see (1.2), (3.4), (5.3))
the following

######  Theorem 4.

For every fixed and admissible $U,L$ there are sets

(5.4) $\overset{(1)}{\Omega}_{l}(U,L),\
\overset{(5)}{\Omega}_{l}(U,L)\subset\mathbb{C},\ l=1,2,3$

such that we have the following formula (transmutation of (1.2))

(5.5)
$\frac{|\zeta(s_{1}^{1})|^{2}}{|\Gamma(s_{1}^{5})|^{2}}-\left\\{1+\mathcal{O}\left(\frac{\ln\ln
L}{\ln
L}\right)\right\\}\frac{|\zeta(s_{2}^{1})|^{2}}{|\Gamma(s_{2}^{5})|^{2}}+\frac{|\zeta(s_{3}^{1})|^{2}}{|\Gamma(s_{3}^{5})|}=0.$

### 5.2.

For the Bessel’s functions

(5.6)
$J_{p_{1}}(s),J_{p_{2}}(s),J_{p_{3}}(s),s\in\mathbb{C},(p_{1},p_{2},p_{3})\in\mathbb{Z}^{3}$

we define the level curves

(5.7) $\overset{(6)}{\Omega}_{l}(U,L,p_{l}),\ l=1,2,3$

as the loci

(5.8) $\begin{split}&|J_{p_{1}}(s_{1}^{6})|=|\sin\alpha_{0}^{1,1}|,\\\
&|J_{p_{2}}(s_{2}^{6})|=|\cos\alpha_{0}^{2,1}|,\\\
&|J_{p_{3}}(s_{3}^{6})|=\cos 2\alpha_{0}^{3,1},\end{split}$

for every fixed and admissible $U,L,p_{1},p_{2},p_{3}$. Now we have (see
(1.2), (3.4), (5.8)) the following.

######  Theorem 5.

For every fixed and admissible $U,L,p_{1},p_{2},p_{3}$ there are sets

(5.9)
$\overset{(1)}{\Omega}_{l}(U,L),\overset{(6)}{\Omega}_{l}(U,L,p_{l})\subset\mathbb{C},l=1,2,3$

such that we have the following formula (transmutation of (1.2))

(5.10)
$\begin{split}&|\zeta(s_{1}^{1})|^{2}|J_{p_{1}}(s_{1}^{6})|-\left\\{1+\mathcal{O}\left(\frac{\ln\ln
L}{\ln L}\right)\right\\}|\zeta(s_{2}^{1})|^{2}|J_{p_{2}}(s_{2}^{6})|^{2}+\\\
&+|\zeta(s_{3}^{1})|^{2}|J_{p_{3}}(s_{3}^{6})|=0.\end{split}$

######  Remark 13.

Here we have the infinite (enumerable) set of transmutations of transmutations
of the mother formula (1.2), comp. Remark 12.

### 5.3.

For the Jacobi’s elliptic functions (their relief-surfaces with $k=0.8$ see in
[1], pp. 202, 203)

(5.11) $\operatorname{sn}(s,k_{1}),\ \operatorname{cn}(s,k_{2}),\
\operatorname{dn}(s,k_{3}),\quad[(k_{1})^{2},(k_{2})^{2},(k_{3})^{2}]\in(0,1)^{3}$

($k_{1},k_{2},k_{3}$ stand for the moduli) we define the level curves

(5.12) $\overset{(7)}{\Omega}_{l}(U,L,k_{l}),\ l=1,2,3$

as the loci

(5.13)
$\begin{split}&|\operatorname{sn}(s_{1}^{7},k_{1})|=|\sin\alpha_{0}^{1,1}|,\\\
&|\operatorname{cn}(s_{2}^{7},k_{2})|=|\cos\alpha_{0}^{2,1}|,\\\
&|\operatorname{dn}(s_{3}^{7},k_{3})|=\cos 2\alpha_{0}^{3,1}\end{split}$

for every fixed and admissible $U,L,k_{1},k_{2},k_{3}$. Now we have (see
(1.2), (3.4), (5.13)) the following.

