The distribution of values of logarithmic derivatives of real L-functions. Mariam Mourtada

Transcription

1 The distribution of values of logarithmic derivatives of real L-functions by Mariam Mourtada A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto Copyright c 203 by Mariam Mourtada

2 Abstract The distribution of values of logarithmic derivatives of real L-functions Mariam Mourtada Doctor of Philosophy Graduate Department of Mathematics University of Toronto 203 We prove in this thesis three main results, involving the distribution of values of L /L(σ, χ D ), D being a fundamental discriminant, and χ D the real character attached to it. We prove two Omega theorems for L /L(, χ D ), compute the moments of L /L(, χ D ), and construct, under GRH, for each σ > /2, a density function Q σ such that #{D fundamental discriminants, such that D Y, and α L /L(σ, χ D ) β} 6 π 2 2π Y β α Q σ (x)dx. ii

3 Dedication To the Man who stood by my side during the hard times To my little angels for the love they brought into our home To my Mom and Dad for their endless sacrifice To my parents in law for their love and support I wouldn t imagine myself graduating today without my husband Hassan s encouragement and patience during the 7 years we have been together, through all the storms, through all the love, our bond is always growing stronger. No matter what happens, I know that he is there, ready to wait, love and care. I wouldn t imagine myself graduating today without my little ones Hadi and Nour smiling at me and giving me full strength to continue forward. I wouldn t imagine myself graduating today without my parents education since my toddler years and the indescribable love of learning they implemented in my heart. Dad, I will always remember you sitting with me, and helping me read carefully and understand more and more stories. Mom I will never forget your unbelievably big heart, and selflessness. I would like to thank my parents in law Khalil and Zeinab for their kindness and prayers, for their support and big heart. iii

4 Acknowledgements I would like, first of all, to thank my great supervisor, Prof. V.Kumar Murty, for everything. Basically, this thesis is due to his great supervision, in terms of pushing me to always try, without mentioning his valuable knowledge and ability to manage all of Ganita members work. You will never see Prof. Murty, the Chair, stressed, or upset. His smile never leaves his face, and he looks calm all the time. I wonder how this could be with all the responsibilities he is tackling. But I have no wonder how this was pushing me to work more and more, regarding him all the way as an example. I will forever cherish every single moment I spent at the Math Department at the University of Toronto, with the great staff in charge, and my colleagues at Ganita. I would love to express all my respects and thanks to Ida who is the living heart of our department. Everyone, without exception, needs her, and she is always ready to help, with such a great kindness! I also would like to thank Pamela Britain for her continuous help, