DOCSLIB.ORG

L-Functions: Siegel-Type Theorems and Structure Theorems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
L-Functions: Siegel-Type Theorems and Structure Theorems Universita` degli Studi di Milano Dipartimento di Matematica Dottorato di Ricerca in Matematica IX Ciclo Tesi di Dottorato L-functions: Siegel-type theorems and structure theorems Giuseppe Molteni Relatore: Alberto Perelli (Dipartimento di Metodi e Modelli Matematici, Universit`adi Genova) Giuseppe Molteni Dipartimento di Matematica Universit`adegli Studi di Milano Via Saldini, 50 20133 Milano — Italy email: [email protected] Acknowledgments I wish to thank my thesis advisor, Prof. Alberto Perelli. During these years he has been an helpful teacher and a sure guide. I wish to thank Prof. C. Viola and R. Dvornicich of University of Pisa, Prof. U. Zannier of University of Venezia and Prof. J. Kaczorowski of University of Pozna`nfor many interesting and useful discussions. I wish to thank dr. A. Languasco of University of Padova and dr. A. Zaccagnini of University of Parma for their suggestions and friendship. Contents Notation iii Introduction 1 Chapter 1. Arithmetical relations coming from Euler products 3 1.1. Explicit computations 3 1.2. An Ω-result 5 1.3. An interesting identity 6 1.4. Estimates for products of coefficients of Dirichlet series with Euler product 9 1.5. The ”fudge factor” and Rankin-Selberg convolution 13 Chapter 2. The Siegel zero 21 2.1. Introduction 21 2.2. The axiomatic L∗-classes 26 2.3. Siegel-type theorems 44 Chapter 3. Existence of a singularity for certain functions of degree 1 55 3.1. Introduction 55 3.2. Definitions and results 55 3.3. Proof of lemma 58 3.4. Appendix 62 Chapter 4. About the Selberg class Sd, 0 ≤ d ≤ 1 65 4.1. Introduction and results 65 4.2. First proof 66 4.3. Second proof 67 Bibliography 69 i Notation s = σ + it with s ∈ C, σ = <s, t = =s. [x],{x} interger and fractional part of x. Q 1 φ(·) Euler totient function: φ(n) = n p|n(1 − p ). #A cardinality of the set A. f(x) = O(g(x)) for x → x0 means that there exists a positive constant c such that |f(x)| ≤ c|g(x)| in a neighborhood of x0; f(x) g(x) is an equivalent expression for this statement. Moreover, if f depends of a parameter ρ, f(x) = Oρ(g(x)) means that the O-estimate holds with c = c(ρ). f(x) = Ω(g(x)) for x → x0 means that there exists a positive constant c and a sequence xn → x0 such that |f(xn)| ≥ c|g(xn)|. f(x) g(x) for x → x0 means that there exist positive constants c1 and c2 such that c1|f(x)| ≤ |g(x)| ≤ c2|f(x)| in a neighborhood of x0. (Z/qZ)∗ group of invertible elements of Z/qZ. (a, b) for a, b ∈ N, the greatest common divisor of a and b. χ Dirichlet character. χ0 principal character: χ0(n) = 1 for every n ∈ N. τ(χ) Gauss sum for the character χ. χD Kronecker symbol; when D is a fundamental discriminant, the Kronecker sym- bol is a real primitive character modulo D. d degree of a polynomial Euler product. sn(α) the n-th elementary symmetric polynomial of the α1, . , αd variables: s (α) := P α ··· α . n 1≤j1<···<jn≤d j1 jn a+m−1 P Pm(a) the polynomial m−1 , equal to c1+···+cm=a 1 when a ∈ N. ci≥0 A, B complex square matrices. F(s, A, B) the ”fudge factor”. QA, WA main parameter and weight of L(s, A). dA, dA arithmetic and analytic degree of L(s, A). iii iv NOTATION PA,B exceptional set for the Rankin-Selberg convolution L(s, A ⊗ B). Λn(A) generalized von Mangoldt function. ¯ RA regular component of L(s, A ⊗ A). sym2 sym2 symmetric square map: GL(2) → GL(3), diag (α, β) −→ diag(α2, αβ, β2). Cd a class of L-functions with arithmetic degree d. S, Sd Selberg class and subset of elements with analytic degree d. ] ] S , Sd a class of L-functions similar to S and Sd respectively, without Euler product. R σ>a ds integral over a vertical line σ = σ0 with σ0 > a. a b Γ0(N) Hecke congruence group of level N: {( c d ) ∈ SL(2, Z), with c ≡ 0(N)}. Introduction The theory of L-functions provides powerful methods to study various arithmetical problems, and hence several such functions have been associated with arithmetical, alge- braic and geometric objects. The Hecke and Artin L-functions associated with algebraic number fields, the L-functions associated with automorphic representations of GL(r) and the L-functions of elliptic curves are some examples. It has been observed that most L-functions arising in number theory share some basic analytic properties, namely: ∗ when normalized, a representation as Dirichlet series ∞ X −s L(s) := ann n=1 exists for <s > 1, and a