######  Theorem 6.

For every fixed and admissible $U,L,k_{1},k_{2},k_{3}$ there are sets

(5.14)
$\overset{(1)}{\Omega}_{l}(U,L),\overset{(7)}{\Omega}_{l}(U,L,k_{l})\subset\mathbb{C},l=1,2,3$

such that we have the following formula (transmutation of (1.2))

(5.15)
$\begin{split}&|\zeta(s_{1}^{1})|^{2}|\operatorname{sn}(s_{1}^{7},k_{1})|-\left\\{1+\mathcal{O}\left(\frac{\ln\ln
L}{\ln
L}\right)\right\\}|\zeta(s_{2}^{1})|^{2}|\operatorname{cn}(s_{2}^{7},k_{2})|^{2}+\\\
&+|\zeta(s_{3}^{1})|^{2}|\operatorname{dn}(s_{3}^{7},k_{3})|=0.\end{split}$

######  Remark 14.

About the potency of the set of transmutations in (5.15) see Remark 2.

I would like to thank Michal Demetrian for his moral support of my study of
Jacob’s ladders.

## References

  * [1] E. Jahnke, F. Emde, ‘Funktionentafeln‘, Moscow, 1952.
  * [2] J. Moser, ‘Jacob’s ladders and almost exact asymptotic representation of the Hardy-Littlewood integral‘, Math. Notes 88, (2010), 414-422, arXiv: 0901.3937.
  * [3] J. Moser, ‘Jacob’s ladders, structure of the Hardy-Littlewood integral and some new class of nonlinear integral equations‘, Proc. Steklov Inst. 276 (2011), 208-221, arXiv: 1103.0359.
  * [4] J. Moser, ‘Jacob’s ladders, reverse iterations and new infinite set of $L_{2}$-orthogonal systems generated by the Riemann zeta-function, arXiv: 1402.2098.
  * [5] J. Moser, ‘Jacob’s ladders, factorization and metamorphoses as an appendix to the Riemann functional equation for $\zeta(s)$ on the critical line‘, Proc. Steklov Inst. 296 (2017), pp. 92-102, arXiv: 1506.00442v1.
  * [6] J. Moser, ’Jacob’s ladders, interactions between $\zeta$-oscillating systems and $\zeta$-analogue of an elementary trigonometric identity’, arXiv: 1609.09293v1, Proc. Steklov Inst. 299, 189-204, 2017.
  * [7] J. Moser, ‘Jacob ladders, crossbreeding, secondary crossbreeding and synergetic phenomena generated by the Riemann’s zeta-function and some elementary functions on disconnected sets of the critical line‘, arXiv: 1806.07095v1.
  * [8] J. Moser, ‘Jacob’s ladders and grafting of the complete hybrid formulas into $\zeta$-synergetic meta-functional equations for the Riemann’s zeta-function‘, arXiv:1809.05327v1.
  * [9] E.C. Titchmarsh, ‘The theory of functions‘ , Oxford University Press, 1968.


# Chemical Properties from Graph Neural Network-Predicted Electron Densities

Ethan M. Sunshine  [  Muhammed Shuaibi  [  Zachary W. Ulissi  [  John R.
Kitchin  [<EMAIL_ADDRESS>
###### Abstract

According to density functional theory, any chemical property can be inferred
from the electron density, making it the most informative attribute of an
atomic structure. In this work, we demonstrate the use of established physical
methods to obtain important chemical properties from model-predicted electron
densities. We introduce graph neural network architectural choices that
provide physically relevant and useful electron density predictions. Despite
not training to predict atomic charges, the model is able to predict atomic
charges with an order of magnitude lower error than a sum of atomic charge
densities. Similarly, the model predicts dipole moments with half the error of
the sum of atomic charge densities method. We demonstrate that larger data
sets lead to more useful predictions in these tasks. These results pave the
way for an alternative path in atomistic machine learning, where data-driven
approaches and existing physical methods are used in tandem to obtain a
variety of chemical properties in an explainable and self-consistent manner.

OC20short = OC20, long = Open Catalyst 2020 VASPshort = VASP, long = Vienna Ab
Initio Simulation Package DFT short = DFT, long = density functional theory
GNN short = GNN, long = graph neural network OCP short = OCP, long = Open
Catalyst Project GPU short = GPU, long = graphics processing unit RPBE short =
RPBE, long = Revised Perdew-Burke-Ernzerhof MAE short = MAE, long = mean
absolute error PAW short = PAW, long = projector-augmented-wave SCF short =
SCF, long = self-consistent field SACD short = SACD, long = superposition of
atomic charge densities CMU]Chemical Engineering, Carnegie Mellon University,
Pittsburgh, PA FAIR]Fundamental Artificial Intelligence Research, Meta
Platforms, Inc., Menlo Park, CA CMU]Chemical Engineering, Carnegie Mellon
University, Pittsburgh, PA [FAIR]Fundamental Artificial Intelligence Research,
Meta Platforms, Inc., Menlo Park, CA CMU]Chemical Engineering, Carnegie Mellon
University, Pittsburgh, PA

##  1 Introduction

Global implementation of sustainable energy technologies requires the
discovery of efficient and economical catalysts for chemical reactions such as
water oxidation, carbon dioxide reduction, and ammonia synthesis. In these
reactions, it is common to use a solid surface as a heterogeneous catalyst.
Because the design space of material surfaces is extremely vast, computational
modeling techniques are employed to identify promising catalysts for further
experimental study. Broadly speaking, these methods can be divided into two
categories: physics-based and data-driven.

First, there are physics-based methods. Of these, DFT is the most popular
choice for heterogeneous catalysis. In DFT, the valence electrons in a
chemical structure are treated as a density field rather than individual
particles described by wavefunctions. This method is based on the two
Hohenberg-Kohn Theorems, which state that 1) All chemical properties are a
function of electron density, and 2) the true electron density minimizes the
energy function1, 2. Thus, in a typical DFT calculation, an initial electron
density is guessed. This density is updated iteratively until energy is
minimized. Many properties can then be easily calculated from the converged
electron density, such as atomic charges, dipole moment, and atomic forces.

More recently, machine learning methods have been developed as a
computationally cheaper alternative to DFT. One popular machine learning
method for chemistry is the GNN. In this paradigm, atoms are represented as
nodes in a graph, and information is exchanged along the edges between them in
a scheme called ”message passing”. Assuming a constant volume, the
computational cost of GNNs scales with the square of the number of atoms,
while the cost of DFT generally scales with the cube. When assuming a constant
atomic density, the comparison is even more favorable for GNNs: the cost
scales linearly with increasing volume for GNNs but cubically for DFT.

However, in comparison to DFT, GNNs have yet to demonstrate competitive
accuracy in heterogeneous catalysis. The largest and most diverse data set for
heterogeneous catalysis is OC203. Each structure in this data set is an
adsorbate placed at a site on a catalyst surface, and the target is the
associated adsorption energy. However, as of yet, no machine learning model
achieves an adsorption energy MAE below 0.3 eV (29 kJ/mol) on the test data4.
By propagating this uncertainty through a simple Arrhenius model at a
temperature of 300K, one can see an uncertainty of more than five orders of
magnitude in kinetic rate constants. Thus, there is a pressing need for more
accurate models in this field.

To bridge the gap between the accurate physics-based models and machine
learning models, machine learning can be used to predict electron density
directly. An accurate prediction of electron density can be used in tandem
with established methods, including DFT, to predict many physical quantities.
For example, the Hellmann-Feynman Theorem asserts that electron density alone
is sufficient to determine atomic forces without the need to assume any
particular functional 5, 6. By integrating these physical methods, machine
learning models can become more useful, explainable, and transferable.

Because the electron density exists in three dimensions throughout the atomic
structure, a significant amount of information is required to represent it
accurately. It can be projected onto a basis set, but more often in
heterogeneous catalysis, electron density data is available at a grid of
points. The density is sampled at a large number of points (in our data,
$5\times 10^{6}$ on average) throughout the atomic structure. This huge amount
of data is largely under-utilized by machine learning methods to date. In
fact, most machine learned potentials do not use electron density data at all.

Several machine learning methods do exist for electron density prediction. In
2018, Grisafi et al. demonstrated that Gaussian process regression can be used
to predict electron density in terms of atom-centered basis functions7, 8, 9,
10. This low-dimensional representation of electron density is natural for
molecules. On the other hand, it is not trivial to project electron density
onto atom-centered basis functions in periodic systems such as surfaces, where
periodic basis functions such as planewaves are common.

More recently, Rackers et al. introduced an equivariant GNN to predict
electron density11, 12. They also demonstrate that increasing the order of the
atom-centered basis function set is necessary for accurate short-range force
prediction. Increasing basis set order rapidly increases the computational
cost of a GNN as well as the difficulty in fitting an appropriate atom-
centered basis to a planewave-based electron density. Considering these
factors, this method is not well-suited for heterogeneous catalysis at
present.

On the other hand, pointwise methods do not rely on atom-centered basis
function representations of the electron density. Instead, the model predicts
electron density at predetermined points in space called probes. This allows
electron density prediction at arbitrary resolutions, and the cost increases
only linearly with the number of probes. This method is particularly
attractive for periodic systems because the electron density does not have to
be represented by atom-centered basis functions. Furthermore, popular DFT
codes for heterogeneous catalysis, such as VASP13, 14, 15, readily provide
point-wise samplings of electron density. In the method introduced by
Jørgensen and Bhowmik16, the probes are added to the graph in the GNN as
information receivers (but not senders). Similarly, Achar et al.17 introduced
a pointwise approach to electron density predictions for solid materials,
although this method is not a GNN. We decided to use a pointwise approach in
this work due to its natural integration with planewave DFT codes.

##  2 Methods

###  2.1 Modeling

Our model is based on that of Jørgensen and Bhowmik. We make two key
enhancements to the model that enable the use of its predictions with physical
methods such as DFT. First, many probes are sufficiently far from any atoms
such that they receive no information in the message-passing procedure. Our
model assumes the electron density at these points to be zero. The electron
density decays slowly as the distance to the atoms increases, so probe points
outside the cutoff radius often still carry some electron density. Since the
model has no information for these points, it is forced to predict the same
density for all of them. After training, this results in a non-physical
uniform negative charge in the void space, which can have undesired
consequences when these predictions are used in downstream applications such
as charge partitioning and DFT calculations. Instead, our model constrains
this density to be zero.

Second, our model enforces a charge balance. Since our systems are periodic, a
prediction where the total charge is not balanced exactly is unsuitable for
many physical models. Thus, our model takes as an input the total number of
valence electrons in the atomic structure. We implement the following
assumption: a random sampling of probes should have a proportional share of
the valence electron density. That is, if the probes are randomly divided into
ten batches, each batch should contain one tenth of the valence electron
density. This assumption is necessary because treating all probes in a single
batch is computationally infeasible. It also means that after predictions are
made on all batches, the total electron density in the atomic structure is
guaranteed to sum to the correct number of valence electrons exactly.

Outside of the model differences, several additional features differentiate
this work from previous work. Our implementation is built as an extension of
the OCP modeling software. This enables the integration of existing message-
passing models, including the re-use of models trained on other tasks. For
example, model weights from a machine-learned potential can be used. We
anticipate further integration with this software will allow us to leverage
future developments of the OCP code repository.

Another notable feature is the probe edge-finding algorithm. Previous
implementations had a computational cost of $O((A+P)^{2})$, where $A$ is the
number of atoms and $P$ is the number of probes. This is because no
distinction was drawn between atom and probe nodes, and a general neighbor-
finding algorithm is used. Because we are not interested in edges between
probes, the cost of this algorithm can be reduced to $O(A(A+P))$, which is
significantly better because $A$ is several orders of magnitude smaller than
$P$. To achieve this, we modified the neighbor-finding algorithm available in
the OCP modeling software. The standard procedure is to consider every
possible edge between two nodes, determine the distance, and remove any which
are longer than some specified cutoff radius. Because we are not interested in
probe-probe edges, we can save a significant amount of memory by not computing
these distances. We partition the graph into two sets, the set of atoms and
the set of probes. Distances are only computed from nodes in the set of atoms.
This procedure is written in Pytorch code that runs efficiently on both CPU
and GPU. It also respects periodic boundary conditions in one, two, or three
directions. Under previous implementations of the probe graph training
procedure, the number of probes used in training was limited by graph
construction speed. In contrast, our implementation is limited by the memory
requirements of message passing on large graphs. This also enables rapid
inference with significant speedup compared to previous implementations.

###  2.2 Data Set

We prepared a large and diverse data set of charge density in heterogeneous
catalysis. The atomic structures in this data set are from OC20. 32794 random
structures from the OC20 train split comprise the training set, and 897 random
structures from the OC20 ”both out of distribution” split comprise the
validation set. One hundred random structures from the OC20 ”both out of
distribution” split comprise the test data set. No adsorbate or surface that
is present in the training set is in the validation or test set. Self-
consistent electronic structure optimization was performed in VASP using the
RPBE functional.

The electron density in this data set is obtained directly from the CHGCAR
output file and represents a pseudo-valence electron density. The density is
sampled on a uniform, three-dimensional grid of points. The number of electron
density probe points varies depending on the size of the unit cell. On
average, each atomic structure in this data set contains about 5 million probe
points.

###  2.3 Training Procedure

The models are trained to minimize the normalized mean absolute error
introduced by Jørgensen and Bhowmik. We note that there is no universally
accepted error metric for electron density. We acknowledge that this metric
may not translate well across data sets, and is biased towards high-density
probe points. The metric is defined as:

$\displaystyle\varepsilon_{MAE}=\frac{\int_{\vec{r}\in
V}{|\hat{\rho}(\vec{r})-\rho(\vec{r})}|}{\int_{\vec{r}\in
V}{|\rho(\vec{r})|}}$ (1)

where $\rho$ is a ground-truth electron density obtained from a converged
self-consistent DFT calculation, $\hat{\rho}$ is a model-predicted electron
density, and $V$ is the simulation volume. This integral is evaluated
numerically by assuming probe points represent equal-volume voxels, each of
uniform density which matches the density at the probe point.

At each training step, one atomic structure is chosen from the training data
set and converted to a geometric graph structure. Then, a subsample of probes
is randomly chosen from a uniform distribution. The number of probes chosen at
this step can be thought of as the ”batch size” and is generally limited by
the GPU memory capacity. For these models, the batch size used for training is
$10^{4}$ or more. The geometric graph is augmented with the probe nodes and
their respective edges. Then, the model makes a prediction of electron density
at each probe point. The error metric is computed, and back-propagation is
used to obtain gradients for each model parameter. Model parameters are
adjusted according to the Adam algorithm18. This procedure is repeated until
each structure in the training data set has been chosen once, which comprises
one epoch. The model is trained until no further improvements are seen on the
validation set, which typically takes about 20 epochs.

In this work, we will focus on one model, a Schnet-based model19 which is akin
to Jørgensen and Bhowmik’s ”Invariant DeepDFT”19. Our implementation is built
from the OCP implementations of Schnet. The main architectural differences
between this models and DeepDFT are charge conservation and the long-range
zero density constraint as described in the modeling section of this work. The
hyperparameters of each of the model are available in the supporting
information of this work.

###  2.4 Interface with DFT Calculator

To make predictions of physical properties such as forces, it is advantageous
to use an existing quantum mechanical calculator. For instance, one may wish
to perform a SCF calculation using model-predicted electron densities as an
initial guess. Thus, we have established methods to use predictions from our
electron density models as inputs to VASP. To do this, it is necessary to have
electron density predictions on the same grid that is sampled in the CHGCAR
file. Obtaining these predictions on a single GPU takes about 1-2 minutes for
a typical structure in our data set.

