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On the Zero-Free Region of Certain Families of Dedekind Zeta-Functions

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On the Zero-Free Region of Certain Families of Dedekind Zeta-Functions ON THE ZERO-FREE REGION OF CERTAIN FAMILIES OF DEDEKIND ZETA-FUNCTIONS by Nicholas Jian Hao Lai B.Math., University of Waterloo, 2016 AN ESSAY SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate and Postdoctoral Studies (Mathematics) The University of British Columbia (Vancouver) February 2019 c Nicholas Jian Hao Lai, 2019 Abstract In this thesis, we will study the analytical properties of subfamilies of Dedekind zeta-functions satisfying certain counting conditions. This was motivated by a result of Pierce, Turnage-Butterbaugh and Wood, giving a Chebotarev density result for subfamilies of Dedekind zeta-functions satisfying a zero-free region assumption. Their result is then applied to remove the requirement to assume the Generalised Riemann Hypothesis in a result of Ellenberg and Venkatesh towards the `-Torsion Conjecture. The Chebotarev density result of Pierce{Turnage-Butterbaugh{Wood is still conditional upon the validity of Strong Artin Conjecture. In particular, the reliance to the Strong Artin Conjecture is due to the application of an analytical method of Kowalski and Michel in their argument, which requires that the irreducible factors of the Dedekind zeta-functions be automorphic cuspidal. The goal of this thesis is to work towards the removal of the reliance to the Strong Artin Conjecture, by condensing down the conditional requirement to a requirement that poles (of pieces) of Dedekind zeta-functions satisfy some mild analytical condition in a family. This will pave a path towards an unconditional effective version of the result of Pierce{Turnage-Butterbaugh{Wood. We will establish directly that certain families of Artin L-functions are ameanable to the analytical method of Kowalski{Michel under the aforementioned mild analytical condition. Applying Kowalski{ Michel, we will prove the analog of an effective Chebotarev density result for certain subfamilies of Dedekind zeta-functions. With regards to the result of Pierce{Turnage-Butterbaugh{Wood, this will show that the Strong Artin Conjecture can be replaced with a mild analytical condition in the assumption of Pierce{Turnage-Butterbaugh{Wood. ii Table of Contents Abstract . ii Table of Contents . iii 1 Introduction . 1 1.1 Historical Background . 1 1.2 Summary of Results . 4 2 A Brief Overview of Artin L-functions . 7 2.1 L-Functions Before Artin . 7 2.1.1 Riemann Zeta-Function . 7 2.1.2 Dedekind Zeta-Functions . 8 2.1.3 Dirichlet Series and Dirichlet L-Functions . 9 2.1.4 Hecke L-Functions . 9 2.2 Artin L-Functions . 10 2.2.1 Decomposition Groups, Inertia Groups and Ramification Groups . 10 2.2.2 Unramified Factors of Artin L-Functions . 14 2.2.3 Ramified and Archimedean Factors of Artin L-Functions . 16 2.2.4 Convolution of Artin L-Functions . 19 2.3 Analyticity of Artin L-Functions . 23 2.3.1 Induction Theorems of Artin and Brauer . 23 2.3.2 Artin and Dedekind Conjecture . 25 2.3.3 Aramata{Brauer Theorem . 26 2.4 Artin L-Functions and Automorphic L-Functions . 28 2.4.1 Automorphic L-Functions . 29 2.4.2 Strong Artin Conjecture . 30 2.5 Convexity Bound on Artin L-Functions . 30 3 Kowalski{Michel Ameanable Families . 32 3.1 Families of Artin L-Functions . 32 3.1.1 Definition of Families of Artin L-Functions . 32 3.1.2 Kowalski{Michel Ameanable Families . 34 3.2 Components of Kowalski{Michel Estimate . 36 3.2.1 Dirichlet Series Expansion . 37 3.2.2 Pseudocharacters . 38 3.2.3 A Zero-Detector . 47 3.3 Zero-Region of Kowalski{Michel Q-Ameanable Families . 52 IV A New Family . 54 4.1 ν-Regular Representations . 54 4.1.1 Socle of a Group . 54 4.1.2 Faithful and Relatively Faithful Representations . 56 iii TABLE OF CONTENTS 4.1.3 ν-Regular and Relative ν-Regular Representations . 59 4.1.4 Variations on a Theorem of Aramata{Brauer . 61 4.2 Family of Artin L-Functions Associated to ν-Regular Representation . 63 4.2.1 A New Family and Its Kowalski{Michel Ameanability . 63 4.2.2 Zero-Free Region For Subfamilies of Dedekind Zeta-Functions . 65 4.2.3 Sterile Subfamilies of F 1 ............................... 69 Q;G 5 An Application: Effective Chebotarev Density Theorem . 72 5.1 Previously Known Chebotarev Density Results . 72 5.2 Effective Chebotarev Density Theorem of Pierce{Turnage-Butterbaugh{Wood . 73 5.2.1 Statement of Effective Chebotarev Density Theorem of Pierce{Turnage-Butterbaugh{ Wood .......................................... 74 5.2.2 Sketch of the Proof . 75 5.3 Removal of Strong Artin Conjecture . 80 6 Conclusion . 83 Bibliography . 