Cover

Title page

Contents

Preface

1. Conference overview

2. M. Ram Murty’s mathematical legacy

3. Acknowledgements

On the Greatest Prime Factor of Some Divisibility Sequences

1. Introduction and results

2. Polynomial Case

3. Elliptic Curve Case

References

A Number Field Extension of a Question of Milnor

1. Introduction

2. The analogous case for the space generated by the values of 𝐿(1,𝜒)

3. Extended Milnor conjecture over number fields intersecting \Q(𝜁_{𝑞}) trivially

4. Extended Milnor conjecture over number fields intersecting \Q(𝜁_{𝑞}) non-trivially

5. Space generated by normalised Hurwitz zeta values

References

Mixing Rates of Random Walks with Little Backtracking

1. Introduction

2. Proofs of main results

3. Comparing the mixing rates of simple, non-backtracking and cliquewise non-backtracking random walks

4. Examples

Appendix

Acknowledgment

References

Additive and Multiplicative Functions with Similar Global Behavior

1. Introduction

2. Additive and multiplicative function functions with the same limit distribution

3. Additive and multiplicative functions with same mean and variance

4. Additive and multiplicative functions whose mean value behaves like loglog𝑛

5. Multiplicative functions with a large variance

6. Numerical computations

7. Final remarks

References

Multidimensional Sequences Uniformly Distributed Modulo 1 Created from Normal Numbers

1. Introduction

2. Main result

3. Final remarks

References

The Index of 𝑎 Modulo 𝑝

1. Introduction

2. Proof of Theorem 1.3

3. Proof of Theorem 1.5

References

Determining Optimal Test Functions for Bounding the Average Rank in Families of 𝐿-Functions

1. Introduction

2. Extension of the Conditions of [ILS]

3. Smoothness Almost Everywhere

4. A System of Integral Equations

5. Finding Coefficients

6. Conclusion and Future Work

References

Familles d’équations de Thue associées à un sous-groupe de rang 1 d’unités totalement réelles d’un corps de nombres

1. Le résultat principal

2. Un lemme élémentaire

3. Démonstration du théorème 1.2

4. Familles d’exemples

Références

Cyclicity of Quotients of Non-CM Elliptic Curves Modulo Primes

1. Introduction

2. Notations

3. Preliminary lemmas

4. Main tools for the proof of Theorem 1.1

5. Proof of Theorem 1.1

6. Elliptic Curves with CM

7. Final remarks

Acknowledgements

References

On the Euler Kronecker Constant of a Cyclotomic Field, II

1. Introduction

2. Preliminary Results

3. Proof of the Theorem

4. Conjectural Argument

References

The Generalized Dedekind Determinant

1. Introduction

2. A lemma from elementary linear algebra

3. The Dedekind determinant and proof of Theorem 1.1

4. A combinatorial analog of the Dedekind determinant and proof of Theorem 1.2

5. A modular analogue of the Dedekind determinant

6. The Redheffer matrix revisited

7. Link with chromatic polynomials

8. Concluding remarks

References

A Remark on Elliptic Curves with a Given Number of Points over Finite Fields

1. Introduction

2. Preliminaries

3. Proof of Theorem 1.5

4. Proof of Lemma 3.1

References

Recovering Cusp Forms on 𝐺𝐿(2) from Symmetric Cubes

Introduction

1. Preliminaries

2. Two Lemmas

3. Proof of Theorem A

4. Proof of Corollary B

References

Arithmetic Nature of Some Infinite Series and Integrals

1. Some results on the transcendence of infinite series

2. Integrals

Acknowledgement

References

Points on Varieties over Finite Fields in Small Boxes

1. Introduction

2. Examples of Varieties

3. Hyperbolas

4. Plane Curves

5. Weierstraß Equations

6. Erdős–Graham Equation

7. Variations and Generalisations

Acknowledgements

References

Bounds for the Lang-Trotter Conjectures

1. Introduction

2. Chebotarev bounds

3. Galois representations

4. Proof of Theorem 1.2

5. Proof of Theorem 1.3

6. Proof of Corollary 1.4

7. Proof of Theorem 2.2

References

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