The electron density prediction is written to a CHGCAR file. To use this file
as a VASP input, one must also write PAW occupancies. In this work, we reuse
PAW occupancies obtained from a DFT calculation with zero electronic
minimization steps. This non-SCF calculation is relatively fast and is only
necessary to obtain a reasonable guess of the augmentation occupancies. In
principle one may obtain reasonable augmentation occupancies by another
method, such as reading the pseudopotential information directly. Once a
CHGCAR is written with the atomic structure, electron density, and PAW
occupancies, VASP can read the file with the INCAR flag ICHARG=1. This will
cause VASP to use the predicted density as an initial guess for the electronic
minimization loop instead of the default method.

The default method VASP uses to obtain an initial guess of electron density is
SACD. This is an easy and inexpensive way to obtain an electron density by
adding the electron density that would be obtained from each atom if it were
isolated. It ignores the effects of inter-atomic interactions on electronic
structure. When discussing our results, we sometimes refer to the SACD method
as a baseline.

##  3 Results

###  3.1 Electron Density Predictions

Across the 100-structure test data set with 4.94 million probe points per
structure, the model achieves an MAE of 2.54%. In comparison, the SACD method
only achieves an MAE of 8.30%. We show parity between the model-predicted and
ground-truth electron densities on a parity heat map. This allows
visualization of model performance across different regimes in the test data
set. Figure 1 shows the performance achieved by our best model. We see that
our model achieves tighter parity in the high-density regime when compared
with the SACD method. This high-density regime includes many points in the
data set, as evidenced by the thin warm-colored line in this portion of the
plot.

Figure 1: Parity heat maps comparing the results of the machine learning model
and SACD method. The model achieves tighter parity in the high-density region
but not necessarily in the low-density region. This plot does not show points
with zero or negative pseudo-valence electron density.

Examining the low-density regime, it is difficult to claim that the machine
learning model has significantly better or worse performance than the SACD
method. It seems that the model is unable to effectively distinguish between
points with electron densities below $10^{-4}$ electrons per cubic Angstrom.
This is to be expected because such points have little impact on the error
metric by which the model is trained. Furthermore, many points in this regime
seem to have disappeared from the plot. This is because these points are
beyond the cutoff radius of any atoms, so the model is forced to predict zero
electron density. Thus these points are not shown on the logarithmic plot.
Ultimately, precision in this low-density regime is likely unnecessary for
many applications.

###  3.2 Learning Curve

To investigate the limitations of the model, we trained three additional
models on smaller subsets of the training data set. The performance of each
trained model is shown in Figure 2. From this learning curve, it appears that
we are still in a regime where the accuracy of the model is limited not by the
model architecture itself, but by the size of the training data set.

Figure 2: Learning curve with respect to training set size. Boxes represent
50% of the test data and whiskers represent 90%. Further increases to data set
size are likely to result in even lower error measurements.

###  3.3 DFT Initialization

One potential use of these model-predicted electron densities is to initialize
self-consistent DFT calculations. Because these calculations gradually
converge an electron density guess to its ground state, it stands to reason
that an initial guess which is close to the actual ground state electron
density will result in faster convergence. In reality, this is only one piece
of the self-consistency puzzle. In a PAW calculation, it is necessary to
converge the wavefunctions and augmentation occupancies in addition to the
electron density. Thus, even a perfect guess of electron density will require
a significant amount of electronic minimization steps to obtain self-
consistency if the initial guesses of wavefunctions and augmentations are
poor. Despite this limitation, we find that improving the error on electron
density improves the overall convergence speed of a DFT calculation. This
result is demonstrated in Figure 3.

Figure 3: VASP performance is improved as the training set size increases.
This is attributed to more accurate electron densities. The number of
Hamiltonian evaluations is normalized by the number required when using the
ground-state (i.e. perfectly accurate) electron density as an initial guess.
Thus, the blue line represents the ”best case scenario” for a perfect electron
density model. Significant progress is still needed to approach this limit.
Interestingly, the machine learning models do not consistently outperform the
baseline SACD method in this task. Boxes represent 50% of the test data and
whiskers represent 95% of the test data.

