# Virtuoso: An Open-Source, Comprehensive and Modular Simulation Framework for
Virtual Memory Research

Konstantinos Kanellopoulos1 Konstantinos Sgouras2 Onur Mutlu1 ETH Zürich1
University of Athens2

††copyright: none Table 1. Virtual memory components supported by existing
simulators and Virtuoso

Simulator/ |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
Component | TLB |  |  |  |  |  |  |
Subsystem | Page Table |  |  |  |  |  |  |
Design | Contiguity |  |  |  |  |  |  |
Schemes | Intermediate |  |  |  |  |  |  |
Address Space | Hash-based |  |  |  |  |  |  |
Translation | Metadata |  |  |  |  |  |  |
Management | Memory |  |  |  |  |  |  |
Management | Page Fault |  |  |  |  |  |  |
Modelling |  |  |  |  |  |  |  |
SimpleScalar (T. Austin, 2002) | L1 D-TLB & L1 D-ITLB |  ✖ | ✖ | ✖ |  ✖ | ✖ | ✖ |  ✖
PTLSim (Yourst, 2007) | L1 D-TLB & L1 D-ITLB |  Simplistic Walker | ✖ | ✖ |  ✖ | ✖ | ✖ |  ✓
Scarab (Hpsresearchgroup, [n. d.]) | ✖ |  ✖ | ✖ | ✖ |  ✖ | ✖ | ✖ |  ✖
Ramulator (Kim et al., 2015) | ✖ |  ✖ | ✖ | ✖ |  ✖ | ✖ | ✖ |  ✖
ZSim (Sanchez and Kozyrakis, 2013) | ✖ |  ✖ | ✖ | ✖ |  ✖ | ✖ | ✖ |  ✖
Multi2Sim (Ubal et al., 2012) | L1 D-TLB & L1 D-ITLB |  ✖ | ✖ | ✖ |  ✖ | ✖ | ✖ |  ✖
Gem5 SE (Binkert et al., 2011) | 2-Level TLB Hierarchy |  4-level Radix-Tree Walker | ✖ | ✖ |  ✖ | ✖ | ✖ |  ✖
Gem5 FS (Binkert et al., 2011) | 2-Level TLB Hierarchy |  4-level Radix-Tree Walker | ✖ | ✖ |  ✖ | ✖ |  THP (Corbet, 2011) |  ✓
Champsim (Gober et al., 2022) | 2-Level TLB Hierarchy |  4-level Radix-Tree Walker | ✖ | ✖ |  ✖ | ✖ | ✖ |  ✖
Sniper (Carlson et al., 2011) | 2-Level TLB Hierarchy |  Static PTW latency | ✖ | ✖ |  ✖ | ✖ | ✖ |  ✖
Virtuoso | Configurable |  |  |  |  |  |  |
TLB hierarchy |  Memory-Efficient Hash Table (Stojkovic et al., 2023) |  Offset-based translation with Redundant Memory Mappings (Karakostas et al., 2015) |  Virtual Block Interface with configurable past-LLC translation (Hajinazar et al., 2020) |  Hybrid address mapping with Utopia (Kanellopoulos et al., 2023a) |  Memory tagging with XMem (Vijaykumar et al., 2018) | Memory Management Emulation | Bi-directional inter-process communication between Virtuoso and custom OS Software to emulate the functionality of PF and simulate the architectural events of PF |
| Multi-page Size TLBs |  |  |  |  |  |  |
(Serial probing) |  Hash Table with Open-Addressing and PTE Clustering (Yaniv and Tsafrir, 2016) |  |  |  |  | Buddy Allocator with pre-created memory allocation snapshots |  |
| Page-size Predictors |  Elastic Cuckoo |  |  |  |  |  |
Hash Table (Skarlatos et al., 2020) |  |  |  |  |  Eager Paging to allocate large contiguous blocks (Karakostas et al., 2015) |  Eager Paging to allocate large contiguous physical blocks (Karakostas et al., 2015) |  |
| TLB prefetching techniques |  Configurable Radix |  |  |  |  |  |
Table with PWCs |  Direct Segments (Basu et al., [n. d.]) |  Midgard-based translation with trace-based VMA tracking (Gupta et al., 2021) |  Hash-based translation with NMT (Picorel et al., 2017) and Utopia (Kanellopoulos et al., 2023a) |  Mondriaan Data Protection (Witchel et al., 2002) |  Reservation-based THP (Navarro et al., 2002) |  |  |
|  Software-based TLBs (Ryoo et al., 2017) |  Support for Nested MMUs (MMU with Nested TLB, Nested Walk etc.) |  |  |  |  | Artificial Fragmentation Generator |
| Storing TLB entries |  |  |  |  |  |  |
in data caches (Kanellopoulos et al., 2023b) |  |  |  |  |  |  |  |

##  1\. Introduction & Background

Virtual memory is a cornerstone of modern computing systems. Introduced as one
of the earliest instances of hardware-software co-design, VM facilitates
programmer-transparent memory management, data sharing, process isolation and
memory protection. In light of current application trends with large data sets
and irregular memory access patterns (Hwang et al., 2020; Gupta et al., [n.
d.]; Graph 500, [n. d.]; Subramaniyan et al., 2021; Luszczek et al., 2006) and
as we transition to much larger address spaces (Intel Corp., [n. d.]) (e.g.,
hybrid memory systems with both volatile and non-volatile memories (Liu et
al., 2019; Alverti et al., [n. d.])) and heterogeneous memory systems (Li et
al., 2017), the overheads of virtual memory are likely to increase (Yan et
al., 2019). Research span across various solutions to reduce the overheads of
virtual memory in the context of (i) improving the efficiency of the TLB
subsystem (Ryoo et al., 2017; Pham et al., 2012; Vavouliotis et al., 2021;
Marathe et al., 2017; Barr et al., 2011) (e.g., software-based TLB (Ryoo et
al., 2017) etc.), (ii) the efficiency of page table structures (Skarlatos et
al., 2020; Stojkovic et al., 2023; Yaniv and Tsafrir, 2016; Gandhi et al.,
2016; Park et al., 2022) (e.g., hash-based page tables (Skarlatos et al.,
2020; Stojkovic et al., 2023)), (iii) accelerating address translation in
virtualized environments (Gandhi et al., 2016; Margaritov et al., 2021;
Stojkovic et al., 2022; Bhargava et al., 2008; Gandhi et al., 2014), (iv)
leveraging contiguity between virtual/physical addresses to enable offset-
based address translation (Zhao et al., 2023; Yan et al., 2019; Alverti et
al., 2020; Karakostas et al., 2015; Pham et al., 2012) (e.g., range-based
translation (Karakostas et al., 2015)), (v) enabling hash-based virtual-to-
physical address mapping (Picorel et al., 2017; Kanellopoulos et al., 2023a;
Gosakan et al., [n. d.]), (vi) enabling intermediate address spaces to perform
address translation only upon LLC misses (Gupta et al., 2021; Basu et al.,
2012), (vii) designing efficient large page allocation techniques (Kwon et
al., [n. d.]; Panwar et al., 2019; Corbet, 2011; Du et al., 2015; Zhao et al.,
2023) (e.g., Transparent Huge Pages (Corbet, 2011)) and (viii) reducing the
overheads of minor page faults (Tirumalasetty et al., 2022; Lee et al., 2020).

##  2\. Lack of Comprehensive Virtual Memory Simulation Framework

Evaluating the efficiency of various virtual memory (VM) designs is crucial
(i) given their significant impact on the system, including the CPU caches,
the main memory, and the storage device and (ii) given that different system
architectures might benefit from various VM techniques. Such an evaluation is
not straightforward, as it heavily hinges on modeling the interplay between
different VM techniques and the interactions of VM with the system
architecture. To better illustrate the interplay between VM techniques, for
example, the memory management policies can directly affect TLB performance
(Pham et al., 2012), contiguity-aware VM solutions significantly influence
memory fragmentation (Alverti et al., 2020), affecting disk accesses, while
the contiguity between virtual and physical addresses is pivotal for the
efficiency of prefetching (Vavouliotis et al., [n. d.]). As also highlighted
in prior works (Yaniv and Tsafrir, 2016; Park et al., 2022), the design of the
page table can negatively impact main memory efficiency. With respect to
system architecture, mobile devices, with their resource constraints, might
benefit from lightweight VM designs that minimize energy overheads. In
contrast, cloud architectures, with abundant resources, might favour VM
designs that maximize throughput and scalability (Hyun et al., [n. d.]). In
contrast, commercial multi-core architectures, with multiple threads running
concurrently, might favour VM designs that reduce contention and ensure
efficient memory bandwidth distribution among cores.

In the face of these intricacies, assessing the merits and drawbacks of old,
modern and futuristic VM proposals becomes a highly-challenging task without a
comprehensive simulation tool. Modern simulators, however, struggle to keep up
with the rapid VM research developments, lacking the capability to model a
wide range of contemporary VM techniques and their interactions. Table 1
summarizes the VM techniques that are supported by existing simulators. For
example, Sniper (Carlson et al., 2011), does not include a realistic model for
the page table walk latency, neither emulates nor simulates the page fault
handler and does not emulate memory management. The system call emulation mode
of gem5 (Binkert et al., 2011) does not include detailed TLB models (e.g., all
TLBs are considered fully-associative), and neither emulates nor simulates
page fault handling and large page allocation. At the same time, the full
system simulation of gem5 emulates and simulates the actual OS including
realistic memory management, large page allocation but at the cost of (i) high
programmability effort, e.g., enabling contiguity-based translation requires
modifying the kernel code in a functional manner and co-design it with the
simulated hardware and (ii) high simulation time (e.g., simulating multi-
programmed workloads in parallel is not supported). Given the wide range of
research directions and the synergies between different VM components, our
goal in this work is to provide researchers with an open-source, comprehensive
and modular tool to study and evaluate various VM techniques and memory
management schemes.

##  3\. Our Approach: Virtuoso

To this end, we present Virtuoso, an open-source, comprehensive and modular
simulation framework that models various VM designs to establish a common
ground for virtual memory research. Virtuoso is built on top of sniper
(Carlson et al., 2011) and, as show in Table 1, extends it with (i) state-of-
the-art TLB techniques and organizations, (ii) four different page table
designs from cutting-edge academic proposals, (iii) support for easily
configurable Nested MMUs, required in virtualized environments(e.g., MMU with
Nested TLB, Nested Walk etc.), (iv) translation techniques that exploit
virtual and physical memory contiguity (Karakostas et al., 2015; Basu et al.,
[n. d.]) (v) support for address translation schemes that rely on intermediate
address spaces (Hajinazar et al., 2020; Gupta et al., 2021), (vi), metadata
management schemes that enhance security and performance (Vijaykumar et al.,
2018; Witchel et al., 2002), (vii) a complete memory management emulator that
includes a reservation-based programmer-transparent large page allocator and
(viii) a new simulation methodology that involves bi-directional inter-process
communication between Virtuoso and software to estimate the performance of OS
kernel code with a focus on the minor page fault handler. Even though Virtuoso
is built on top of Sniper, it is highly-modular, avoiding the use of
simulator-specific semantics. The only requirement to integrating Virtuoso in
any architectural simulator, is attaching it to the simulator’s memory
subsystem model (e.g. interface to access the L1 data cache).

To demonstrate the versatility and utility of Virtuoso we examine four new
example case studies.

  * •

We analyze the performance, memory and cache footprint of different page table
designs under different system workloads (e.g, memory fragmentation and memory
bandwidth) and different execution environments (native vs virtualized
execution).

  * •

We perform a head-to-head comparison between large-page, intermediate address
space and contiguity based schemes taking into consideration (i) address
translation latency and (ii) impact on memory fragmentation.

  * •

We examine the performance and microarchitectural impact of Virtuoso’s
reservation-based large page allocator across different fragmentation levels
and compare it against Linux THP.

  * •

We evaluate the microarchitectural impact caused by minor page faults in the
presence of different page allocation policies and contiguity-aware
translation approaches.

##  4\. Our Contributions

In this work, we make the following contributions:

  * •

We present Virtuoso, an open-source, comprehensive and modular simulator that
models various virtual memory components and schemes to establish a common
ground for virtual memory research.

  * •

We demonstrate the versatility and the potential of Virtuoso with four new
case studies. A minimal, alpha version of the simulator is used in two recent
research works (Kanellopoulos et al., 2023b, a). Virtuoso is freely open-
source and can be found at https://github.com/CMU-SAFARI/Virtuoso.

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# On Signs of Hecke eigenvalues of Ikeda Lifts

Nagarjuna Chary Addanki

##  1 Introduction

Siegel modular forms of genus $n$ and weight $k$ with respect to
$\Gamma_{n}=Sp_{2n}(\mathbb{Z})$ are holomorphic functions on the Siegel upper
half space $\mathbb{H}_{n}$. They are generalization of elliptic modular
forms, the case $n=1$ gives us the classical modular forms over
$SL_{2}(\mathbb{Z})$. Let $M_{k}(\Gamma_{n})$ denote the space of Siegel
modular forms and $S_{k}(\Gamma_{n})$ denote the subspace of cuspidal forms.
$S_{k}(\Gamma_{n})$ has a special basis called Hecke eigenforms, these are
eigenvectors with respect to operators on $S_{k}(\Gamma_{n})$ called the Hecke
operators. W. Duke and Ö. Imamo$\bar{g}$lu conjectured that given $f\in
S_{2k-n}(\Gamma_{1})$, there exists $F\in S_{2k}(\Gamma_{n})$, such that the
L-functions associated to $F$ and $f$ satisfy certain relations. For further
details, check [BK00]. Lifts from $n=1$ to $2$ are known as Saito-Kurokawa
lifts. A construction of these lifts can be found in [Maa79]. Ikeda
constructed lifts to higher genus. Theorem $3.2$ of [Ike01] explains the
construction of $F_{f}\in S_{k+n}(\Gamma_{2n})$ from $f\in S_{2k}(\Gamma_{1})$
for $n\equiv k(mod\ 2)$. These lifts are called Ikeda lifts and in this
article we focus on Ikeda lifts for $n=4$ and higher.

For each positive integer $m$ there is a Hecke operator associated to it,
denoted by $T(m)$. For a Hecke eigenform $F$, let $\lambda_{F}(m)$ denote the
eigenvalue of $T(m)$. In 2013, [KLSW10] proved that for two normalized Hecke
eigenforms $F,G\in S_{k}(\Gamma_{1})$, if
$\text{sign}(\lambda_{F}(p^{r}))=\text{sign}(\lambda_{G}(p^{r}))$ for almost
all $p$ and $r$ then $F=G$. A similar result is proved in [GKP21] for $n=2$.
Theorem $4$ of the article states that under a specific condition, if $F\in
S_{k_{1}}(\Gamma_{2})$ and $G\in S_{k_{2}}(\Gamma_{2})$ are in orthogonal
compliment of their respective Maass subspaces (the subspace of
$S_{k}(\Gamma_{2})$ generated by Saito-Kurokawa lifts) then half of the non
zero coefficients in the sequence $\\{\lambda_{F}(m)\lambda_{G}(m)\\}_{m\geq
1}$ are positive and half are negative. Saito-Kurokawa lifts can be
characterized by the sign behavior of eigenvalues. Breulmann, in [Bre99],
showed that $F\in S_{k}(\Gamma_{2})$ is a Saito-Kurokawa lift if and only if
$\lambda_{F}(m)>0$ for all $m\geq 1$. Kohnen, in [Koh07], showed that a Hecke
eigenform $F\in S_{k}(\Gamma_{2})$ is in the orthogonal compliment of Maass
space if and only if there are infinitely many sign changes in the sequence
$\\{\lambda_{F}(m)\\}_{m\geq 1}$. Kohnen’s and Breulmann’s results can be
combined to conclude that the space $S_{k}(\Gamma_{2})$ breaks into two
disjoint subspaces depending on the signs of eigenvalues of Hecke eigenforms.
These are a few examples of incredible information stored in the sign behavior
of the sequence $\\{\lambda_{F}(m)\\}_{m\geq 1}.$

Mathematicians have studied this sign behavior through various questions. For
example, Kohnen and Sengupta, in [KS07], proved that for $n=2$, the first
negative eigenvalue occurs for some $m\ll k^{2}log^{20}k$. Pitale and Schmidt,
in [PS08], proved that there are infinitely many prime numbers $p$ such that
the sequence of Hecke eigenvalues $\\{\lambda_{F}(p^{r})\\}_{r\geq 1}$ has
infinitely many sign changes. In this article, we focus on eigenvalues of
Ikeda lifts of genus $4$ and higher. From Breulmann’s result, $F\in
S_{k}(\Gamma_{2})$ is a Saito-Kurokawa lift if and only if $\lambda_{F}(m)\geq
0$ for all $m.$ We ask if a similar result is expected for Ikeda lifts. In
this article we prove two theorems on the signs of eigenvalues of Ikeda lifts.
The first result is true for Ikeda lifts of all genus.

######  Theorem 1.1.

If $F_{f}\in S_{k+n}(\Gamma_{2n})$ is the Ikeda lift of $f\in
S_{2k}(\Gamma_{1})$ then for all large enough primes $p$,
$\lambda_{F_{f}}(p)\geq 0.$

The second result is for $n=4$.

######  Theorem 1.2.

$F_{f}\in S_{k+2}(\Gamma_{4})$ be the Ikeda lift of $f\in S_{2k}(\Gamma_{1})$.
Given $r$ there exists a positive constant $C_{r}$ such that
$\lambda_{F_{f}}(p^{r})\geq 0$ for all $p>C_{r}$.

In [And74], it is proved that for a general $n$, $-\lambda_{F}(p)$ is the
coefficient of $p^{-s}$ in the spin L-function of $F$. Hence the spin
L-function is enough to determine the sign of $\lambda_{F}(p)$. Schmidt, in
[Sch03], found an explicit expression for the spin L-function of the Ikeda
lift $F_{f}$ in terms of the L-function of $f$. In section $2$, we talk about
the basics of Siegel modular forms. We define objects like cuspidal forms,
Hecke operators, spin L-functions, Petterson inner product and so on. Using
Schmidt’s formula for the spin L-function for Ikeda lifts and Andrianov’s
result, we compute $\lambda_{F}(p)$ and prove Theorem $1.1.$

To compute $\lambda_{F}(p^{r})$ for higher powers of $p$, we need more
information. Andrianov, in [And74], proved that for any $F\in
S_{k}(\Gamma_{n})$, there are polynomials $Q_{F,p}(x)$ and $P_{F,p}(x)$ such
that

$\sum_{r=0}^{\infty}\lambda_{F}(p^{r})p^{-rs}=\frac{P_{F,p}(p^{-s})}{Q_{F,p}(p^{-s})}$

and $Q_{F,p}(p^{-s})$ is the inverse of the spin L-function. Hence we need
$P_{F,p}(x)$ and $Q_{F,p}(x)$ to compute $\lambda_{F}(p^{r})$. $P_{F,p}(x)$
are known only up to $n=4$, and the case of $n=4$ has been recently shown by
Vankov in [Van11]. In section $3$ we compute $\lambda_{F}(p^{r})$ for an Ikeda
lift $F\in S_{k}(\Gamma_{4})$. We use Vankov’s result to compute the
polynomial $P_{F,p}(x)$ and Schmidt’s result to compute $Q_{F,p}(x)$. We use
Andrianov’s equation to find an explicit formula for $\lambda_{F}(p^{r})$ and
prove Theorem $1.2.$

##  2 Siegel Modular Forms and Signs of $\lambda_{F}(p)$ for Ikeda lifts

For any ring $R$, let

$G(R)\coloneqq GSp_{2n}(R)=\Big{\\{}g=\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\in GL_{2n}(R):g^{t}Jg=\mu(g)J,\ J=\begin{pmatrix}0&1_{n}\\\
-1_{n}&0\end{pmatrix}\Big{\\}}$

where $\mu$ is the similitude homomorphism, $1_{n}$ is identity matrix of size
$n$ and $A,B,C,D\in M_{n}(R)$.

$Sp_{2n}(R)\coloneqq\\{g\in G(R):\mu(g)=1\\}\quad\Gamma_{n}\coloneqq
Sp_{2n}(\mathbb{Z}).$

A Siegel modular form F, of genus $n$ and weight $k$, with respect to
$\Gamma_{n}$, is a holomorphic function on the Siegel upper half space

$\mathbb{H}_{n}\coloneqq\\{Z:Z\in M_{n}(\mathbb{C}),\ \prescript{t}{}{Z}=Z\
\text{and}\ Im(Z)>0\\}$

satisfying

  1. 1.

modularity condition

$F((AZ+B)(CZ+D)^{-1})=det(CZ+D)^{k}F(Z)\quad\forall\ \begin{pmatrix}A&B\\\
C&D\end{pmatrix}\in\Gamma_{n}\ \text{and}\ Z\in\mathbb{H}_{n}$

  2. 2.

and for $n=1$, $F(Z)$ is bounded on $\\{Z=X+iY:Y\geq Y_{0}\\}\ \forall\
Y_{0}>0.$

Holomorphy and modularity imply that a Siegel modular form has a Fourier
expansion of the form

$F(Z)=\sum_{\begin{subarray}{c}T=T^{t}\ T\geq 0\\\ T\ \text{half
integral}\end{subarray}}A(T)e^{2\pi itr(TZ)}.$

Let $M_{k}(\Gamma_{n})$ denote the space of Siegel modular forms of genus $n$
and weight $k$. $F$ is called cuspidal if $A(T)=0$ unless $T>0$ and let
$S_{k}(\Gamma_{n})$ denote the subspace of cusp forms. For $F_{1}\in
S_{k}(\Gamma_{n})$ and $F_{2}\in M_{k}(\Gamma_{n})$ we define the Petterson
inner product

$\langle
F,G\rangle=\int_{\Gamma_{n}/\mathbb{H}_{n}}F_{1}(Z)\overline{F_{2}(Z)}\frac{dXdY}{det{Y}^{n+1-k}}.$

For each $g\in G(\mathbb{Q})\cap M_{2n}(\mathbb{Z})$ with a positive $\mu(g)$,
a Hecke operator $T(g)$ on $M_{k}(\Gamma_{n})$ is defined as follows,

$\text{for F}\in M_{k}(\Gamma_{n}),\
T(g)F\coloneqq\sum_{i}F|_{k}g_{i}\quad\text{where}\
\Gamma_{n}g\Gamma_{n}=\sqcup_{i}\Gamma_{n}g_{i},\quad
g_{i}=\begin{pmatrix}A_{i}&B_{i}\\\ C_{i}&D_{i}\\\ \end{pmatrix}$ $\text{and}\
F|_{k}g_{i}(Z)=\mu(g)^{nk-\frac{n(n+1)}{2}}det(C_{i}Z+D_{i})^{-k}F((AZ+B)(CZ+D)^{-1}).$

For every positive integer $m$, the above definition is extended to define
$T(m)$ as follows

$T(m)F\coloneqq\sum_{\mu(g)=m}T(g)F.$

In Theorem $4.7$ of [And74], it is shown that there exists a basis for
$M_{k}(\Gamma_{n})$ which are eigenforms with respect to all the Hecke
operators simultaneously. Since $\displaystyle{\langle T(g)F,G\rangle=\langle
F,T(g)G\rangle}$ we conclude that the eigenvalues must be real. For a Hecke
eigenform $F\in S_{k}(\Gamma_{n})$, denote $\lambda_{F}(g)$ as the eigenvalue
for the operator $T(g)$. Classically $\lambda_{F}(g)$ can be expressed in
terms of $Satake\ p-parameters$. For any $g$ with $\mu(g)=p^{r}$, depending on
$F$ there are $n+1$ complex numbers $(a_{0,p},a_{1,p},\dots,a_{n,p})$
satisfying

$\lambda(g)=a_{0,p}^{r}\sum_{i}\prod_{j=1}^{n}(a_{j,p}^{-j})^{d_{ij}}\
\text{where}\ \Gamma_{n}g\Gamma_{n}=\sqcup_{i}\Gamma_{n}g_{i},$
$g_{i}=\begin{pmatrix}A_{i}&B_{i}\\\ 0&D_{i}\\\ \end{pmatrix}\ \text{and}\
B_{i}=\begin{pmatrix}p^{d_{i1}}&&\ast\\\ &\ddots&\vdots\\\
0&\dots&p^{d_{in}}\end{pmatrix}.$

The $a_{j,p}$ are called Satake p-parameters of the eigenform $F$. See first
$3$ chapters of [Pit19] for more on classical Siegel modular forms.

Theorem $1.3.2$ of [And74] states that there are polynomials $P_{F,p}(x)$ and
$Q_{F,p}(x)$ such that

$\sum_{r=0}^{\infty}\lambda_{F}(p^{r})p^{-rs}=\frac{P_{F,p}(p^{-s})}{Q_{F,p}(p^{-s})}$
(1)

and

$Q_{F,p}(p^{-s})=(1-a_{0,p}p^{-s})\Big{(}\prod_{j=1}^{n}\prod_{1\leq\delta_{1}<\delta_{2}<..<\delta_{j}\leq
n}(1-a_{0,p}a_{{\delta_{1}},p}...a_{{\delta_{j}},p}p^{-s})\Big{)}.$

The denominator defines the spinor L-function of $F$, $L(s,F,spin)$ as follows

$L(s,F,spin)\coloneqq\prod_{p}L_{p}(s,F,spin)\quad
L_{p}(s,F,spin)=Q_{F,p}(p^{-s})^{-1}.$

From equation $(1)$ it is clear that to compute $\lambda_{F}(p^{r})$ both
$P_{F,p}(x)$ and $L_{p}(s,F,spin)$ are necessary. If $f\in S_{k}(\Gamma_{1})$
with Satake parameters $ap^{\frac{k-1}{2}}$ and $a^{-1}p^{\frac{k-1}{2}}$ then
it is well known that $P_{f,p}(x)\equiv 1$ and

$L_{p}(s,f,spin)=Q_{f,p}(p^{-s})^{-1}=\Big{(}(1-ap^{\frac{k-1}{2}}p^{-s})(1-a^{-1}p^{\frac{k-1}{2}}p^{-s})\Big{)}^{-1}.$

For genus 2, in [And74], Andrianov showed that

$Q_{F,p}(p^{-s})=1-\lambda_{F}(p)p^{-s}+(\lambda_{F}(p)^{2}-\lambda_{F}(p^{2})-p^{2k-4})p^{-2s}-\lambda_{F}(p)p^{2k-3}p^{-3s}+p^{4k-6}p^{-4s}$

and

$P_{F,p}(p^{-s})=1-p^{4k+2}p^{-2s}.$

Vankov found the polynomials $P_{F,p}(x)$ for genus $3$ and $4$ in [Van11].
For higher genus they are still unknown to the best of our knowledge.

Siegel modular forms are uniquely defined by their Fourier coefficients. Ikeda
defines coefficients $A(B)$ for $F_{f}$ using the Fourier coefficients of $f$
and proves the following theorem in [Ike01].

######  Theorem (Ikeda).

Assume that $n\equiv k(mod\ 2)$. Then

$F_{f}(Z)=\sum_{B\in\mathcal{S}_{2n}(\mathbb{Z})^{+}}A(B)e^{2\pi
itr(BZ)},\quad Z\in\mathbb{H}_{2n}$

is a Hecke eigen form in $S_{k+n}(\Gamma_{2n})$ and the standard L-function of
$F$ is equal to

$\zeta(s)\prod_{i=1}^{2n}L(s+k+n-i,spin,f).$

Here $\mathcal{S}_{2n}(\mathbb{Z})^{+}$ is set of all positive definite half-
integral matrices of size $2n$ and $\zeta(s)$ is Riemann zeta function.