representation as Euler product Y L(s) = Lp(s) p holds, where p runs over the primes; ∗ L(s) has analytic continuation to C as a meromorphic function with a pole at s = 1 at most, and satisfies a functional equation of the following standard type Qsγ(s)L(s) = ωQ1−sγ¯(1 − s)L¯(1 − s), γ(s) being a product of Γ-factors, Q > 0 and |ω| = 1. The well known Selberg class S, see [56], and the L-class introduced by Carletti, Monti-Bragadin and Perelli [7] are attempts to axiomatically define a class including most L-functions of number-theoretical interest. In S and L the above properties of the L-functions are postulated, but some further conditions are imposed. For example, in S the Ramanujan hypothesis in the form an n is assumed, and in L ⊂ S, moreover, the Euler product has a polynomial form, i.e., −1 −s Lp(s) = Pp (p ) with Pp(x) ∈ C[x]. The Ramanujan hypothesis is conjectured for every L-function of number-theoretical interest, but it is not proved in general. Therefore, it is interesting to investigate an axiomatic class of L-functions where such hypothesis is replaced by weaker conditions. This will be done in Chapter 2, in the framework of certain L∗-classes. 1 2 INTRODUCTION The results that in L are proved independently of the Ramanujan hypothesis extend almost immediately to L∗. However, the usual approach to some important results, such as upper-bounds of type L(1) Q , depends heavily on the Ramanujan hypothesis: new techniques are necessary in the setting of L∗-classes. An essential ingredient of these techniques is the particular form of the multiplicative law satisfied by the coefficients an, depending on the polynomial form of the Euler product. In Chapter 1, a preliminary chapter for the whole thesis, this law is extensively investigated. A classical zero-free region holds for L∗-classes and, as in the original case of Dirichlet L-functions, the ”Siegel zero phenomenon” arises. Actually, it is conjectured that the Siegel zero β˜ does not exist, and this has recently been proved for cuspidal automor- phic forms on GL(2) and GL(3) by Goldfeld, Hoffstein and Lieman [20], Hoffstein and Ramakrishnan [25] and Banks [1]. Moreover, several results show that its existence is an extraordinary event. Therefore, it is an important and non-trivial problem to find a ”Siegel zero-free region”, as large as possible. For the classical Dirichlet L-functions, the best result in this direction is the famous Siegel theorem, stating that 1 − β˜ q− . In Chapter 2 we prove Siegel-type theorems for L∗-classes. These theorems ex- tend several known results, such as the classical Siegel theorem, that of Golubeva and Fomenko [21, 22] for holomorphic cusp forms and Hoffstein and Lockhart’s [24] result for the symmetric square of a Maass forms. Moreover, new results for the symmetric cube and other L-functions connected with holomorphic or Maass cusp forms are deduced from general principles. In Chapter 3, another class, denoted by C and inspired by the work of Duke and Iwaniec [14], is defined and investigated. In C, the exact form of the functional equation is not important and only some conditions about the growth on vertical lines are assumed. Moreover, the existence of the twisted functions ∞ X −s Lχ(s) := χ(n)ann χ primitive Dirichlet character n=1 is assumed. A polynomial Euler product is also assumed, so that Cd denotes the subclass of C of the elements with deg Pp(x) ≤ d, for every p. In C we prove an interesting phenomenon: the existence of an L-function L(s) of Cd, for some d ≥ 2, for which Lχ(s) is entire for every primitive character, is an obstruction to the existence of an analogous L-function of C1. In other words: every element of C1 has a twist with a pole at s = 1. This agrees with the conjecture that the Dirichlet L-functions and their shifts exhaust C1. Finally, it is well known that in the Selberg class an analytic notion of degree can be introduced; it is an important and unproved conjecture that the degree actually assumes only integer values. In accordance with this conjecture, it has been proved by several authors that the Selberg class Sd, the subset of functions with analytic degree d, is empty for 0 < d < 1. In Chapter 4 we give two new proofs of this result. CHAPTER 1 Arithmetical relations coming from Euler products The Dirichlet series arising from number theory usually have an Euler product of polynomial type, i.e., they satisfy an identity of the form ∞ d −1 X an Y Y αj(p) L(s) = = 1 − ns ps n=1 p j=1 in some right half-plane σ > σ0.
