**CBMS Regional Conference Series in Mathematics**

Volume: 129;
2019;
95 pp;
Softcover

MSC: Primary 11; 05;

**Print ISBN: 978-1-4704-4900-1
Product Code: CBMS/129**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

**Electronic ISBN: 978-1-4704-5192-9
Product Code: CBMS/129.E**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

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#### Supplemental Materials

# Zeta and \(L\)-functions in Number Theory and Combinatorics

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*Wen-Ching Winnie Li*

A co-publication of the AMS and CBMS

Zeta and \(L\)-functions play a central
role in number theory. They provide important information of
arithmetic nature. This book, which grew out of the author's teaching
over several years, explores the interaction between number theory and
combinatorics using zeta and \(L\)-functions as a central theme. It
provides a systematic and comprehensive account of these functions in
a combinatorial setting and establishes, among other things, the
combinatorial counterparts of celebrated results in number theory,
such as the prime number theorem and the Chebotarev density theorem.

The spectral theory for finite graphs and higher dimensional
complexes is studied. Of special interest in theory and applications
are the spectrally extremal objects, called Ramanujan graphs and
Ramanujan complexes, which can be characterized by their associated
zeta functions satisfying the Riemann Hypothesis. Explicit
constructions of these extremal combinatorial objects, using
number-theoretic and combinatorial means, are presented.

Research on zeta and \(L\)-functions for complexes other
than graphs emerged only in recent years. This is the first book for
graduate students and researchers offering deep insight into this
fascinating and fast developing area.

#### Readership

Graduate students and researchers interested in Zeta and \(L\)-functions.

#### Table of Contents

# Table of Contents

## Zeta and $L$-functions in Number Theory and Combinatorics

- Cover Cover11
- Title page i2
- Preface v6
- Lecture 1. Number theoretic zeta and 𝐿-functions 110
- Lecture 2. The Selberg zeta function 1524
- Lecture 3. 𝐿-functions in geometry 2736
- Lecture 4. The Ihara zeta function 3544
- Lecture 5. Spectral graph theory 4352
- Lecture 6. Explicit constructions of Ramanujan graphs 4756
- 6.1. Expanding constant and spectral gap 4756
- 6.2. Some nice features of Ramanujan graphs 4857
- 6.3. Cayley graphs 4958
- 6.4. 𝑃𝐺𝐿₂(ℚ_{𝕡})/ℙ𝔾𝕃₂(ℤ_{𝕡}) as a (𝕡+1)-regular tree 5059
- 6.5. Ramanujan graphs as finite quotients of 𝑃𝐺𝐿₂(ℚ_{𝕡}) 5160
- 6.6. Hashimoto’s class number formula 5362
- 6.7. Biregular bipartite Ramanujan graphs 5665

- Lecture 7. Artin 𝐿-functions and prime distributions for graphs 6170
- Lecture 8. Zeta and 𝐿-functions of complexes 6978
- 8.1. The building attached to 𝑃𝐺𝐿_{𝑛}(𝐹) 6978
- 8.2. Spectral theory of regular complexes from ℬ_{𝓃}(ℱ) 7079
- 8.3. Ramanujan complexes as finite quotients of ℬ_{𝓃}(ℱ) 7079
- 8.4. Zeta and 𝐿-functions of finite quotients of ℬ₃(ℱ) 7483
- 8.5. Zeta functions of finite quotients of the building Δ(𝐹) of 𝑆𝑝₄(𝐹) 8089
- 8.6. Distribution of primes in finite quotients of ℬ₃(ℱ) and Δ(ℱ) 8392

- Bibliography 8796
- Index 93102
- Back Cover Back Cover1106