We omit the details of construction of $A(B)$, check introduction of [Ike01]
for more details. Let $F_{f}\in S_{k+n}(\Gamma_{2n})$ be the Ikeda lift of
$f\in S_{2k}(\Gamma_{1})$. To compute the spin L-functions of Ikeda lifts one
has to move to the adelic language of modular forms. For each Hecke eigenform
$F\in S_{k}(\Gamma_{n})$ there is an associated automorphic representation
$\Pi_{F}$ of $G(\mathbb{A})$, where $\mathbb{A}$ is ring of adeles of
$\mathbb{Q}$. See Section $4$ of [AS01] for details. Say $\Pi_{F_{f}}$,
$\pi_{f}$ are automorphic representations corresponding to $F_{f}$ and $f$
respectively. Let $a,a^{-1}$ be the Satake-p-parameters for $\pi_{f}$. By Prop
$2.2.1$ of [Sch03], the spin L-function of $\Pi_{F_{f}}$ at a non archimedian
place $p$ is given by

$L_{p}(s,\Pi_{F_{f}},spin)=\prod_{j=0}^{n}\prod_{\begin{subarray}{c}r=j(j-2n)\\\
Step2\end{subarray}}^{j(2n-j)}L_{p}(s+\frac{r}{2},\pi_{f},Sym^{n-j})^{\beta(r,j,n)}$
(2)

where

$L_{p}(s,\pi_{f},Sym^{n})=\prod_{i=0}^{n}(1-a^{i}a^{-(n-i)}p^{-s})^{-1},$

$\beta(r,j,n)=\alpha(r,j,n)-\alpha(r,j-2,n)$ and $\alpha(r,j,n)$ is the number
of ways to pick $j$ numbers from the set
$\\{1-2n,3-2n,...,-1,1,...,2n-3,2n-1\\}$ such that they add up to $r$. The
relation between $L(s,F_{f},spin)$ and $L(s,\Pi_{F_{f}},spin)$ is explained by
Lemma $10$ of [AS01]. In the case of Ikeda lifts $F_{f}\in
S_{k+n}(\Gamma_{2n})$, if

$L_{p}(s,\Pi_{F_{f}},spin)^{-1}=(1-a_{0}p^{-s})\prod_{j=1}^{n}\prod_{1\leq\delta_{1}<\delta_{2}<..<\delta_{j}\leq
n}(1-a_{0}a_{\delta_{1}}...a_{\delta_{j}}p^{-s})$

then

$L_{p}(s,F_{f},spin)^{-1}=(1-p^{nk-\frac{n}{2}}a_{0}p^{-s})\prod_{j=1}^{n}\prod_{1\leq\delta_{1}<\delta_{2}<..<\delta_{j}\leq
n}(1-p^{nk-\frac{n}{2}}a_{0}a_{\delta_{1}}...a_{\delta_{j}}p^{-s}).$

######  Theorem 2.1.

For $n\equiv k(mod\ 2)$, let $F_{f}\in S_{k+n}(\Gamma_{2n})$ be the Ikeda lift
of $f\in S_{2k}(\Gamma_{1})$. For all large enough primes $p$,
$\lambda_{F_{f}}(p)\geq 0$.

###### Proof.

By the statement of Theorem $1.3.2$ of [And74] the coefficient of $x$ in
$Q_{F_{f},p}(x)$ is $-\lambda_{F_{f}}(p)$. Using Schmidt’s formula for
L-function and Andrianov’s result we can compute $\lambda_{F_{f}}(p)$ for an
Ikeda lift. In case of Ikeda lifts, $-p^{\frac{n}{2}-nk}\lambda_{F_{f}}(p)$
would be the coefficient of $p^{-s}$ in

$L_{p}(s,\Pi_{F_{f}},spin)^{-1}=\prod_{j=0}^{n}\prod_{\begin{subarray}{c}r=j(j-2n)\\\
Step2\end{subarray}}^{j(2n-j)}L_{p}(s+\frac{r}{2},\pi_{f},Sym^{n-j})^{-\beta(r,j,n)}$

For a fixed $j$, let us compute the coefficient of $p^{-s}$ in
$\displaystyle{L_{p}(s+\frac{r}{2},\pi_{f},Sym^{n-j})^{-1}}$

$L_{p}(s+\frac{r}{2},\pi_{f},Sym^{n-j})^{-1}=\prod_{i=0}^{n-j}(1-a^{i}(a^{-1})^{n-j-i}p^{-(s+\frac{r}{2})})=\prod_{i=0}^{n-j}(1-a^{2i+j-n}p^{\frac{-r}{2}}p^{-s}).$

Coefficient of $p^{-s}$ is

$-\sum_{i=0}^{n-j}a^{2i+j-n}p^{-\frac{r}{2}}.$

Hence the coefficient of $p^{-s}$ in
$\displaystyle{\prod_{j=0}^{n}\prod_{\begin{subarray}{c}r=j(j-2n)\\\
step2\end{subarray}}^{j(2n-j)}L_{p}(s+\frac{r}{2},\pi_{f},Sym^{n-j})^{-\beta(r,j,n)}}$
is

$-\sum_{j=0}^{n}\Bigg{(}\sum_{i=0}^{n-j}a^{2i+j-n}\Bigg{(}\sum_{\begin{subarray}{c}r=j(j-2n)\\\
step2\end{subarray}}^{j(2n-j)}p^{\frac{-r}{2}}\beta(r,j,n)\Bigg{)}\Bigg{)}$
$\Rightarrow\lambda_{F_{f}}(p)=p^{nk-\frac{n}{2}}\sum_{j=0}^{n}\Bigg{(}\sum_{i=0}^{n-j}a^{2i+j-n}\Bigg{(}\sum_{\begin{subarray}{c}r=j(j-2n)\\\
step2\end{subarray}}^{j(2n-j)}p^{\frac{-r}{2}}\beta(r,j,n)\Bigg{)}\Bigg{)}$
(3)

Since the above expression is a polynomial in $p^{\pm\frac{1}{2}}$, for a
large enough $p$ we can determine the sign by calculating the leading term,
i.e, the term with highest power of $p$. In the expression $(3)$, the highest
power of $p$ appears when $r$ is the smallest. Every $r$ is of the form
$j^{2}-2nj+b$ for some non negative integer $b$. Smallest value of
$j^{2}-2nj+b$ is when $j=n$ and $b=0$. The leading term is
$\beta(-n^{2},n,n)p^{\frac{n^{2}}{2}}.$

$\beta(-n^{2},n,n)=\alpha(-n^{2},n,n)-\alpha(-n^{2},n-2,n),$

and the only way to pick $r$ numbers from
$\\{1-2n,3-2n,...,-1,1,...,2n-3,2n-1\\}$ to add up to $-n^{2}$ is to pick all
$n$ negative numbers and no positive numbers. We can conclude that
$\alpha(-n^{2},n,n)=1$, $\alpha(-n^{2},n-2,n)=0$ and $\beta(-n^{2},n,n)=1$.

$\Rightarrow\lambda_{F_{f}}(p)=p^{\frac{n}{2}-nk}\big{(}p^{\frac{n^{2}}{2}}+\sum_{r=-n^{2}}^{n^{2}-1}c_{r}p^{\frac{r}{2}}\big{)}$

For a fixed $r$ let us compute the coefficient $c_{r}$. From equation $(3)$ it
is clear that $c_{r}=\sum_{j=0}^{n}c_{j,r}$, where $c_{j,r}$ is the
coefficient of $p^{\frac{r}{2}}$ in

$\sum_{i=0}^{n-j}a^{2i+j-n}\Bigg{(}\sum_{\begin{subarray}{c}r=j(j-2n)\\\
step2\end{subarray}}^{j(2n-j)}p^{\frac{-r}{2}}\beta(r,j,n)\Bigg{)}$
$\Rightarrow c_{j,r}=\sum_{i=0}^{n-j}a^{2i+j-n}\beta(-r,j,n).$ $\Rightarrow
c_{r}=\sum_{j=0}^{n}c_{j,r}=\sum_{j=0}^{n}\bigg{(}\sum_{i=0}^{n-j}a^{2i+j-n}\beta(-r,j,n)\bigg{)}$

Since $\lvert a\rvert\leq 1$ and $\beta(r,j,n)$ are bounded, we conclude that
$c_{r}$ are bounded and the bound is independent of $p$. Hence for a large
enough prime

$p^{\frac{n^{2}}{2}}>\lvert\sum_{r=-n^{2}}^{n^{2}-1}c_{r}p^{\frac{r}{2}}\rvert.$

This implies that $\lambda_{F_{f}}(p)$ is positive for all large enough primes
$p$.

∎

##  3 $\lambda_{F}(p^{r})$ for Ikeda lifts of genus $4$

In this section, using equation $(1)$ we compute $\lambda_{F_{f}}(p^{r})$ for
Ikeda lifts of genus $4$. Vankov calculated the polynomials $P_{F,p}(x)$ and
$Q_{F,p}(x)$ as polynomials over the Hecke algebra. Hecke algebra,
$\mathcal{H}_{p}$, over $G(\mathbb{Q})$ is the ring generated by the following
Hecke operators

$T_{i}(p^{2})=T(diag(\underbrace{1,\dots,1}_{n-i},\underbrace{p,\dots,p}_{i},\underbrace{p^{2},\dots,p^{2}}_{n-i},\underbrace{p,\dots,p}_{i}))\
\text{for}\ 1\leq i\leq n$ $\text{and}\
T(p)=T(diag(\underbrace{1,\dots,1}_{n},\underbrace{p,\dots,p}_{n})).$

Andrianov, in [And68], proved that there exists polynomials $P_{p}(x)$ and
$Q_{p}(x)$ in $\mathcal{H}_{p}[x]$ such that

$\sum_{r=0}^{\infty}T(p^{r})x^{r}=\frac{P_{p}(x)}{Q_{p}(x)}.$ (4)

Vankov, in [Van11], calculated these polynomials explicitly for genus $4$.
$P_{F_{f},p}(x)$ and $Q_{F_{f},p}(x)$ can be retrieved from $P_{p}(x)$ and
$Q_{p}(x)$ by substituting the operators appearing in the coefficients with
their respective eigenvalues of $F_{f}$. For example, in case of genus $1$.

$Q_{p}(x)=1-T(p)x+p^{k-1}x^{2}\ \text{and}\
Q_{f,p}(x)=1-\lambda_{f}(p)x+p^{k-1}x^{2}$ $\Rightarrow
L_{p}(s,f,spin)=(1-\lambda_{f}(p)p^{-s}+p^{k-1-2s})^{-1}.$

Hence we compute spin L-function for genus $4$ Ikeda lift to obtain
$Q_{F_{f},p}(x)$, which will give us the eigenvalues for $F_{f}$ for the Hecke
operators $T_{1}(p^{2}),\ T_{2}(p^{2}),\ T_{3}(p^{2})$ and $T(p)$.

######  Lemma 3.1.

If $F_{f}\in S_{k+2}(\Gamma_{4})$ is the Ikeda lift of $f\in
S_{2k}(\Gamma_{1})$ and the Satake parameters for $f$ are $ap^{\frac{k-1}{2}}$
and $a^{-1}p^{\frac{k-1}{2}}$ then

$Q_{F_{f},p}(x)=(1-ap^{2k-1}p^{\frac{1}{2}}x)(1-a^{-1}p^{2k-1}p^{\frac{1}{2}}x)(1-ap^{2k-1}p^{\frac{-1}{2}}x)(1-a^{-1}p^{2k-1}p^{\frac{-1}{2}}x)$
$(1-ap^{2k-1}p^{\frac{3}{2}}x)(1-a^{-1}p^{2k-1}p^{\frac{3}{2}}x)(1-ap^{2k-1}p^{\frac{-3}{2}}x)(1-a^{-1}p^{2k-1}p^{\frac{-3}{2}}x)(1-p^{2k-1}p^{2}x)(1-p^{2k-1}px)$
$(1-a^{2}p^{2k-1}x)(1-a^{-2}p^{2k-1}x)(1-p^{2k-1}p^{-1}x)(1-p^{2k-1}p^{-2}x)(1-p^{2k-1}x)^{2}.$

###### Proof.

Since $Q_{F_{f},p}(p^{-s})^{-1}=L_{p}(s,F_{f},spin)$, we use $(2)$ to
calculate the spin L-function of $F$ and by extension $Q_{F_{f},p}(p^{-s})$.
In the case of Ikeda lift $F_{f}$ of genus $4$ we substitute $n=2$ in $(2)$.
We get three sets of values for $\beta(r,j,2)$ depending on $j$. These can be
found in appendix of [Sch03] or can be calculated using the definition of
$\alpha(r,j,n)$ for small values of $n$. Let us compute
$L_{p}(s,\pi_{f},Sym^{j})^{\beta(r,jn)}$ for each $j$.

  1. 1.

For $j=0$, $r=0$ and $\beta(0,0,2)=1.$

$L_{p}(s,\pi_{f},Sym^{2})=\Big{(}\
(1-a^{2}p^{-s})(1-a^{-2}p^{-s})(1-p^{-s})\Big{)}\ ^{-1}$

  2. 2.

For $j=1$, $r=-3,-1,1,3$ and
$\beta(3,1,2)=\beta(-3,1,2)=\beta(1,1,2)=\beta(-1,1,2)=1$

$L_{p}(s+\frac{-3}{2},\pi_{f},Sym^{1})L_{p}(s+\frac{-1}{2},\pi_{f},Sym^{1})L_{p}(s+\frac{1}{2},\pi_{f},Sym^{1})L_{p}(s+\frac{3}{2},\pi_{f},Sym^{1})=$
$\Big{(}(1-ap^{\frac{3}{2}-s})(1-a^{-1}p^{\frac{3}{2}-s})(1-ap^{\frac{1}{2}-s})(1-a^{-1}p^{\frac{1}{2}-s})(1-ap^{\frac{-1}{2}-s})(1-a^{-1}p^{\frac{-1}{2}-s})$
$(1-ap^{\frac{-3}{2}-s})(1-a^{-1}p^{\frac{-3}{2}-s})\Big{)}^{-1}$

  3. 3.

For $j=2$, $r=-4,-2,0,2,4$ and $\beta(r,j,2)=1$ for all $r$.