Load more

Recommended publications
  • Class Numbers of Quadratic Fields Ajit Bhand, M Ram Murty
    Class Numbers of Quadratic Fields Ajit Bhand, M Ram Murty To cite this version: Ajit Bhand, M Ram Murty. Class Numbers of Quadratic Fields. Hardy-Ramanujan Journal, Hardy- Ramanujan Society, 2020, Volume 42 - Special Commemorative volume in honour of Alan Baker, pp.17 - 25. hal-02554226 HAL Id: hal-02554226 https://hal.archives-ouvertes.fr/hal-02554226 Submitted on 1 May 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Hardy-Ramanujan Journal 42 (2019), 17-25 submitted 08/07/2019, accepted 07/10/2019, revised 15/10/2019 Class Numbers of Quadratic Fields Ajit Bhand and M. Ram Murty∗ Dedicated to the memory of Alan Baker Abstract. We present a survey of some recent results regarding the class numbers of quadratic fields Keywords. class numbers, Baker's theorem, Cohen-Lenstra heuristics. 2010 Mathematics Subject Classification. Primary 11R42, 11S40, Secondary 11R29. 1. Introduction The concept of class number first occurs in Gauss's Disquisitiones Arithmeticae written in 1801. In this work, we find the beginnings of modern number theory. Here, Gauss laid the foundations of the theory of binary quadratic forms which is closely related to the theory of quadratic fields.
    [Show full text]
  • Li's Criterion and Zero-Free Regions of L-Functions
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Journal of Number Theory 111 (2005) 1–32 www.elsevier.com/locate/jnt Li’s criterion and zero-free regions of L-functions Francis C.S. Brown A2X, 351 Cours de la Liberation, F-33405 Talence cedex, France Received 9 September 2002; revised 10 May 2004 Communicated by J.-B. Bost Abstract −s/ Let denote the Riemann zeta function, and let (s) = s(s− 1) 2(s/2)(s) denote the completed zeta function. A theorem of X.-J. Li states that the Riemann hypothesis is true if and only if certain inequalities Pn() in the first n coefficients of the Taylor expansion of at s = 1 are satisfied for all n ∈ N. We extend this result to a general class of functions which includes the completed Artin L-functions which satisfy Artin’s conjecture. Now let be any such function. For large N ∈ N, we show that the inequalities P1(),...,PN () imply the existence of a certain zero-free region for , and conversely, we prove that a zero-free region for implies a certain number of the Pn() hold. We show that the inequality P2() implies the existence of a small zero-free region near 1, and this gives a simple condition in (1), (1), and (1), for to have no Siegel zero. © 2004 Elsevier Inc. All rights reserved. 1. Statement of results 1.1. Introduction Let (s) = s(s − 1)−s/2(s/2)(s), where is the Riemann zeta function. Li [5] proved that the Riemann hypothesis is equivalent to the positivity of the real parts E-mail address: [email protected].