84 iv Chapter 1 Introduction 1.1 Historical Background To understand where the branch of mathematics that is analytic number theory comes from, it is perhaps most informative to start from the beginning of the subject. Let us consider the positive integers N and its multiplicative structure. One of the first things one learns is the existence of prime numbers, and how they are the building block for N via the unique prime factorisation of integers. As such, to study N, the most basic and visible of mathematical objects, a study of prime numbers would yield many properties of N. Given their importance, the natural question one might ask is the following. Question. How many prime numbers are there in N? Of course, with a basic course in number theory, one would be able to reproduce the arguments originally utilised by Euclid to show that there are infinitely many prime numbers. Instead, we will like to direct the readers' attention to the proof of Euler regarding the infinitude of prime numbers. Central in Euler's argument is the divergence of the harmonic series 8 1 fpkq “ nk n“1 ¸ at k “ 1. In particular, Euler's proof exploits the analytical properties of the function f to deduce an arithmetic result. This is the essence of analytical number theory: using analytical properties to deduce arithmetic results. This line of investigation bore a number of arithmetical results under the study of Dirichlet, who was interested in primes in arithmetic progressions. Utilising character theory, Dirichlet rephrase the question of infinitude of primes in arithmetic progression to the question of divergence of Dirichlet series associated to Dirichlet characters, which will then yield results of infinitude of primes in arithmetic progression. This study of Dirichlet would then firmly establish analytical methods in multiplicative number theory. With the infinitude of prime numbers, one might then ask the following question. Question. What is the distribution of prime numbers in N? Let us rephrase the question: Let πpxq be the prime-counting function, i.e. πpxq “ |t primes p ¤ x u| How does πpxq grows as x grows? In a landmark paper [Rie59], motivated by his study for formulas for the prime-counting function, Riemann did groundbreaking work on the study of ζpsq, the Riemann zeta-functions. In his paper, Riemann expanded fpkq as a function on the complex plane C, which he denotes as ζpsq, and proceed 1 CHAPTER 1. INTRODUCTION to deduce many analytical properties of ζpsq. Of particular note is Riemann's sketch, later proved by von Mangoldt, of his explicit formula for the (normalised) prime-counting function involving non-trivial zeros of ζpsq. Riemann thus divised a general program for studying πpxq, by studying the non-trivial zeros of ζpsq. This is greatly expanded upon by Hadamard and de la Val´eePoussin, culminating in the celebrated Prime Number Theorem. Prime Number Theorem. Let πpxq be the number of primes less than or equal to x. Then x πpxq „ log x Returning to Riemann's paper [Rie59], Riemann proceeded to study the non-trivial zeros of ζpsq. Riemann calculated the first few zeroes of Zpsq, and discovered that they all lie on the critical line 1 Repsq “ 2 . From this, Riemann conjectured the most famous open conjecture in mathematics. 1 Riemann Hypothesis. All the non-trivial zeroes of ζpsq has real part 2 . The fame of Riemann Hypothesis is not unwarranted. Besides its notoriety as an extremely inadmissible open problem, its veracity also greatly improve the Prime Number Theorem, by explicitly ? quantifying the error term in the Prime Number Theorem to be of the magnitude of x log x. This cements the role of the study of non-trivial zeroes of ζpsq in analytical number theory as the premier direction towards the study of the distribution of prime numbers. Let us shift our focus. In light of the development of algebraic number theory, we can do an analogous study that we have been doing for N to a more general setting. Let K be a number field, and let OK be the integral closure of Z in K. Even though there is no notion of prime element in OK , we still have the notion of prime ideal in OK and that the unique prime factorisation of ideals holds in OK . In this general case, analogous to the Q case, we have now Dedekind zeta-functions ζK psq playing the role of ζpsq. Dedekind zeta-functions are very natural generalisation of ζpsq to general number fields, and they share similar properties. In fact, a version of Riemann Hypothesis for Dedekind zeta-functions is conjectured to hold. 1 Generalised Riemann Hypothesis. All the non-trivial zeroes of ζK psq has real part 2 . The analogous question of prime distribution in OK will be the following. Question. What is the distribution of prime ideals in OK ? We will rephrase the question similar to the case we have done for Z, but more generally, as follows: Let F be a number field, and consider a normal extension K{F with Galois group GalpK{F q.
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