$L_{p}(s-2,\pi_{f},Sym^{0})L_{p}(s-1,\pi_{f},Sym^{0})L_{p}(s,\pi_{f},Sym^{0})L_{p}(s+1,\pi_{f},Sym^{0})L_{p}(s+2,\pi_{f},Sym^{0})$
$=\Big{(}(1-p^{-s+2})(1-p^{-s+1})(1-p^{-s})(1-p^{-s-1})(1-p^{-s-2})\Big{)}^{-1}$

Combining all the cases we get $L_{p}(s,\Pi_{F_{f}},spin)$ as

$\Big{(}(1-a^{2}p^{-s})(1-a^{-2}p^{-s})(1-p^{-s})(1-ap^{\frac{3}{2}-s})(1-a^{-1}p^{\frac{3}{2}-s})$
$(1-ap^{\frac{1}{2}-s})(1-a^{-1}p^{\frac{1}{2}-s})(1-ap^{\frac{-1}{2}-s})(1-a^{-1}p^{\frac{-1}{2}-s})(1-ap^{\frac{-3}{2}-s})$
$(1-a^{-1}p^{\frac{-3}{2}-s})(1-p^{-s+2})(1-p^{-s+1})(1-p^{-s})(1-p^{-s-1})(1-p^{-s-2})\Big{)}^{-1}$

Replacing $p^{-s}$ in $L_{p}(s,\Pi_{F_{f}},spin)$ with $p^{2k-1}p^{-s}$ we get
$L_{p}(s,F_{f},spin)$ and hence the formula of $Q_{F_{f},p}(x)$ as in the
statement of the lemma. ∎

Coefficients of the polynomial $P_{p}(x)$ computed by Vankov in [Van11]
contain $T_{1}(p^{2}),\ T_{2}(p^{2}),\ T_{3}(p^{2})$ and $T(p)$. To retrieve
$P_{F_{f},p}(x)$ from $P_{p}(x)$ we need eigenvalues of the aforementioned
operators. Say $\lambda_{F_{f}}(T_{i}(p^{2}))$ is the eigenvalue for $F$ with
respect to $T_{i}(p^{2}).$

######  Lemma 3.2.

The eigenvalues $\lambda_{F_{f}}(p)$ and $\lambda_{F_{f}}(T_{i}(p^{2}))$ for
$i=1,2,3$ are

$\lambda_{F_{f}}(T_{3}(p^{2}))=\frac{\left(p^{3}+p^{2}+p+1\right)\left(a^{2}p^{5/2}+a(p-1)+p^{5/2}\right)p^{4(k-3)}}{a}$
$\lambda_{F_{f}}(T_{2}(p^{2}))=p^{4k-2}+(a+\frac{1}{a})(p^{4k-\frac{7}{2}}+p^{4k-\frac{9}{2}}+p^{4k-\frac{11}{2}}-p^{4k-\frac{15}{2}}-p^{4k-\frac{17}{2}}-p^{4k-\frac{19}{2}})$
$(a+\frac{1}{a})^{2}(p^{4k-3}+2p^{4k-4}+p^{4k-6}+p^{4k-7})+p^{4k-4}(a^{2}+\frac{1}{a^{2}}+3)-p^{4k-8}-p^{4k-10}$
$\lambda_{F_{f}}(T_{1}(p^{2}))=p^{4k}+p^{4k-10}+p^{4k-4}+(a+\frac{1}{a})(p^{4k+\frac{1}{2}}+2p^{4k-\frac{1}{2}}-p^{4k-\frac{11}{2}}-p^{4k-\frac{13}{2}}-p^{4k-\frac{15}{2}})$
$+(a+\frac{1}{a})^{2}(p^{4k-1}+p^{4k-2}+p^{4k-3}-p^{4k-5})-p^{4k-6}-p^{4k-7})(a^{2}+\frac{1}{a^{2}}+3)-p^{4k-8}-p^{4k-10}$
$+p^{4k-\frac{7}{2}}(a+\frac{1}{a})^{3}-p^{4k-6}(a^{2}+3+\frac{1}{a^{2}})+(p^{4k-\frac{5}{2}}+p^{4k-\frac{3}{2}})(a^{3}+4a+4\frac{1}{a}+\frac{1}{a^{3}})$
$\lambda_{F_{f}}(p)=p^{2k-1}(a^{2}+\frac{1}{a^{2}}+\frac{a\left(p^{3}+p^{2}+p+1\right)}{p^{3/2}}+\frac{p^{3}+p^{2}+p+1}{ap^{3/2}}+p^{2}+\frac{1}{p^{2}}+p+\frac{1}{p}+2)$

###### Proof.

Since coefficients of $Q_{p}(x)$ also contain all the operators $T_{i}(p^{2})$
and $T(p)$, we can compute the eigenvalues by comparing coefficient of
$Q_{F_{f},p}(x)$ and $Q_{p}(x).$ ∎

Above lemma gives an explicit expression for $\lambda_{F_{f}}(p)$. Using it we
can improve the result of Theorem $2.1$ and prove that $\lambda_{F_{f}}(p)$ is
positive for all primes $p$ when $n=4.$

######  Proposition 3.3.

If $F_{f}\in S_{k+2}(\Gamma_{4})$ is the Ikeda lift of $f\in
S_{2k}(\Gamma_{1})$ then $\lambda_{F_{f}}(p)$ is positive for all primes $p$.

###### Proof.

The formula of $\lambda_{F_{f}}(p)$ from Lemma $3.2$ can be rewritten as

$p^{2k-1}\Big{(}p^{\frac{3}{2}}\Big{(}\sqrt{p}+a+\frac{1}{a}\Big{)}+p^{\frac{1}{2}}\Big{(}\sqrt{p}+a+\frac{1}{a}\Big{)}+p^{\frac{-3}{2}}\Big{(}\sqrt{p}+a+\frac{1}{a}\Big{)}+$
$2+p^{\frac{-1}{2}}\Big{(}a+\frac{1}{a}\Big{)}+\frac{1}{p^{2}}+a^{2}+\frac{1}{a^{2}}\Big{)}$

Say $a+\frac{1}{a}=u$, then $\lambda_{F}(p)$

$=p^{2k-1}\Big{(}p^{\frac{3}{2}}\Big{(}\sqrt{p}+u\Big{)}+p^{\frac{-1}{2}}u+p^{\frac{1}{2}}\Big{(}\sqrt{p}+u\Big{)}+p^{\frac{-3}{2}}\Big{(}\sqrt{p}+u\Big{)}+\frac{1}{p^{2}}+u^{2}\Big{)}$

Considering the expression as a quadratic in $u$, we can easily check when it
is positive.

  1. 1.

Assume $p\geq 5$. Since $u\geq-2$, $\sqrt{p}+u\geq\sqrt{5}-2>0.236$ and
$\frac{u}{\sqrt{p}}\geq-\frac{2}{\sqrt{5}}\geq-0.9.$ These inequalities imply
that

$p^{2k-1}(p^{\frac{3}{2}}(\sqrt{p}+u)+\frac{u}{\sqrt{p}})\geq
p^{2k-1}(5^{\frac{3}{2}}(0.236)-0.9)>0$

and

$p^{2k-1}(p^{\frac{1}{2}}(\sqrt{p}+u)+p^{\frac{-3}{2}}(\sqrt{p}+u)+\frac{1}{p^{2}}+u^{2})>0.$

Adding the above two inequalities, we conclude that $\lambda_{F_{f}}(p)\geq 0$
for all $p>5.$

  2. 2.

If $p=3$ then

$\lambda_{F_{f}}(3)=3^{2k-1}(u^{2}+\frac{40}{3\sqrt{3}}u+\frac{112}{9})$

  3. 3.

If $p=2$ then

$\lambda_{F_{f}}(2)=2^{2k-1}(u^{2}+\frac{15}{2\sqrt{2}}u+\frac{27}{4})$

In cases of $p=2,3$ we get quadratic equations which are positive for
$u\geq-2$. This shows $\lambda_{F_{f}}(p)$ is positive for all primes. ∎

Substituting the eigenvalues found in Lemma $3.2$ into $P_{p}(x)$ computed by
Vankov, we derive $P_{F_{f},p}(x)$. And Lemma $3.1$ calculates
$Q_{F_{f},p}(x)$. With this information we can prove the following theorem.

######  Theorem 3.4.

Let $F_{f}$ be a Hecke eigenform in $S_{k+2}(\Gamma_{4})$ and the Ikeda lift
of $f\in S_{2k}(\Gamma_{1})$. For a fixed $r$, $\lambda_{F_{f}}(p^{r})$ is
positive for all large enough primes $p$.

###### Proof.

To compute $\lambda_{F_{f}}(p^{r})$ we mimic Bruelmann’s idea([Bre99]) of
writing $1/Q_{F_{f},p}(x)$ as partial fractions and comparing the powers of
$x^{r}$ on both sides.

$\frac{1}{Q_{F_{f},p}(x)}=\frac{a_{1}}{(-1+p^{2k-1}x)^{2}}+\frac{a_{2}}{(-1+p^{2k-1}x)}+\frac{a_{3}}{(-a\sqrt{p}+p^{2k-1}x)}+\frac{a_{4}}{(-ap^{\frac{3}{2}}+p^{2k-1}x)}$
$+\frac{a_{5}}{(-p^{2}+p^{2k-1}x)}+\frac{a_{6}}{(-\sqrt{p}+p^{2k-1}x)}+\frac{a_{7}}{(-1+a^{2}p^{2k-1}x)}+\frac{a_{8}}{(-a+\sqrt{p}p^{2k-1}x)}+$
$\frac{a_{9}}{(-1+a\sqrt{p}p^{2k-1}x)}+\frac{a_{10}}{(-1+pp^{2k-1}x)}+\frac{a_{11}}{(-a+p^{\frac{3}{2}}p^{2k-1}x)}+\frac{a_{12}}{(-1+ap^{\frac{3}{2}}p^{2k-1}x)}$
$+\frac{a_{13}}{(-1+p^{2}p^{2k-1}x)}+\frac{a_{14}}{(-a^{2}+p^{2k-1}x)}+\frac{a_{15}}{(-p+p^{2k-1}x)}+\frac{a_{16}}{(-p^{\frac{3}{2}}+ap^{2k-1}x)}$

Precise values of $a_{i}^{\prime}s$ are given in appendix. Above fractions can
be written as power series, for example

$\frac{1}{-a+bx}=-\sum_{r=0}^{\infty}a^{-(r+1)}b^{r}x^{r}.$

Writing every fraction as a power series and collecting like terms together,
we get the coefficient of $x^{r}$ as

$p^{(2k-1)r}(a_{1}(r+1)-a_{2}-a_{3}a^{-r-1}p^{-(r+1)/2}-a_{4}a^{-r-1}p^{-3(r+1)/2}-a_{5}p^{-2(r+1)}-a_{6}a^{r}p^{-(r+1)/2}$
$-a_{7}a^{2r}-a_{8}a^{-r-1}p^{r/2}-a_{9}a^{r}p^{r/2}-a_{10}p^{r}-a_{11}a^{-r-1}p^{3r/2}-a_{12}a^{r}p^{3r/2}-a_{13}p^{2r}-a_{14}a^{-2r-2}$
$-a_{15}p^{-(r+1)}-a_{16}a^{r}p^{-3(r+1)/2}).$

Let us denote it by $g(r)$. We can rewrite $(1)$ as

$\sum_{r=0}^{\infty}\lambda_{F_{f}}(p^{r})x^{r}=P_{F_{f}}(x)\sum_{r=0}^{\infty}g(r)x^{r}.$

Comparing coefficients of $x^{r}$ on both sides, we have

$\lambda_{F}(p^{r})=\sum_{i=0}^{14}e_{i}g(r-i)$ (5)

where it is understood that $g(r-i)=0$ for $r-i<0$. Here $e_{i}$ are defined
by $\displaystyle{P_{F_{f}}(x)=\sum_{i=0}^{14}e_{i}x^{i}}$.

$\lambda_{F_{f}}(p^{r})$ is expression of the form $f_{1}(p)r+f_{2}(p)$ where
$f_{i}$ are rational functions in $p^{\frac{1}{2}}$ with coefficients in
$\mathbb{Z}[a,a^{-1}]$. Hence for a fixed $r$, $\lambda_{F_{f}}(p^{r})$ is a
rational function in $p^{\frac{1}{2}}$ with coefficients in
$\mathbb{Z}[a,a^{-1}]$. The denominator on right side of $(5)$ is

$a^{2r}p^{6k+2r+14}(a^{2}-1)^{3}(a^{2}+1)(p-1)^{5}(p+1)^{3}(p^{2}+1)(p^{2}+p+1)$
$a^{17}(p+(a+\frac{1}{a})\sqrt{p}+1)(p-(a+\frac{1}{a})\sqrt{p}+1)^{2}(p-(a^{3}+\frac{1}{a^{3}})\sqrt{p}+1)(p^{3}-(a+\frac{1}{a})p^{\frac{3}{2}}+1)^{3}$
$(p^{3}+(a+\frac{1}{a})p^{\frac{3}{2}}+1)(p^{4}-(a^{2}+\frac{1}{a^{2}})p^{2}+1)(p^{3}-(a^{3}+\frac{1}{a^{3}})p^{\frac{3}{2}}+1)^{3}$
$(p^{3}+(a+\frac{1}{a})p^{\frac{3}{2}}+1)(p^{5}-(a+\frac{1}{a})p^{\frac{5}{2}}+1)(p^{7}-(a+\frac{1}{a})p^{\frac{7}{2}}+1)$

Expressions of the form
$\displaystyle{p^{m}+(\mu+\frac{1}{\mu})p^{\frac{m}{2}}+1}$ are always
positive for $p\geq 3$ and $\mu\in\mathbb{S}^{1}$.
$(p-1)^{5}(p+1)^{3}(p^{2}+1)(p^{2}+p+1)p^{6k+2r+14}$ is also positive for all
primes. Let us collect all these terms and call it $D(r)$,

$D(r)=p^{6k+2r+14}(p-1)^{5}(p+1)^{3}(p^{2}+1)(p^{2}+p+1)(p+(a+\frac{1}{a})\sqrt{p}+1)(p-(a+\frac{1}{a})\sqrt{p}+1)^{2}$
$(p-(a^{3}+\frac{1}{a^{3}})\sqrt{p}+1)(p^{3}-(a+\frac{1}{a})p^{\frac{3}{2}}+1)^{3}(p^{3}+(a+\frac{1}{a})p^{\frac{3}{2}}+1)(p^{4}-(a^{2}+\frac{1}{a^{2}})p^{2}+1)$
$(p^{3}-(a^{3}+\frac{1}{a^{3}})p^{\frac{3}{2}}+1)^{3}(p^{3}+(a+\frac{1}{a})p^{\frac{3}{2}}+1)(p^{5}-(a+\frac{1}{a})p^{\frac{5}{2}}+1)(p^{7}-(a+\frac{1}{a})p^{\frac{7}{2}}+1)$

Multiply $D(r)$ on both sides of equation $(5)$ to get

$D(r)\lambda_{F}(p^{r})=D(r)\sum_{i=0}^{14}e_{i}g(r-i)$ (6)

Since $D(r)$ is positive for all prime $p\geq 3$, to determine the sign of
$\lambda_{F_{f}}(p^{r})$, it is enough to check the sign on the right side of
the above expression. The numerator of $(6)$ is a polynomial in
$p^{\frac{1}{2}}$ with coefficients in $\mathbb{Z}[a,a^{-1}]$ and leading term
is

$p^{61+6k+3r+2kr}(a^{2r}(a^{25}-2a^{23}+2a^{19}-a^{17})).$

The denominator of $(6)$ is $a^{2r+17}(a^{2}-1)^{3}(a^{2}+1)$. This implies
that for a fixed $r$ the right side of equation $(6)$ is a polynomial in
$p^{\frac{1}{2}}$ with the leading term $p^{61+6k+3r+2kr}$.