    [Show full text]
  • And Siegel's Theorem We Used Positivi
    Math 259: Introduction to Analytic Number Theory A nearly zero-free region for L(s; χ), and Siegel's theorem We used positivity of the logarithmic derivative of ζq to get a crude zero-free region for L(s; χ). Better zero-free regions can be obtained with some more effort by working with the L(s; χ) individually. The situation is most satisfactory for 2 complex χ, that is, for characters with χ = χ0. (Recall that real χ were also the characters that gave us the most difficulty6 in the proof of L(1; χ) = 0; it is again in the neighborhood of s = 1 that it is hard to find a good zero-free6 region for the L-function of a real character.) To obtain the zero-free region for ζ(s), we started with the expansion of the logarithmic derivative ζ 1 0 (s) = Λ(n)n s (σ > 1) ζ X − − n=1 and applied the inequality 1 0 (z + 2 +z ¯)2 = Re(z2 + 4z + 3) = 3 + 4 cos θ + cos 2θ (z = eiθ) ≤ 2 it iθ s to the phases z = n− = e of the terms n− . To apply the same inequality to L 1 0 (s; χ) = χ(n)Λ(n)n s; L X − − n=1 it it we must use z = χ(n)n− instead of n− , obtaining it 2 2it 0 Re 3 + 4χ(n)n− + χ (n)n− : ≤ σ Multiplying by Λ(n)n− and summing over n yields L0 L0 L0 0 3 (σ; χ ) + 4 Re (σ + it; χ) + Re (σ + 2it; χ2) : (1) ≤ − L 0 − L − L 2 Now we see why the case χ = χ0 will give us trouble near s = 1: for such χ the last term in (1) is within O(1) of Re( (ζ0/ζ)(σ + 2it)), so the pole of (ζ0/ζ)(s) at s = 1 will undo us for small t .
    [Show full text]
  • Zeta Functions of Finite Graphs and Coverings, III
    Graph Zeta Functions 1 INTRODUCTION Zeta Functions of Finite Graphs and Coverings, III A. A. Terras Math. Dept., U.C.S.D., La Jolla, CA 92093-0112 E-mail: [email protected] and H. M. Stark Math. Dept., U.C.S.D., La Jolla, CA 92093-0112 Version: March 13, 2006 A graph theoretical analog of Brauer-Siegel theory for zeta functions of number …elds is developed using the theory of Artin L-functions for Galois coverings of graphs from parts I and II. In the process, we discuss possible versions of the Riemann hypothesis for the Ihara zeta function of an irregular graph. 1. INTRODUCTION In our previous two papers [12], [13] we developed the theory of zeta and L-functions of graphs and covering graphs. Here zeta and L-functions are reciprocals of polynomials which means these functions have poles not zeros. Just as number theorists are interested in the locations of the zeros of number theoretic zeta and L-functions, thanks to applications to the distribution of primes, we are interested in knowing the locations of the poles of graph-theoretic zeta and L-functions. We study an analog of Brauer-Siegel theory for the zeta functions of number …elds (see Stark [11] or Lang [6]). As explained below, this is a necessary step in the discussion of the distribution of primes. We will always assume that our graphs X are …nite, connected, rank 1 with no danglers (i.e., degree 1 vertices). Let us recall some of the de…nitions basic to Stark and Terras [12], [13].
    [Show full text]
  • SPEECH: SIEGEL ZERO out Line: 1. Introducing the Problem of Existence of Infinite Primes in Arithmetic Progressions (Aps), 2. Di
    SPEECH: SIEGEL ZERO MEHDI HASSANI Abstract. We talk about the effect of the positions of the zeros of Dirichlet L-function in the distribution of prime numbers in arithmetic progressions. We point to effect of possible existence of Siegel zero on above distribution, distribution of twin primes, entropy of black holes, and finally in the size of least prime in arithmetic progressions. Out line: 1. Introducing the problem of existence of infinite primes in Arithmetic Progressions (APs), 2. Dirichlet characters and Dirichlet L-functions, 3. Brief review of Dirichlet's proof, and pointing importance of L(1; χ) 6= 0 in proof, 4. About the zeros of Dirichlet L-functions: Zero Free Regions (ZFR), Siegel zero and Siegel's theorem, 5. Connection between zeros of L-functions and distribution of primes in APs: Explicit formula, 6. Prime Number Theorem (PNT) for APs, 7. More about Siegel zero and the problem of least prime in APs. Extended abstract: After introducing the problem of existence of infinite primes in the arithmetic progression a; a + q; a + 2q; ··· with (a; q) = 1, we give some fast information about Dirichlet characters and Dirichlet L-functions: mainly we write orthogonality relations for Dirichlet characters and Euler product for L-fucntions, and after taking logarithm we arrive at 1 X X (1) χ(a) log L(s; χ) = m−1p−ms '(q) χ p;m pm≡a [q] where the left sum is over all '(q) characters molulo q, and in the right sum m rins over all positive integers and p runs over all primes.
    [Show full text]
  • Standard Zero-Free Regions for Rankin–Selberg L-Functions
    STANDARD ZERO-FREE REGIONS FOR RANKIN–SELBERG L-FUNCTIONS VIA SIEVE THEORY PETER HUMPHRIES, WITH AN APPENDIX BY FARRELL BRUMLEY Abstract. We give a simple proof of a standard zero-free region in the t- aspect for the Rankin–Selberg L-function L(s,π × πe) for any unitary cuspidal automorphic representation π of GLn(AF ) that is tempered at every nonar- chimedean place outside a set of Dirichlet density zero. 1. Introduction Let F be a number field, let n be a positive integer, and let π be a unitary cuspidal automorphic representation of GLn(AF ) with L-function L(s, π), with π normalised such that its central character is trivial on the diagonally embedded copy of the positive reals. The proof of the prime number theorem due to de la Valle´e-Poussin gives a zero-free region for the Riemann zeta function ζ(s) of the form c σ> 1 − log(|t| + 3) for s = σ + it, and this generalises to a zero-free region for L(s, π) of the form c (1.1) σ ≥ 1 − (n[F : Q])4 log(q(π)(|t| + 3)) for some absolute constant c> 0, where q(π) is the analytic conductor of π in the sense of [IK04, Equation (5.7], with the possible exception of a simple real-zero βπ < 1 when π is self-dual. A proof of this is given in [IK04, Theorem 5.10]; the method requires constructing an auxiliary L-function having a zero of higher order than the order of the pole at s = 1, then using an effective version of Landau’s lemma [IK04, Lemma 5.9].
    [Show full text]
  • THE TRUTH ABOUT TORSION in the CM CASE 1. Introduction the Aim of This Note Is to Prove the Following Result. Theorem 1. There I
    THE TRUTH ABOUT TORSION IN THE CM CASE PETE L. CLARK AND PAUL POLLACK Abstract. We show that the upper order of the size of the torsion subgroup of a CM elliptic curve over a degree d number field is d log log d. 1. Introduction The aim of this note is to prove the following result. Theorem 1. There is an absolute, effective constant C such that for all number fields F of degree d ≥ 3 and all elliptic curves E=F with complex multiplication, #E(F )[tors] ≤ Cd log log d: It is natural to compare this result with the following one. Theorem 2 (Hindry{Silverman [HS99]). For all number fields F of degree d ≥ 2 and all elliptic curves E=F with j-invariant j(E) 2 OF , we have #E(F )[tors] ≤ 1977408d log d: Every CM elliptic curve E=F has j(E) 2 OF , but only finitely many j 2 OF are j-invariants of CM elliptic curves E=F . Thus Theorem 1 has a significantly stronger hypothesis and a slightly stronger conclusion than Theorem 2. But the improvement of log log d over log d is interesting in view of the following result. Theorem 3 (Breuer [Br10]). Let E=F be an elliptic curve over a number field. There exists a constant c(E; F ) > 0, integers 3 ≤ d1 < d2 < : : : < dn < : : : and + number fields Fn ⊃ F with [Fn : F ] = dn such that for all n 2 Z we have ¨ c(E; F )dn log log dn if E has CM; #E(Fn)[tors] ≥ p c(E; F ) dn log log dn otherwise: Thus if TCM(d) is the largest size of a torsion subgroup of a CM elliptic curve defined over a number field of degree d, its upper order is d log log d: T (d) lim sup CM 2 ]0; 1[: d d log log d To our knowledge this is the first instance of an upper order result for torsion points on a class of abelian varieties over number fields of varying degree.