This was measured by running a DFT self-consistency calculation (single point)
on each structure in the test data set. This was repeated with electron
densities predicted by each model. Additionally, we ran the same experiment
using for initialization the actual ground-state electron densities obtained
from DFT. To measure performance we take the total number of evaluations of
the Hamiltonian acting on a wavefunction, which is reported in VASP’s output
as ”ncg”. The performance is then normalized relative to the case where the
ground-state initialization is used as the initial guess.

The figure shows a clear correlation between the training set size and
resulting DFT performance. However, it is also clear that this scheme offers
no practical value at this time. Even the best model presented in this work is
unable to consistently yield electron density guesses that lead to faster
convergence than the SACD method. It appears that simply increasing the
training set size is not a practical path towards optimal performance on this
task.

###  3.4 Atomic Charge Predictions

Another common use of the electron density is to determine atomic charges.
This is done by a charge partitioning scheme. These methods divide the
structure into volumes, each of which contains one atom. Then, the total
charge within that volume is summed up, including the negative charge from the
electron density and the positive charge from the nucleus. These total charges
are assigned to each atom. Quantitatively, atomic charges can be used to
investigate electrostatic interactions between atoms20. Qualitatively, they
can help explain the movement of electrons between atoms and investigate
chemical reaction mechanisms. Furthermore, after dividing the space into these
volumes, additional topological analysis can be performed to study chemical
bonds21.

One common such scheme is Bader charge partitioning, in which volumes are
defined by zero-flux surfaces22, 23, 24. We performed this analysis on the
electron densities predicted by each of our models on the test data set, as
well as the ground-state and SACD electron densities25. We evaluate the
performance of each model by calculating the absolute error between each
predicted atomic charge in the test data set, as compared to the charges
obtained from the ground-state electron density. The results are summarized in
Figure 4.

The model performs significantly better than the SACD method, and performance
is improved with the size of the training data set. Several previous works
have used message-passing GNNs for atomic charge prediction and achieved
significantly better performance than this26, 27, 28. However, it is worth
considering that the models presented today were not explicitly trained on
atomic charge data. In this work we are able to achieve this performance
simply by training on electron density data and using existing methods to
determine atomic charges. This enforces a conservation of total charge, which
is not guaranteed by all existing methods. Moreover, it demonstrates that
accurate electron density predictions can be used to obtain other properties
by established, physics-based methods.

Figure 4: Error in Bader charges and electrostatic dipole moments. In each
case, the machine learning models offer marked improvement over the SACD
method. Boxes represent 50% of the test data and whiskers represent 95% of the
test data. Training on larger data sets appears to offer slight improvements.
Note that the models are not trained to predict either of these properties
directly. Rather, existing methods are used to calculate them from the
predicted electron densities.

###  3.5 Dipole Moment Predictions

Electrostatic dipole moments were calculated from the electron densities and
pseudo-ion charges. Like the Bader analysis, this calculation was done with
the SACD density, ground-state density, and each model-predicted density for
each structure in the test data set. The result is shown in Figure 4. The
results show a similar trend to that of the Bader analysis. There is a marked
improvement in the performance of the machine learning models compared to the
SACD method, and training on larger data sets results in further decreases in
error.

##  4 Discussion

###  4.1 Current Limitations

With any machine learning method, the resultant model is greatly limited by
the training data which is used in its generation. As such, although the model
can be used as-is to obtain predictions for out-of-domain systems, these
predictions have no guaranteed degree of accuracy. Such out-of-domain systems
include but are not limited to oxide materials, zeolite materials, metal-
organic frameworks, amorphous materials, and liquid systems. It is possible to
use the tools and techniques presented in this work to create an equivalent
model for each of these domains, provided sufficient training data is
available. Similarly, we have only trained on the pseudo-valence electron
density (available in the CHGCAR) and make no claims about model performance
with all-electron density or similar quantities.

This model is not accurate enough to directly use the predicted electron
density to obtain an estimate of forces, as has been done by Rackers et al 11.
We attribute this to the diversity of our data set in comparison to that used
in other work, and we hope that future training on larger but equally diverse
data sets will achieve sufficient accuracy for this task.