$\lambda_{F_{f}}(p^{r})=p^{61+6k+3r+2kr}+\sum_{i=0}^{121+12k+6r+4kr}c_{i}p^{\frac{i}{2}}\
\text{s.t}\ c_{i}\in\mathbb{Z}[a,a^{-1}]$

Since $c_{i}$ are bounded and the bound is dependent on $r$, for a fixed $r$

$\lambda_{F_{f}}(p^{r})\geq 0\ \text{for all large enough $p$.}$

∎

## References

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Appendix

$a_{1}$ |  $-a^{6}p^{7}((a^{2}-1)^{2}(p-1)^{4}(p+1)^{2}(a-\sqrt{p})^{2}(a\sqrt{p}-1)^{2}(a-p^{3/2})^{2}(ap^{3/2}-1)^{2})^{-1}$
---|---
$a_{2}$ |  $7a^{6}p^{7}((a^{2}-1)^{2}(p-1)^{4}(p+1)^{2}(a-\sqrt{p})^{2}(a\sqrt{p}-1)^{2}(a-p^{3/2})^{2}(ap^{3/2}-1)^{2})^{-1}$
$a_{3}$ |  $a^{2}p^{9/2}((a^{2}-1)(p-1)^{3}(p+1)(a-\sqrt{p})^{3}(a+\sqrt{p})(a\sqrt{p}-1)^{3}(a\sqrt{p}+1)(a^{3}\sqrt{p}-1)(a-p^{3/2})(ap^{3/2}-1)(a^{2}p^{2}-1)(ap^{5/2}-1))^{-1}$
$a_{4}$ |  $-a^{2}p^{2}((a^{2}-1)(p-1)^{3}(p+1)(p^{2}+p+1)(a-\sqrt{p})(a\sqrt{p}-1)^{3}(a\sqrt{p}+1)(a-p^{3/2})(ap^{3/2}-1)^{3}(a^{2}p^{2}-1)(ap^{5/2}-1)(a^{3}p^{5/2}+a^{2}p^{2}+a^{2}p+ap^{3/2}+a\sqrt{p}+1)(ap^{7/2}-1))^{-1}$
$a_{5}$ |  $-a^{6}p^{2}((p-1)^{5}(p+1)^{3}(p^{2}+1)(p^{2}+p+1)(a-\sqrt{p})(a\sqrt{p}-1)(a-p^{3/2})(ap^{3/2}-1)(a^{2}-p^{2})(a^{2}p^{2}-1)(a-p^{5/2})(ap^{5/2}-1)(a-p^{7/2})(ap^{7/2}-1))^{-1}$
$a_{6}$ |  $a^{17}p^{9/2}((a^{2}-1)(p-1)^{3}(p+1)(a-\sqrt{p})^{3}(a^{3}-\sqrt{p})(a+\sqrt{p})(a\sqrt{p}-1)^{3}(a\sqrt{p}+1)(a-p^{3/2})(ap^{3/2}-1)(a^{2}-p^{2})(a-p^{5/2}))^{-1}$
$a_{7}$ |  $-a^{32}p^{7}((a^{2}-1)^{3}(a^{2}+1)(a-\sqrt{p})^{3}(a^{3}-\sqrt{p})(a+\sqrt{p})(a\sqrt{p}-1)^{3}(a\sqrt{p}+1)(a^{3}\sqrt{p}-1)(a^{2}+a\sqrt{p}+p)(a^{2}p+a\sqrt{p}+1)(a-p^{3/2})(ap^{3/2}-1)(a^{2}-p^{2})(a^{2}p^{2}-1))^{-1}$
$a_{8}$ |  $-a^{2}p^{12}((a^{2}-1)(p-1)^{3}(p+1)(a-\sqrt{p})^{3}(a^{3}-\sqrt{p})(a+\sqrt{p})(a\sqrt{p}-1)^{3}(a\sqrt{p}+1)(a-p^{3/2})(ap^{3/2}-1)(a^{2}-p^{2})(a-p^{5/2}))^{-1}$
$a_{9}$ |  $-a^{17}p^{12}((a^{2}-1)(p-1)^{3}(p+1)(a-\sqrt{p})^{3}(a+\sqrt{p})(a\sqrt{p}-1)^{3}(a^{4}p+a^{3}\sqrt{p}-a\sqrt{p}-1)(a-p^{3/2})(ap^{3/2}-1)(a^{3}p^{9/2}-a^{2}p^{2}-ap^{5/2}+1))^{-1}$
$a_{10}$ |  $-a^{6}p^{18}((p-1)^{5}(p+1)(p^{2}+p+1)(a-\sqrt{p})^{3}(a+\sqrt{p})(a\sqrt{p}-1)^{3}(a\sqrt{p}+1)(a-p^{3/2})(ap^{3/2}-1)(a-p^{5/2})(ap^{5/2}-1))^{-1}$
$a_{11}$ |  $a^{2}p^{49/2}((a^{2}-1)(p-1)^{3}(p+1)(p^{2}+p+1)(a\sqrt{p}-1)(a^{2}-ap^{3/2}-a\sqrt{p}+p^{2})^{3}(a^{4}+a^{3}p^{3/2}+2a^{3}\sqrt{p}+2a^{2}p^{2}+2a^{2}p+ap^{3/2}+2ap^{5/2}+p^{3})(a^{5}p^{3/2}-a^{4}p^{5}-a^{4}p^{4}-a^{4}+a^{3}p^{5/2}+a^{3}p^{15/2}+a^{2}p^{7}+a^{2}p^{2}-ap^{9/2}-ap^{11/2}-ap^{19/2}+p^{8})^{-1}$
$a_{12}$ |  $a^{17}p^{49/2}((a^{2}-1)(p-1)^{3}(p+1)(p^{2}+p+1)(a-\sqrt{p})(a\sqrt{p}-1)^{3}(a\sqrt{p}+1)(a^{2}p+a\sqrt{p}+1)(a-p^{3/2})(ap^{3/2}-1)^{3}(ap^{3/2}+1)(a^{2}p^{2}-1)(ap^{5/2}-1)(ap^{7/2}-1)^{-1}$
$a_{13}$ |  $a^{6}p^{32}((p-1)^{5}(p+1)^{3}(p^{4}+p^{3}+2p^{2}+p+1)(a^{7}p^{3/2}+a^{5}p^{3/2}-a^{9}p^{5/2}+a^{7}p^{5/2}+a^{5}p^{5/2}-a^{3}p^{5/2}-a^{9}p^{7/2}+2a^{7}p^{7/2}+2a^{5}p^{7/2}-a^{3}p^{7/2}-a^{9}p^{9/2}+3a^{7}p^{9/2}+3a^{5}p^{9/2}-a^{3}p^{9/2}-2a^{9}p^{11/2}+3a^{7}p^{11/2}+3a^{5}p^{11/2}-2a^{3}p^{11/2}-a^{11}p^{13/2}-3a^{9}p^{13/2}+3a^{7}p^{13/2}+3a^{5}p^{13/2}-3a^{3}p^{13/2}-ap^{13/2}-a^{11}p^{15/2}-4a^{9}p^{15/2}+3a^{7}p^{15/2}+3a^{5}p^{15/2}-4a^{3}p^{15/2}-ap^{15/2}-a^{11}p^{17/2}-4a^{9}p^{17/2}+3a^{7}p^{17/2}+3a^{5}p^{17/2}-4a^{3}p^{17/2}-ap^{17/2}-a^{11}p^{19/2}-4a^{9}p^{19/2}+4a^{7}p^{19/2}+4a^{5}p^{19/2}-4a^{3}p^{19/2}-ap^{19/2}-a^{11}p^{21/2}-4a^{9}p^{21/2}+4a^{7}p^{21/2}+4a^{5}p^{21/2}-4a^{3}p^{21/2}-ap^{21/2}-a^{11}p^{23/2}-4a^{9}p^{23/2}+3a^{7}p^{23/2}+3a^{5}p^{23/2}-4a^{3}p^{23/2}-ap^{23/2}-a^{11}p^{25/2}-4a^{9}p^{25/2}+3a^{7}p^{25/2}+3a^{5}p^{25/2}-4a^{3}p^{25/2}-ap^{25/2}-a^{11}p^{27/2}-3a^{9}p^{27/2}+3a^{7}p^{27/2}+3a^{5}p^{27/2}-3a^{3}p^{27/2}-ap^{27/2}-2a^{9}p^{29/2}+3a^{7}p^{29/2}+3a^{5}p^{29/2}-2a^{3}p^{29/2}-a^{9}p^{31/2}+3a^{7}p^{31/2}+3a^{5}p^{31/2}-a^{3}p^{31/2}-a^{9}p^{33/2}+2a^{7}p^{33/2}+2a^{5}p^{33/2}-a^{3}p^{33/2}-a^{9}p^{35/2}+a^{7}p^{35/2}+a^{5}p^{35/2}-a^{3}p^{35/2}+a^{7}p^{37/2}+a^{5}p^{37/2}+a^{7}p^{39/2}+a^{5}p^{39/2}-a^{6}p^{20}-a^{6}p^{19}-2a^{6}p^{18}-3a^{6}p^{17}+a^{10}p^{16}-4a^{6}p^{16}+a^{2}p^{16}+a^{10}p^{15}+a^{8}p^{15}-4a^{6}p^{15}+a^{4}p^{15}+a^{2}p^{15}+2a^{10}p^{14}+a^{8}p^{14}-5a^{6}p^{14}+a^{4}p^{14}+2a^{2}p^{14}+2a^{10}p^{13}+a^{8}p^{13}-6a^{6}p^{13}+a^{4}p^{13}+2a^{2}p^{13}+2a^{10}p^{12}+a^{8}p^{12}-7a^{6}p^{12}+a^{4}p^{12}+2a^{2}p^{12}+3a^{10}p^{11}+2a^{8}p^{11}-6a^{6}p^{11}+2a^{4}p^{11}+3a^{2}p^{11}+a^{12}p^{10}+4a^{10}p^{10}+3a^{8}p^{10}-6a^{6}p^{10}+3a^{4}p^{10}+4a^{2}p^{10}+p^{10}+3a^{10}p^{9}+2a^{8}p^{9}-6a^{6}p^{9}+2a^{4}p^{9}+3a^{2}p^{9}+2a^{10}p^{8}+a^{8}p^{8}-7a^{6}p^{8}+a^{4}p^{8}+2a^{2}p^{8}+2a^{10}p^{7}+a^{8}p^{7}-6a^{6}p^{7}+a^{4}p^{7}+2a^{2}p^{7}+2a^{10}p^{6}+a^{8}p^{6}-5a^{6}p^{6}+a^{4}p^{6}+2a^{2}p^{6}+a^{10}p^{5}+a^{8}p^{5}-4a^{6}p^{5}+a^{4}p^{5}+a^{2}p^{5}+a^{10}p^{4}-4a^{6}p^{4}+a^{2}p^{4}-3a^{6}p^{3}-2a^{6}p^{2}-a^{6}p-a^{6}+a^{7}\sqrt{p}+a^{5}\sqrt{p})^{-1}$
$a_{14}$ |  $a^{2}p^{7}((a^{2}-1)^{3}(a^{2}+1)(a-\sqrt{p})^{3}(a^{3}-\sqrt{p})(a+\sqrt{p})(a\sqrt{p}-1)^{3}(a\sqrt{p}+1)(a^{3}\sqrt{p}-1)(a^{2}+a\sqrt{p}+p)(a^{2}p+a\sqrt{p}+1)(a-p^{3/2})(ap^{3/2}-1)(a^{2}-p^{2})(a^{2}p^{2}-1))^{-1}$
$a_{15}$ |  $a^{6}p^{3}((p-1)^{5}(p+1)(p^{2}+p+1)(a-\sqrt{p})^{3}(a+\sqrt{p})(a\sqrt{p}-1)^{3}(a\sqrt{p}+1)(a-p^{3/2})(ap^{3/2}-1)(a-p^{5/2})(ap^{5/2}-1))^{-1}$
$a_{16}$ |  $-a^{17}p^{2}((a^{2}-1)(p-1)^{3}(p+1)(p^{2}+p+1)(a-\sqrt{p})^{3}(a+\sqrt{p})(a\sqrt{p}-1)(a^{2}+a\sqrt{p}+p)(a-p^{3/2})^{3}(a+p^{3/2})(ap^{3/2}-1)(a^{2}-p^{2})(a-p^{5/2})(a-p^{7/2}))^{-1}$


# Divisor Functions: Train-like Structure
and Density Properties

Evelina Dubovski
Staten Island Technical High School
<EMAIL_ADDRESS>
###### Abstract

We investigate the density properties of generalized divisor functions
$\displaystyle f_{s}(n)=\frac{\sum_{d|n}d^{s}}{n^{s}}$ and extend the analysis
from the already-proven density of $s=1$ to $s\geq 0$. We demonstrate that for
every $s>0$, $f_{s}$ is locally dense, revealing the structure of $f_{s}$ as
the union of infinitely many $trains$—specially organized collections of
decreasing sequences—which we define. We analyze Wolke’s conjecture that
$|f_{1}(n)-a|<\frac{1}{n^{1-\varepsilon}}$ has infinitely many solutions and
prove it for points in the range of $f_{s}$. We establish that $f_{s}$ is
dense for $0<s\leq 1$ but loses density for $s>1$. As a result, in the latter
case the graphs experience ruptures. We extend Wolke’s discovery
$\displaystyle|f_{1}(n)-a|<\frac{1}{n^{0.4-\varepsilon}}$ to all $0<s\leq 1$.
In the last section we prove that the rational complement to the range of
$f_{s}$ is dense for all $s>0$. Thus, the range of $f_{1}$ and its complement
form a partition of rational numbers to two dense subsets. If we treat the
divisor function as a uniformly distributed random variable, then its
expectation turns out to be $\zeta(s+1)$. The theoretical findings are
supported by computations. Ironically, perfect and multiperfect numbers do not
exhibit any distinctive characteristics for divisor functions.