    [Show full text]
  • Landau-Siegel Zeros and Their Illusory Consequences Kyle Pratt University of Illinois
    University of Mississippi eGrove NSF-CBMS Conference: L-functions and 2019 NSF-CBMS Conference: L-functions and Multiplicative Number Theory Multiplicative Number Theory May 20th, 2:30 PM - 3:30 PM Landau-Siegel zeros and their illusory consequences Kyle Pratt University of Illinois Follow this and additional works at: https://egrove.olemiss.edu/cbms2019 Part of the Number Theory Commons Recommended Citation Pratt, Kyle, "Landau-Siegel zeros and their illusory consequences" (2019). NSF-CBMS Conference: L-functions and Multiplicative Number Theory. 18. https://egrove.olemiss.edu/cbms2019/2019/schedule/18 This Presentation is brought to you for free and open access by the Mathematics, Department of at eGrove. It has been accepted for inclusion in NSF- CBMS Conference: L-functions and Multiplicative Number Theory by an authorized administrator of eGrove. For more information, please contact [email protected]. Landau-Siegel zeros and their illusory consequences Kyle Pratt May 20, 2019 Prelude: effective and ineffective constants I Since the terms effective and ineffective will arise often, let us briefly define them I A constant, implied or otherwise, is effective if, given enough time and patience, one could go through a proofp and write down an actual value for the constant like π 17=1234 or 10−1010 etc I A constant is ineffective if it is not possible (!) to make the constant effective. That is, the very mechanism of the proof does not allow one to compute the constant. Ineffective constants usually arise from invoking the law of the excluded middle Landau-Siegel zeros and class numbers I Dirichlet (1837) introduced his eponymous characters χ (mod D) to prove there are infinitely many primes p ≡ a (mod D), (a; D) = 1.
    [Show full text]
  • Arxiv:2001.02405V1 [Math.NT]
    ′ ON ( L (1, χ)) AND ZERO-FREE REGIONS NEAR s =1 ℜ L CHRISTIAN TAFULA´ Abstract. Let q 2 be an integer, χ (mod q) a primitive Dirichlet character, ≥ and f : Z≥2 R a function satisfying 2 f(q) log(q). We show that, if L(s,χ) has no zeros→ in the region ≤ ≪ 1 1 σ + it C σ> 1 , t < 1 , ∈ − f(q) | | − ( f(q) log(q) ) ′ L p then ( L (1,χ)) f(q) log(q) uniformly for primitive χ (mod q). As an exam- ple ofℜ an application,≪ we show that the uniform abc-conjecture implies a strong p version of “no Siegel zeros” for odd real characters of qo(1)-smooth moduli, by using our result in T. [9] together with a theorem of Chang [1] on zero-free regions. 1. Introduction Write s = σ + it C (with σ = (s), t = (s)), and consider the critical strip s C 0 <σ< ∈1 . A family ofℜ regions ℑ = (q) s C 0 <σ< 1 , { ∈ | } Q Q ⊆ { ∈ | } for q Z 2, shall be called quasi zero-free (in the q-aspect) for primitive Dirichlet ∈ ≥ s characters χ (mod q) if the Dirichlet L-function L(s, χ) = n 1 χ(n)n− has no zeros in (q) when χ is complex (i.e., not real), and has at most≥ one real simple Q1 P zero β > 2 in (q) when χ is real (the so-called Siegel zero). Note that this excludes principal characters,Q for the only primitive principal character is the trivial character modulo 1.
    [Show full text]
  • Arxiv:2010.01308V2 [Math.NT] 3 Nov 2020 L Eee O Nipoe Presentation
    CONDITIONAL BOUNDS ON SIEGEL ZEROS GAUTAMI BHOWMIK AND KARIN HALUPCZOK Dedicated to Melvyn Nathanson. Abstract. We present an overview of bounds on zeros of L- functions and obtain some improvements under weak conjectures related to the Goldbach problem. 1. Introduction The existence of non-trivial real zeros of a Dirichlet L-function would contradict the Generalised Riemann Hypothesis. One possible counter- example, called the Landau–Siegel zero, is real and simple and the re- gion in which it could eventually exist is important to determine. In 1936 Siegel gave a quantitative estimate on the distance of an excep- tional zero from the line s = 1. The splitting into cases depending on whether such an exceptionalℜ zero exists or not happens to be an im- portant technique often used in analytic number theory, for example in the theorem of Linnik. In the first section we discuss properties of the Siegel zero and results assuming classical and more recent hypotheses. This part of the paper is expository. In the second section we present a conditional bound. In 2016 Fei improved Siegel’s bound for certain moduli under a weakened Hardy– Littlewood conjecture on the Goldbach problem of representing an even number as the sum of two primes. In Theorem 11 and Corollary 1 we further weaken this conjecture and enlarge the set of moduli to include arXiv:2010.01308v2 [math.NT] 3 Nov 2020 more Dirichlet characters. 2. Background Consider a completely multiplicative, periodic arithmetic function χ : Z C where for q 1 there exists a group homomorphism → ≥ χ˜ : (Z/qZ)× C× such that χ(n)=χ ˜(n(mod q)) for n coprime to q and χ(n) = 0→ if not.