Predicting the full electron density of a structure in this data set with this
model takes about 1-2 GPU-minutes. This is because the number of probe points
is too large for the GPU to handle in one batch. Instead, the probe points are
divided into many batches which are transferred to the GPU in succession. In
comparison, machine-learned potentials can be evaluated in a fraction of a
second. Bridging this speed gap will require optimizations, such as predicting
on fewer probe points and efficiently up-sampling to a full grid29.

###  4.2 Paths Towards Improved Performance

As shown in Figure 2, it appears that model performance can still be improved
by training on larger data sets. It is relatively easy to generate an electron
density data set that is one or two orders of magnitude larger using DFT.
Based on the learning curve, it seems that this would lead to models with an
MAE near or below 2%. However, training on such a data set remains a
challenge. Practically speaking, the current size of the data set is well over
500 GB. This is comparable in size to the full OC20 data set despite
containing four orders of magnitude fewer structures. In fact, due to the
batching procedure, the models presented today have not truly been trained on
every data point. An electron density data set with the same number of
structures as OC20 would contain petabytes of data, placing it among the
largest atomistic data sets created to date. Training on such large data sets
requires extensive GPU resources and specialized parallelization techniques.
Thus there is a need for efficient implementations for storing and training on
such large data sets. This path shows the most promise to improve on the
accuracy of the models presented today.

On the other hand, one might consider training with a more advanced GNN model.
The SchNet model used in this work is not considered state-of-the-art in
OC204, 30. PaiNN-based architectures have already been used for electron
density prediction16. This could also become very compute-expensive as more
and more advanced architectures are used. Furthermore, not all GNN
architectures are suitable for this task. For instance, a model such as GemNet
that enumerates three- and four-body interactions31, 32 is highly impractical
since a typical structure will have millions of ”bodies”: the probe points. It
is critical to choose or create GNN message-passing architectures with cost
that scales linearly (or better) with the amount of probe points in the batch.

Finally, we discuss some practical paths towards improved DFT performance. As
mentioned previously, a self-consistent PAW calculation requires wavefunction
and augmentation occupancies in addition to the electron density. Although it
is possible to use the default methods for these quantities, improving the
initial guesses of these properties will amplify any performance gains from an
accurate electron density guess. We expect it should be possible to model the
augmentation occupancies using equivariant neural networks that make atom-
centered predictions (such as the electron density models introduced
previously)7, 8, 9, 10, 11, 12. Predicting the periodic wavefunctions promises
further SCF convergence speed improvements, but it is not immediately obvious
how this can be efficiently integrated with current GNN models. Further
increases in training set size are not likely to result in large performance
improvements on this task. Further investigation is necessary to determine why
the SACD method outperforms the machine learning models on this task, despite
significantly worse error metrics.

##  5 Conclusions

In this work, we implemented a message-passing GNN for electron density
prediction. It includes several useful features that enable the predicted
electron density to be used with physical models such as DFT and Bader charge
partitioning. Although fast and accurate machine learning models exist for
many individual properties, this work demonstrates a new paradigm, in which
models can be trained on electron density only, and existing methods can be
used to determine other quantities. We believe that training on larger data
sets and with more advanced models will enable even more accurate models in
this domain.

The authors acknowledge support from Meta Platforms, Inc. via the Open
Catalyst Project collaboration. We thank Arghya Bhowmik and Peter Bjørn
Jørgensen for helpful discussions regarding the DeepDFT code. We are also
grateful for discussions with Open Catalyst Project team members: Jehad Abed,
Lowik Chanussot, Abhishek Das, Adeesh Kolluru, Janice Lan, Kyle Michel, Joe
Musielewicz, Saro Passaro, Ammar Rizvi, Nima Shoghi, Anuroop Sriram, Matt
Uyttendaele, Brook Wander, Brandon Wood, and C. Lawrence Zitnick.

The supporting information contains more details about the code and data
availability, as well as the hyperparameters of the models and training
procedures presented in this work.

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