Divisor Functions: Train-like Structure and
Density Properties
Evelina Dubovski111E-mail<EMAIL_ADDRESS>
Staten Island Technical High School

##  1 Introduction

Perfect numbers are one of the most unique phenomena in number theory. Dating
back to ancient Greece, perfect numbers first appeared in Euclid’s Elements
and have been the focus of many great mathematicians, including Leonardo of
Pisa (Fibonacci) and Euler. But what are perfect numbers? A perfect number is
a positive integer that is equal to the sum of its positive divisors,
excluding the number itself.

Despite the attention that perfect numbers have received over the centuries,
there is still much left to be discovered. The last known perfect number was
discovered in 2018, bringing the total of identified perfect numbers to 51. We
do not even know whether there exist odd perfect numbers or whether their
cardinality is infinite. An analysis of perfect numbers has led us to our
investigation of the divisor function and its properties. The intended
audience is researchers in multiplicative number theory and the broad
community of math lovers.

The function $\sigma(n)=\sum\limits_{d|n}d$ is called the divisor function of
whole number $n$. It is equal to the sum of all divisors $d$ (factors) of $n$,
including 1 and $n$. For example, $\sigma(10)=18$. Clearly, $\sigma(p)=1+p$ if
$p$ is prime. We consider the generalization of perfect numbers in terms of
the divisor function. A number is perfect if $\sigma(n)=2n$. Since
$\sigma(6)=12$, then 6 is perfect along with 28, 496, etc. Scientists also
consider multiperfect numbers, for which divisor function $\sigma$ takes
integer values called abundancies. If $\sigma(n)=kn$, then $n$ is multiperfect
of abundancy $k$. A recent work [1] states that no $k$-perfect odd numbers are
known for any $k\geq 2$.

The motivation for this research was to analyze whether perfect and
multiperfect numbers are really special, leading, as a research tool, to the
investigation of the properties of divisor functions presented in this
article. The theory of the divisor function $\sigma(n)$ and its relation to
other challenging problems can be found, e.g., in books [2] and [3].
Interesting recent results on the sums involving the divisor function are in
[4, 5].

In 1941, Cramer [6] proved that the function $\displaystyle
f(n)=\frac{\sigma(n)}{n}$ is dense on $[1,\infty)$, i.e., for every
$a\in[1,\infty)$ and any positive small $\varepsilon>0$, there is at least one
$n$ such that $|f(n)-a|<\varepsilon$. Since $\varepsilon$ is arbitrary, then
“at least one” actually implies “infinitely many.”

In 1977, Wolke [7] strengthened this result as follows: for every $a\geq 1$
and every $\varepsilon>0$, the inequality

$\left|f(n)-a\right|\leq\frac{1}{n^{0.4-\varepsilon}}$ (1)

has infinitely many solutions in whole numbers $n$. This is further evidence
that the function $\displaystyle f(n)=\frac{\sigma(n)}{n}$ is dense in
$[1,\infty)$. In fact, the above inequality implies that in any small
neighborhood of any real $a\geq 1$ we can find infinitely many values of
$f(n)$. Moreover, in his paper D. Wolke conjectured that constant $0.4$ can be
replaced by 1.

Our goal is to examine the generalization of the divisor
function,$\sigma_{s}(n)=\sum_{d|n}d^{s}$ for $s\geq 0$. If $s=1$, we arrive at
the usual divisor function $\sigma(n)$. If $s=0$, we obtain $\sigma_{0}(n)$,
which is just the number of divisors of $n$.

We investigate the function
$f_{s}(n)=\displaystyle\frac{\sigma_{s}(n)}{n^{s}}$ for different non-negative
values of $s$ to determine whether their ranges are dense (like $f_{1}$) or
not. We explore the structure of functions $f_{s}$ and show that their range
is the union of trains, specially organized linked decreasing sequences, that
are defined in this paper. This observation leads to the proof of Wolke’s
conjecture for the points from $R(f_{s})$, the range of $f_{s}$. That is, we
show that the extended Wolke’s conjecture is valid for any $a\in R(f_{s})$.
Particularly, for $s=1$, the constant 0.4 in inequality (1) can be replaced by
1.

For all reals, not only from the range, we rely on Cramer’s approach [6] and
demonstrate that for $0<s\leq 1$, the function $f_{s}$ is dense and chaotic,
but for $s>1$, $f_{s}$ loses density and ruptures appear.

By extending Wolke’s finding (1) to $0<s\leq 1$, we obtain the following
quantitative measure of the density strength for all $a\geq 1$:

$\left|f_{s}(n)-a\right|\leq\frac{1}{n^{0.4s-\varepsilon}}.$ (2)

We end the paper by proving that the rational complement to the range is also
dense in $[1,\infty)$ for positive integer $s$. The presented analysis is
supported by computations.

##  2 Local density

The most essential property of the divisor function is its multiplicativity
for the products of relatively prime numbers. If $a$ and $b$ are relatively
prime and $c_{i}$ are the factors of $a$ whereas $d_{i}$ are the factors of
$b$, then

$\sigma_{s}(ab)=\sum_{i,j}c_{i}^{s}d_{j}^{s}=\sum_{i}c_{i}^{s}\cdot\sum_{j}d_{j}^{s}=\sigma_{s}(a)\cdot\sigma_{s}(b).$

Particularly, if we consider the prime factorization
$n=\prod_{i=1}^{m}p_{i}^{k_{i}}$, then
$\sigma_{s}(np)=\sigma_{s}(n)\cdot\sigma_{s}(p)$ if prime number $p$ does not
coincide with any of $p_{i}$, i.e, $p\not\in
P_{n}=\\{p_{1},p_{2},\ldots,p_{n}\\}$, the set of all prime factors of $n$.
Similar results hold for functions $f_{s}$, and we have
$f_{s}(np)=f_{s}(n)\cdot f_{s}(p)$. So,

$f_{s}(n)=f_{s}\left(\prod_{i=1}^{m}p_{i}^{k_{i}}\right)=\prod_{i=1}^{m}f_{s}(p_{i}^{k_{i}}).$
(3)

The computations shown below indicate that the graphs of $f_{s}(n)$ are formed
by an infinite union of subgraphs, each of which is a sequence that decreases
as $n\to\infty$. Motivated by these computational results, we consider prime
$p\not\in P_{n}$ and generate the decreasing sequence
$\\{f_{s}(np)\\}_{p\not\in P_{n}}$. In the well-investigated case $s=1$, we
obtain for $n=2$ the sequence
$\\{f_{1}(2p)\\}=\left\\{2,9/5,12/7,18/11,\ldots\right\\}$, where the
corresponding values are $p=3,5,7,11,\ldots$. For $n=6$, we obtain the
sequence $\\{f_{1}(6p)\\}=\left\\{12/5,16/7,24/11,\ldots\right\\}$. Here
$p=5,7,11,\ldots$. Clearly, these sample sequences are decreasing.

######  Lemma 1.

For $s>0$, the sequences $\\{f_{s}(np)\\}$ are decreasing for all sufficiently
large $p$ and $\lim_{p\to\infty}f_{s}(np)=f_{s}(n)$.

Proof. Since $p\to\infty$, we can assume that $p\not\in P_{n}$. Then

$f_{s}(np)=f_{s}(n)f_{s}(p)=f_{s}(n)\cdot\left(1+\frac{1}{p^{s}}\right)\to
f_{s}(n)\quad{\rm as}\ p\to\infty.$ (4)

This proves Lemma 1. ∎

We introduce the following definition to conveniently describe the structure
of the range of $f_{s}$.

######  Definition 1.

We call train an ordered collection of linked sequences such that
(1) any term of the sequence is greater that any term of the preceding
sequence;
(2) the infimum of any sequence is equal to the supremum of the preceding
sequence (i.e., the sequences are linked).
The above sequences, forming the train, are called cars.

This terminology comes from real trains in which the passengers of any car are
ahead of all passengers in the following car.

######  Remark 1.

If $p\in P_{n}$, then sequence $\\{f_{s}(np)\\}$ is not necessarily decreasing
for small $p$. For example, if $n=2$ and $p=2$ then
$\\{f_{1}(np)\\}=\\{1.75,2,1.8,\ldots\\}$.

######  Theorem 1.

(Local density) Let $s>0$. If $r=f_{s}(n)$ is a number from the range
$R(f_{s})$ of function $f_{s}$, then for every $\varepsilon>0$ there are
infinitely many values of $m$ such that $f_{s}(m)\in[f(n),f(n)+\varepsilon]$.

The proof of Theorem 1 follows directly from Lemma 1.

As such, it can be said that the range of $f_{s}$ is “dense from above.” Also,
the function is locally dense: in any neighborhood of any point $r\in
R(f_{s})$, there are infinitely many points from range $R(f_{s})$. Figure 1
clarifies the statement of Theorem 1.

Figure 1:  Three first sequences (“cars”) of the train for $n=6$, $s=1$.

We begin with the blue sequence $f(6p)$, in which the first value of $p$ is 5,
so $f(6\times 5)$ is the highest value in this “car.” The decreasing sequence
$f(6p)\to f(6)$ as $p\to\infty$. Any term of the next “car” is greater than
any term of the previous sequence. As shown, $f(30p)=f(6\times 5\times p)$
decreases to $f(30)$, the highest term of $f(6p)$. Once again, any term of a
“car” is greater than any term of the previous “car.” Just as the cars of a
train are connected but do not overlap, so are these sequences.

According to the proof of Lemma 1, for a given
$n=p_{1}^{k_{1}}p_{2}^{k_{2}}\cdot\ldots\cdot p_{m}^{k_{m}}$, if $q_{1}$ is
the smallest prime outside the set $P_{n}$, then the sequence
$\\{f_{s}(np)\\}$ attains its maximum at $p=q_{1}$, which is $\displaystyle
f_{s}(n)\cdot(1+\frac{1}{q_{1}^{s}})$. Similarly, the sequence
$\\{f_{s}(nq_{1}p)\\}$ attains its maximum at $p=q_{2}$, which is the lowest
prime number outside $P_{nq_{1}}$. By continuing, we arrive at the sequence
$\\{f_{s}(nq_{1}q_{2}\cdot\ldots\cdot q_{j}p)\\}_{p\not\in P_{n}}$. Here, the
primes $q_{i}\not\in P_{n}$ and form the ordered sequence outside of $P_{n}$,
$q_{1}<q_{2}<\ldots q_{j}$. The graphs for $f_{s}$ are described by the
following assertion:

######  Proposition 1.

For every prime $p>q_{j+1}$, $p\not\in P_{n}$,

$f_{s}(np)<f_{s}(nq_{1}p)<\ldots<f_{s}(nq_{1}q_{2}\cdot\ldots\cdot q_{j}p).$

Thus, every term from sequence $\\{f_{s}(nq_{1}q_{2}\cdot\ldots\cdot
q_{j}q_{j+1}p)\\}$ with variable primes $p$ is greater than any term from
sequence $\\{f_{s}(nq_{1}q_{2}\cdot\ldots\cdot q_{j}p)\\}$.

Proof. In view of (4),

$\displaystyle f_{s}(nq_{1}q_{2}\cdot\ldots\cdot
q_{j}p)=f_{s}(nq_{1}q_{2}\cdot\ldots\cdot q_{j})f_{s}(p)$
$\displaystyle\phantom{WW}<f_{s}(nq_{1}q_{2}\cdot\ldots\cdot
q_{j})f_{s}(q_{j+1})=f_{s}(nq_{1}q_{2}\cdot\ldots\cdot q_{j}q_{j+1})$
$\displaystyle\phantom{WW}<f_{s}(nq_{1}q_{2}\cdot\ldots\cdot q_{j}q_{j+1}p).$

Therefore,

$\max_{p}f_{s}(nq_{1}\ldots q_{j}p)\leq f_{s}(nq_{1}\ldots
q_{j}q_{j+1})<f_{s}(nq_{1}\ldots q_{j}q_{j+1}p).$

This proves Proposition 1 ∎

Proposition 1 explains the visible structure of the graphs for functions
$f_{s}$ (see Figures 1, 2, and 3). Each of the sequences can be considered a
“car” on the train, or collection, of sequences
$\\{f_{s}(nq_{1}q_{2}\cdot\ldots\cdot q_{j}p)\\}$.

Relying on the above analysis, we now strengthen Wolke’s estimate (1) for the
points from the range $R(f_{s})$.

######  Theorem 2.

Let $s>0$. For every $a\in R(f_{s})$ there are infinitely many solutions to
inequality

$|f_{s}(n)-a|\leq\frac{c(a)}{n^{s}}.$ (5)

Proof. Let $a=f_{s}(m)$. Then for every $p\not\in{P}_{m}$ we have
$\displaystyle f_{s}(mp)=a\cdot\left(1+\frac{1}{p^{s}}\right)$. Hence, for all
$n=mp$ and $p\not\in{P}_{m}$, we obtain $\displaystyle
f_{s}(n)-a=\frac{a[f_{s}^{-1}(a)]^{s}}{n^{s}}$. This proves Theorem 2. ∎

######  Corollary 1.

Let the conditions of Theorem 2 hold. Then for every $\varepsilon>0$, the
inequality

$|f_{s}(n)-a|\leq\frac{1}{n^{s-\varepsilon}},\quad s>0,$ (6)

has infinitely many solutions.

Proof. The shift by $\varepsilon$ compensates for the constant $c(a)$ and
allows us to replace it by 1. ∎

##  3 Global density

Let $n=p^{k}$ for prime $p$. Then

$f_{s}(p^{k})=\frac{1+p^{s}+\ldots+p^{ks}}{p^{ks}}=\frac{p^{(k+1)s}-1}{(p^{s}-1)p^{ks}}=\frac{p^{s}-p^{-ks}}{p^{s}-1}=\frac{p^{s}}{p^{s}-1}-\frac{p^{-ks}}{p^{s}-1}.$
(7)

In general, for $n=\prod\limits_{i=1}^{m}p_{i}^{k_{i}}$, we use multiplicative
property (3) and obtain

$f_{s}(n)=f_{s}\left(\prod_{i=1}^{m}p_{i}^{k_{i}}\right)=\prod_{i=1}^{m}f_{s}\left(p_{i}^{k_{i}}\right)=\prod_{i=1}^{m}\left(\frac{p_{i}^{s}}{p_{i}^{s}-1}-\frac{p_{i}^{-k_{i}s}}{p_{i}^{s}-1}\right).$
(8)

The second term $p_{i}^{-k_{i}s}\cdot(p_{i}^{s}-1)^{-1}$ becomes arbitrarily
small for big $p_{i}$. The first, principal, term in (8) yields

$f_{s}(n)\approx\prod_{i=1}^{m}\frac{p_{i}^{s}}{p_{i}^{s}-1}=Q_{m}.$ (9)

Let’s investigate how closely $Q_{m}$ approximates $f_{s}(n)$.
It is known [3] that

$\lim_{m\to\infty}\prod_{i=1}^{m}\left(1+\frac{1}{p_{i}-1}\right)=\infty,$
(10)

where $p_{i}$ are all prime numbers.