    [Show full text]
  • Arxiv:2105.14653V1 [Math.NT]
    SIEGEL ZEROS AND SARNAK’S CONJECTURE JAKE CHINIS Abstract. Assuming the existence of Siegel zeros, we prove that there exists an increasing sequence of positive integers for which Chowla’s Conjecture on k-point correlations of the Liouville function holds. This extends work of Germ´an and K´atai, where they studied the case k = 2 under identical hypotheses. An immediate corollary, which follows from a well-known argument due to Sarnak, is that Sarnak’s Conjecture on M¨obius disjointness holds. More precisely, assuming the existence of Siegel zeros, there exists a subsequence of the natural numbers for which the Liouville function is asymptotically orthogonal to any sequence of topological entropy zero. 1. Introduction The Liouville function, λ, is the completely multiplicative function defined by λ(p) := −1 for all primes p. As such, λ(n) = 1 if n has an even number of prime factors, counted with multiplicity, and λ(n) = −1 otherwise. An important problem in analytic number theory is to understand the asymptotic behaviour of the partial sums n≤x λ(n). Viewing {λ(n)}n as a sequence of independent random variables, each taking the value ±1 withP equal probability, we might expect that the partial sums of λ exhibit some cancellation. Indeed, the Prime Number Theorem (PNT) is equivalent to the fact λ(n)= o(x), nX≤x as x → ∞. The Riemann Hypothesis (RH) is equivalent to the fact that the partial sums of λ exhibit “square-root cancellation:” for any ǫ> 0, λ(n) ≪ x1/2+ǫ, nX≤x as x → ∞. A famous conjecture due to Chowla [Cho65] states similar estimates hold for multiple correlations of λ: Conjecture 1.1 (Chowla’s Conjecture).
    [Show full text]
  • The Work of Granville–Stark on ABC and Siegel Zeros
    The work of Granville{Stark on ABC and Siegel zeros Christian T´afula D´epartement de math´ematiques et de statistique (DMS), Universit´ede Montr´eal(UdeM) Math 726: Topics in Number Theory Modular forms and the theory of complex multiplication T´afula,C. (DMS, UdeM) Granville{Stark October 2020 0 / 22 Contents 1 Introduction 2 Lower bounds for h(D) 3 The Siegel zero T´afula,C. (DMS, UdeM) Granville{Stark October 2020 1 / 22 Contents 1 Introduction 2 Lower bounds for h(D) 3 The Siegel zero T´afula,C. (DMS, UdeM) 1. Intro October 2020 2 / 22 Notation p p D < 0 fundamental discriminant (so D2 = R, D = disc Q( D)); p C`(D) = class group of Q( D); p h(D) = jC`(D)j class number of Q( D); j = j-invariant function: 3 1 + 240 P P d3qn n≥1 djn 2πiτ j(τ) := Q n 24 =(τ) > 0; q = e q n≥1(1 − q ) j(i) = 1728; q-expansion of j: j(τ + 1) = j(τ); 2πi=3 j(e ) = 0; 1 j(−1/τ) = j(τ); j(τ) = + 744 + 196884q + ··· \j(i1) = 11"; q T´afula,C. (DMS, UdeM) 1. Intro October 2020 3 / 22 Singular moduli (1/2) Let τ 2 h (=(τ) > 0) A; B; C 2 ; A > 0; CM-point: τ j Aτ 2 + Bτ + C = 0 Z gcd(A; B; C) = 1; unique Singular modulus: j(τ)(j = j-invariant, τ a CM-point) CM-points Quadratic forms F ! τ 2 h Ax2 + Bxy + Cy2 2πi πi e 3 e 3 disc(τ) := B2 − 4AC • ◦ τ ∼ τ 0 () (A; B; C) ∼ (A0;B0;C0) τ 2 F () (A; B; C) reduced -1 -0.5 0 0.5 1 T´afula,C.
    [Show full text]