######  Lemma 2.

For $0<s\leq 1$,

$\prod_{i=1}^{\infty}\left(1+\frac{1}{p_{i}^{s}-1}\right)=\infty$ (11)

whereas for $s>1$ this infinite product is finite

$\prod_{i=1}^{\infty}\left(1+\frac{1}{p_{i}^{s}-1}\right)<\infty.$ (12)

Proof. For $0<s\leq 1$ we obtain

$1+\frac{1}{p_{i}^{s}-1}\geq 1+\frac{1}{p_{i}-1}=\frac{p_{i}}{p_{i}-1}.$

In view of (10) we obtain (11):

$\prod_{i=1}^{\infty}\left(1+\frac{1}{p^{s}_{i}-1}\right)\geq\prod_{i=1}^{\infty}\left(1+\frac{1}{p_{i}-1}\right)=\infty.$
(13)

For $s>1$ we have

$\prod_{i=1}^{\infty}\left(1+\frac{1}{p_{i}^{s}-1}\right)=\exp\left(\sum_{i=1}^{\infty}\ln\left(1+\frac{1}{p_{i}^{s}-1}\right)\right).$
(14)

Since $\ln(1+x)\leq x$, then we consider

$\sum_{i=1}^{\infty}\ln\left(1+\frac{1}{p_{i}^{s}-1}\right)\leq\sum_{i=1}^{\infty}\frac{1}{p_{i}^{s}-1}<1+\sum_{i=2}^{\infty}\frac{1}{p_{i}^{s}-1}.$
(15)

Since $p_{i}^{s}\geq(p_{i-1}+1)^{s}>p_{i-1}^{s}+1$, then
$\displaystyle\frac{1}{p_{i}^{s}-1}<\frac{1}{p_{i-1}^{s}}$, and for (15) we
obtain the upper bound

$\sum_{i=1}^{\infty}\frac{1}{p_{i}^{s}-1}<1+\sum_{i=2}^{\infty}\frac{1}{p_{i-1}^{s}}=1+\sum_{i=1}^{\infty}\frac{1}{p_{i}^{s}}<\sum_{i=1}^{\infty}\frac{1}{i^{s}}<\infty.$

Combining this result with (14), (15), we derive (12). ∎

In Section 2, we showed that the points from range $R(f_{s})$ can be
approximated from above. Now we consider arbitrary real numbers $a$ and show
that $a>1$ can be approximated from below.

######  Lemma 3.

Let $0<s\leq 1$. Then for any real number $a>1$ and any $\varepsilon>0$, there
exist primes $q_{1}<q_{2}<\ldots<q_{m}$ such that $a-\varepsilon<Q_{m}<a$,
where

$Q_{m}=\prod_{i=1}^{m}\left(1+\frac{1}{q_{i}^{s}-1}\right).$

Proof. Let $q_{1}$ be the smallest prime such that $\displaystyle
1+\frac{1}{q^{s}_{1}-1}\leq a$. Since the infinite product (10) diverges and
$\displaystyle 1+\frac{1}{p^{s}_{i}-1}\to 1$ as $p_{i}\to\infty$, we can
choose additional $j_{1}$ consecutive bigger primes $q_{i}$, $1\leq i\leq
j_{1}$ such that the corresponding product

$Q_{j_{1}}=\prod\limits_{i=1}^{j_{1}}\left(1+\frac{1}{q_{i}^{s}-1}\right)$

is still less than $a$. Let $q_{j_{1}}$ be the last $q$ which can be chosen
this way. Then $Q_{j_{1}}<a$ but

$Q_{j_{1}}\cdot\left(1+\frac{1}{\hat{q}^{s}-1}\right)\geq a$ (16)

where $\hat{q}$ is the nearest prime number greater than $q_{j_{1}}$.
Then we skip several prime numbers and select the smallest prime $q_{j_{1}+1}$
such that

$Q_{j_{1}+1}=Q_{j_{1}}\cdot\left(1+\frac{1}{q_{j_{1}+1}^{s}-1}\right)\leq a.$

Once again, we select the next finite set of consecutive primes $q_{j_{1}+1}$,
$q_{j_{1}+2}$, $\ldots$, $q_{j_{2}}$ such that $Q_{j_{2}}\leq a$ but
$\displaystyle Q_{j_{2}}\cdot\left(1+\frac{1}{\hat{q}^{s}-1}\right)\geq a$. As
before, $\hat{q}$ is the nearest prime number greater than $q_{j_{2}}$. Then
we repeat the procedure, finding $q_{j_{3}}$, $q_{j_{4}}$, $\ldots$,
$q_{j_{k}}$ such that the corresponding

$Q_{j_{3}}<Q_{j_{4}}<\ldots<Q_{j_{k}}\leq a$

and, eventually, $Q_{j_{k}}\geq a-\varepsilon$.
Thus, $m=j_{k}$, proving Lemma 3. So, we can arrive arbitrarily close to $a$
from below. ∎

The following lemma is needed in order to analyze the negligibility of the
small term $\displaystyle\frac{p_{i}^{-k_{i}s}}{p_{i}^{s}-1}$ in the infinite
product (8).

######  Lemma 4.

Let $s>0$. For any set of $m$ primes $q_{1}<q_{2}<\ldots q_{m}$ and any
$\varepsilon>0$ there exist exponents $n_{1},n_{2},\ldots,n_{m}$ such that

$Q_{m}-\varepsilon<f_{s}\left(q_{1}^{sk_{1}}q_{2}^{sk_{2}}\times\ldots\times
q_{m}^{sk_{m}}\right)<Q_{m}$ (17)

whenever $k_{i}>n_{i}$.

Proof. Following (7), let us consider $\displaystyle
f_{s}(q^{sx})=\frac{q^{s(x+1)}-1}{q^{sx}(q^{s}-1)}$. We differentiate in $x$
and obtain $\displaystyle\frac{d}{dx}f_{s}(q^{sx})=\frac{s\ln
q}{(q^{s}-1)q^{sx}}>0$. Thus, $f_{s}$ increases with $x$. Since
$\displaystyle\lim_{x\to\infty}f_{s}(q^{sx})=\frac{q^{s}}{q^{s}-1}$, then for
any $\varepsilon>0$, there is an $n$ such that

$\frac{q^{s}}{q^{s}-1}-\varepsilon<f_{s}(q^{sk})<\frac{q^{s}}{q^{s}-1}$

whenever $k>n$. It follows that, for any two primes $q_{1}$ and $q_{2}$ and
any $\varepsilon_{1}>0$ and $\varepsilon_{2}>0$, there are numbers $n_{1}$ and
$n_{2}$ such that for all $k_{1}>n_{1}$ and $k_{2}>n_{2}$ the following
inequalities hold:

$\frac{q_{i}^{s}}{q_{i}^{s}-1}-\varepsilon_{i}<f_{s}\left(q_{i}^{sk_{i}}\right)<\frac{q_{i}^{s}}{q_{i}^{s}-1}.$
(18)

Using the multiplicative property,
$f_{s}(q_{1}^{sk_{1}}q_{2}^{sk_{2}})=f_{s}(q_{1}^{sk_{1}})f_{s}(q_{2}^{sk_{2}})$,
we obtain by multiplication the inequalities (18) for $i=1$ and $i=2$:

$\displaystyle
Q_{2}-\left(\frac{\varepsilon_{1}q^{s}_{2}}{q^{s}_{2}-1}+\varepsilon_{2}Q_{1}-\varepsilon_{1}\varepsilon_{2}\right)<f_{s}(q_{1}^{sk_{1}}q_{2}^{sk_{2}})<Q_{2}.$
(19)

If we choose
$\displaystyle\varepsilon_{1}<\varepsilon\frac{q^{s}_{2}-1}{2q^{s}_{2}}$ and
$\displaystyle\varepsilon_{2}<\frac{\varepsilon}{2Q_{1}}$, then from (19) we
have

$Q_{2}-\varepsilon<f_{s}(q_{1}^{sk_{1}}q_{2}^{sk_{2}})<Q_{2}.$ (20)

Then we multiply (20) by (18) at $i=3$, then by (18) at $i=4$, and so on,
extending this argument to the complete set of $m$ primes by induction. ∎

######  Theorem 3.

Let $0<s\leq 1$. Then there exist infinitely many integers $n$ such that
$f_{s}(n)$ differs from $a$ by the amount less than arbitrary $\varepsilon>0$
where $a$ is any real constant, $a\geq 1$.

Proof. By Lemma 3 we take $m$ primes $q_{1},q_{2},\ldots,q_{m}$ such that
$Q_{m}$ differs from $a$ by an amount less than $\frac{1}{2}\varepsilon$.
Lemma 4 allows us to choose the exponents of these primes such that
$f_{s}\left(q_{1}^{sk_{1}}q_{2}^{sk_{2}}\ldots q_{m}^{sk_{m}}\right)$ differs
from $Q_{m}$ by less than $\frac{1}{2}\varepsilon$. Finally,
$\left|f_{s}\left(q_{1}^{sk_{1}}q_{2}^{sk_{2}}\ldots
q_{m}^{sk_{m}}\right)-a\right|<\varepsilon$, proving Theorem 3. ∎

The computations confirm the structure of the range $R(f_{s})$ as the union of
trains. Figure 1 showed the train $T(6)$. Trains are also observable in
Figures 2 and 3 below.

In Figure 2 ($s=1$), the trains look dense, fitting Theorem 3. In Figure 3 the
computations verify that density fails for $s>1$ as proved in Theorem 4 below.

Figure 2: Dense behavior of $f_{s}(n)$ at $s=1$. A “chaos” structured by
trains (see Fig. 1).

The compressed graph on the right makes density more visible.

######  Theorem 4.

In order for the functions $f_{s}$ to lose the density property and have
ruptures, it is necessary and sufficient that $s>1$.

Proof. The necessity follows from the key point of the proof of Theorem 3, the
divergence of the infinite product (13). For $s>1$, the divergence fails.
To prove sufficiency, let us find the upper bound for range $R(f_{s})$. From
(7),

$\displaystyle
f_{s}(n)=\prod\limits_{i=1}^{m}\frac{1-p_{i}^{-(k+1)s}}{1-p_{i}^{-s}}<\prod\limits_{i=1}^{\infty}\frac{1}{1-p_{i}^{-s}}=\sum_{n=1}^{\infty}\frac{1}{n^{s}}.$

The last step comes from Euler’s identity. Finally, for every $n\in\mathbb{N}$
and $s>1$,

$f_{s}(n)<\zeta(s)<\infty.\hfill\qed$ (21)

Particularly, for $s=2$ we have $f_{2}(n)<\frac{\pi^{2}}{6}$. The computations
(Figure 3) below perfectly fit this estimate and show that density fails for
$s>1$.

Figure 3: Non-dense range of $f_{s}$ at $s=2$. $f_{2}(n)<\frac{\pi^{2}}{6}$.
No density and the appearance of ruptures for $s>1$. Trains are also
observable. The compressed graph of $f_{2}$ on the right makes the ruptures
more visible.

The following Theorem strengthens the density result of Theorem 3 and provides
the quantitative measure of the density. Here, we extend inequality (2) from
$s=1$ to all $0<s\leq 1$.

######  Theorem 5.

Let $0<s\leq 1$. Then for every $a\geq 1$ and $\varepsilon>0$ the inequality

$|f_{s}(n)-a|\leq\frac{1}{n^{0.4s-\varepsilon}}$ (22)

has infinitely many solutions on $[1,\infty)$.

Proof. We follow Wolke’s idea [7] and use the following notations:
$\overline{p}$ is the greatest prime less than prime number $p$ whereas
$\hat{p}$ is the smallest prime greater than $p$. So,
$\overline{p}<p<\hat{p}$. We choose any positive $y$ and find prime $p_{0}>y$
such that

$\prod_{y<p\leq p_{0}}\left(1+\frac{1}{p^{s}}\right)\leq a\quad{\rm
and}\quad\left(1+\frac{1}{\hat{p}^{s}}\right)\cdot\prod_{y<p\leq
p_{0}}\left(1+\frac{1}{p^{s}}\right)>a$ (23)

Let us consider the product of primes

$n_{0}=\prod_{y<p\leq p_{0}}p.$

Similar to the previous constructions, let $p_{1}$ be the smallest prime such
that $\displaystyle f_{s}(n_{0})\left(1+\frac{1}{p_{1}^{s}}\right)<a$. We
define step-by-step the sequence $\\{n_{k}\\}$ as follows:

$n_{k}=n_{0}p_{1}p_{2}\cdot\ldots\cdot p_{k},\quad f_{s}(n_{k})<a,\quad{\rm
but}\ f_{s}(n_{k-1})\cdot\left(1+\frac{1}{\overline{p}_{k}^{s}}\right)>a.$
(24)

Let $p_{k+1}$ be the smallest prime number such that

$f_{s}(n_{k})\cdot\left(1+\frac{1}{p^{s}_{k+1}}\right)\leq a.$

Thus, $p_{k+1}>p_{k}$ and $n_{k+1}=n_{k}p_{k+1}$. Certainly, if
$f_{s}(n_{k+1})=a$, then the construction of the sequence $\\{n_{k}\\}$ ends.
Further, we consider infinite sequence $\\{n_{k}\\}$.
Due to the above construction of $\\{n_{k}\\}$, it is easy to observe

$f_{s}(n_{k+1})=f_{s}(n_{k-1})\left(1+\frac{1}{p^{s}_{k}}\right)\cdot\left(1+\frac{1}{p^{s}_{k+1}}\right)\leq
a\leq f_{s}(n_{k-1})\left(1+\frac{1}{\overline{p}^{s}_{k}}\right).$

Then

$\displaystyle 1+\frac{1}{p^{s}_{k+1}}\leq\frac{\displaystyle
1+\frac{1}{\overline{p}^{s}_{k}}}{\displaystyle
1+\frac{1}{p^{s}_{k}}}=\left(1+\frac{1}{\overline{p}^{s}_{k}}\right)\sum_{i=0}^{\infty}(-1)^{i}\frac{1}{p^{is}_{k}}\leq\left(1+\frac{1}{\overline{p}^{s}_{k}}\right)\left(1-\frac{1}{p^{s}_{k}}+\frac{1}{p^{2s}_{k}}\right).$

Since $\overline{p}_{k}>\frac{1}{2}p_{k}$, then we obtain

$p_{k+1}^{s}\geq\frac{\overline{p}_{k}^{s}p_{k}^{s}}{p_{k}^{s}-\overline{p}_{k}^{s}}\geq\frac{1}{2^{s}}\cdot\frac{p_{k}^{2s}}{p_{k}^{s}-\overline{p}_{k}^{s}}.$
(25)

We introduce $\Delta_{k}$ such that
$f_{s}(n_{k})\cdot\left(1+\frac{1}{\Delta_{k}}\right)=a$. Then from the
inequality

$f_{s}(n_{k})\left(1+\frac{1}{\overline{p}_{k+1}^{s}}\right)>a$

we obtain

$1+\frac{1}{\Delta_{k}}<1+\frac{1}{\overline{p}_{k+1}^{s}}.$

Taking (25) into account, we get

$\Delta_{k}>\overline{p}^{s}_{k+1}\geq\frac{1}{2^{s}}p^{s}_{k+1}\geq\frac{1}{4^{s}}\cdot\frac{p_{k}^{2s}}{p_{k}^{s}-\overline{p}_{k}^{s}}.$

Next,

$\left|f_{s}(n_{k})-a\right|=\left|f_{s}(n_{k})-f_{s}(n_{k})\cdot\left(1+\frac{1}{\Delta_{k}}\right)\right|\leq\frac{f_{s}(n_{k})}{\Delta_{k}}\leq
4^{s}a\cdot\frac{p^{s}_{k}-\overline{p}^{s}_{k}}{p^{2s}_{k}}.$ (26)

As Montgomery [8] showed, for any $\delta>0$ and sufficiently big
$x>x_{0}(\varepsilon)$, there is a prime in the interval
$[x,x+x^{0.6+\delta}]$. Then

$p^{s}-\overline{p}^{s}\leq p^{0.6s+\delta s}$

From (25), for sufficiently big $k\geq k_{0}(\delta)$, we have
$p^{s}_{k+1}\geq p_{k}^{(1.4-\delta)s}$. Then,

$p_{k+2}^{s}\geq p_{k+1}^{(1.4-\delta)s}\geq p_{k}^{(1.4-\delta)^{2}s^{2}},$
(27)

and we have

$p_{k+m}^{s}\geq p_{k}^{(1.4-\delta)^{m}s^{m}}.$

Therefore,

$p_{k-m}^{s}\leq p_{k}^{(1.4s-\delta s)^{-m}}.$

This yields

$n_{k}\leq n_{k_{0}}p_{k}^{(0.4s-\delta s)/(1.4s-\delta s)}.$ (28)

In addition, using (26), (27),

$|f_{s}(n_{k})-a|\leq C\frac{p_{k}^{0.6s+\delta
s}}{p^{2s}_{k}}=Cp_{k}^{-1.4s+\delta s}.$

Then we use (26) and (28) to obtain

$|f_{s}(n_{k})-a|\leq Cp_{k}^{-1.4s+\delta s}\leq
n_{k}^{-0.4s(1+\varepsilon)}\leq\frac{1}{n_{k}^{0.4s-\varepsilon}}.$

Since there are infinitely many $n_{k}$, inequality (22) has been proved.
Thus, we can state that Theorem 5 generalizes Wolke’s discovery. The shadowed
area below demonstrates the essence of Theorem 5 and contains infinitely many
solutions to inequality (22).

Figure 4: The shadowed area contains infinitely many solutions to inequality
(22). The narrower this area, the greater the density strength.

∎

##  4 Density of the complement within rationals

If $s$ is a natural number, then, clearly, the values $f_{s}(n)$ are rational,
$R(f_{s})\subset\mathbb{Q}$. However, there are rational numbers outside of
this range. In fact, if $n$ is composite, then no value $\displaystyle
1+\frac{1}{m}$ belongs to the range of $f_{1}$ because $\displaystyle
f_{1}(n)=1+\frac{1}{n}$ only if $m$ is prime. As proved earlier, the range
$R(f_{1})$ is dense. In this section, we address the question of whether its
complement within rationals is also dense. Since we are operating within the
set of rationals, let us consider only natural values of $s$.

######  Lemma 5.

Let $s\in\mathbb{N}$ and $n$ be composite. Then there is no $m$ such that
$\displaystyle f_{s}(m)=1+\frac{1}{n^{s}}$.

Proof. If $m$ were prime, then $\displaystyle
f_{s}(m)=1+\frac{1}{m^{s}}=1+\frac{1}{n^{s}}$. So, $m=n$ but $n$ is composite.
Thus, $m$ is also composite. Let $m=\prod p_{i}$. In this product, prime
numbers $p_{i}$ may repeat. We have

$f_{s}(m)=\frac{\sum\prod_{i}p_{i}^{s}}{m^{s}}$

where the products in the numerator $\prod_{i}p_{i}^{s}$ include all
combinations of $p_{i}$, the prime factors of $m$. The last expression is
equal to $\displaystyle 1+\frac{1}{n^{s}}$. Then

$n^{s}\sum\prod_{i}p_{i}^{s}=(n^{s}+1)m^{s}.$

Since $n^{s}$ and $n^{s}+1$ are relatively prime, then $n|m$. Therefore the
prime factorization of $n$ contains only the primes $p_{i}$ from the
factorization of $m$ and $f_{s}(n)<f_{s}(m)$. Therefore

$f_{s}(m)=1+\frac{1}{n^{s}}>f_{s}(n)>1+\frac{1}{n^{s}}.$

This contradiction proves Lemma 5. ∎

######  Corollary 2.

Let $n$ be a composite number, which is mutually prime with
$N=\prod\limits_{i=1}^{m}p_{i}^{k_{i}}$. Then number

$\left(1+\frac{1}{n^{s}}\right)f_{s}(N)$

does not belong to the range $R(f_{s})$.

######  Theorem 6.

Let $s\geq 0$. Then the rational complement $\mathbb{Q}\setminus R(f_{s})$ is
dense in $[1,\infty)$.

Proof. If $s$ is fractional, then the statement is trivial since $R(f_{s})$ is
irrational and, consequently, $\mathbb{Q}\setminus R(f_{s})=\mathbb{Q}$. For
$s=0$ the statement is also trivial since $f_{0}(n)$ is the number of divisors
and takes only natural values.

For $s\in\mathbb{N}$ and an arbitrary interval $(a,b)$ we pick up a point
$x_{0}\in R(f_{s})\cap(a,b)$. Then we choose a sufficiently big composite
number $n$, which is mutually prime with $x_{0}$, such that $\displaystyle
x=x_{0}\cdot\bigl{(}1+\frac{1}{n^{s}}\bigr{)}\in(a,b)$. In view of Corollary
2, $x\not\in R(f_{s})$ but $x\in(a,b)\cap\mathbb{Q}$. ∎

##  5 Probabilistic estimates: Expectation and Variance

Let us treat the divisor function $f_{s}(n)$ as a uniformly distributed random
variable defined on a sufficiently large interval $1\leq n\leq N$. Then, this
random variable takes values $f_{s}(n)$ with probability $\frac{1}{N}$. Since
the values of $n$ are large, then with probability $\frac{1}{2}$ number $n$ is
even, with probability $\frac{1}{3}$ number $n$ is divisible by 3, and so on.
Thus, with probability $\frac{1}{k}$ number $n$ has factors $k$ and
$\frac{n}{k}$. Then

$f_{s}(n)\approx\frac{1}{n^{s}}\left(\frac{1}{2}\frac{n^{s}}{2^{s}}+\frac{1}{3}\frac{n^{s}}{3^{s}}+\ldots+\frac{1}{k}\frac{n^{s}}{k^{s}}+\ldots\right).$

Then, as $n\to\infty$,

$f_{s}(n)\to\sum_{k=1}^{\infty}\frac{1}{k^{s+1}}=\zeta(s+1).$ (29)

Particularly, the standard divisor function $f_{1}$ has expectation
$\zeta(2)=\frac{\pi^{2}}{6}$. Thus, we arrived at the following result.

######  Theorem 7.

The expectation of the divisor function $f_{s}$ is equal to $\zeta(s+1)$.

As a remark, we can observe that as $s=0$, then the expectation becomes
infinite, which perfectly fits the sense of $f_{0}$ as the number of factors.

For the computational verification of Theorem 7, we use the law of large
numbers and calculate the average values of $f_{s}(n)$,

$\bar{f}_{s}=\frac{1}{N}\sum_{n=1}^{N}f_{s}(n)\to E(f_{s})\quad{as}\
N\to\infty.$ (30)

The computational results demonstrate the perfect fit for the presented
theory.

Figure 5: Expectation of the divisor function $f_{1}$ is equal to
$\displaystyle\frac{\pi^{2}}{6}$.

The dependence of the expectation on $s$ is given in the graph below.

Figure 6: Expectation of the divisor function $f_{s}$ is equal to
$\zeta(s+1)$.

To calculate the variance of random variable $f_{s}$ for large $n$, we
calculate the average of $f_{s}^{2}$ and subtract the squared expectation.
Then we obtain the decaying to zero function $Var(s)$, which is infinite at
$s=0$ and bounded for $s>0$. Particularly, we obtain $Var(1)=0.3$,
$Var(2)=0.03$, and $Var(3)=0.005$. Since $f_{0}(n)$ is just the number of
divisors of $n$, and $f_{0}$ can be drastically changed from 2, if $n=2^{k}-1$
is prime, to a huge value $k+1$ for $n=2^{k}$, then it is not surprising that
$Var(f_{s})\to\infty$ as $s\to 0$.

##  6 Conclusions

The paper investigates the dense properties of functions $\displaystyle
f_{s}(n)=\frac{\sigma_{s}(n)}{n^{s}}$ where $\sigma_{s}$ is the sum of
$s$-powers of all factors of $n$. Based on the convergence and divergence
properties of the infinite product (10), we prove that for every $s>0$,
$R(f_{s})$, the range of function $f_{s}$, is locally dense, i.e., any
neighborhood of any point from the range contains infinitely many other points
from $R(f_{s})$. Every number $n$ generates its train $T_{s}(n)$ – an ordered
collection of ordered sequences $\\{f_{s}(np)\\}$. The proof of local density
reveals the structure of the range. It turns out that the range is the union
of infinitely many trains. We prove Wolke’s conjecture for the points from the
range $R_{s}$ and show that inequality

$|f_{s}(n)-a|\leq\frac{1}{n^{1-\varepsilon}}$

has infinitely many solutions for any $a\in R_{s}$ and $\varepsilon>0$. This
finding narrows the zone with infinitely many points $f_{s}$, revealing a
stronger quantitative estimate of density strength.

Then, we address the density for all points in $[1,\infty)$, or the global
density. We prove that for $0<s\leq 1$, the range of functions $f_{s}$ is
dense and chaotic in $[1,\infty)$ whereas it loses its density properties and
experiences ruptures at $s>1$. At threshold $s=1$, $f_{s}$ loses its chaotic
properties. In order to measure density, we extend Wolke’s discovery from
$s=1$ to all $s\in(0,1]$ and prove the infinite cardinality of the solutions
to inequality

$|f_{s}(n)-a|\leq\frac{1}{n^{0.4s-\varepsilon}}.$

Our research demonstrates that Wolke’s theorem is a particular case of the
above inequality.

For integer $s\geq 0$, the range is within the set of rationals $\mathbb{Q}$,
and for those $s$, we prove that the rational complement to $R(f_{s})$, i.e.,
$\mathbb{Q}\setminus R(f_{s})$, is also dense in $[1,\infty)$. For $s=1$, this
observation leads to a new partition of the rationals into two dense subsets.

In the last section, we treat the divisor function as a uniformly distributed
random variable and prove that $F(f_{s})=\zeta(z+1)$. Particularly, the
expectation of the standard divisor function $f_{1}$ is
$\displaystyle\frac{\pi^{2}}{6}$. It is worth mentioning that
$E(f_{2})=1.20206$ (Apery’s constant) and $\displaystyle
E(f_{3})=\frac{\pi^{4}}{90}$. The variance decays when $s$ increases. The
decay of both the expectation and the variance remains continuous even when
the range of the divisor function $f_{s}$ switches from dense to ruptured at
$s=1$.

The analysis is supported by computational demonstrations, highlighting the
presented findings.

All of our results show that when considering the dense properties of the
divisor function, perfect numbers are not special in any way. While these
numbers remain a fascinating topic of study, our research concludes that in
this particular context, perfect and multiperfect numbers do not exhibit any
distinctive characteristics.

As a remaining open problem, we should point out the question of whether the
union of all ranges covers the entire $[1,\infty)$, i.e., whether
$\bigcup\limits_{s\geq 0}R(f_{s})=[1,\infty)$. Additionally, drawing on the
problem of the unknown infiniteness of perfect numbers, we can pose the
following question: whether there exists at least one point from the range
$R_{s}(f)$ that repeats infinitely many times.

Next steps could include the investigation of the density properties of a new
function $\displaystyle f_{g}(n)=\sum\limits_{d|n}\frac{g(d)}{g(n)}$ with
increasing function $g$ instead of the power function $\displaystyle
f_{s}(n)=\sum\limits_{d|n}\frac{d^{s}}{n^{s}}$. I expect that if $g(n)$
increases slower than the linear function, then the range is also dense.
However, the immediate application of the theory developed in this paper is
impossible due to the loss of the multiplicative property:
$f_{g}(np)\not=f_{g}(n)\cdot f_{g}(p)$ unless $g(n)=n^{s}$.

Acknowledgment

I would like to express my sincere gratitude to Professor L. Boyadzhiev and
Mrs. L. Asher for their motivation, support, and guidance.

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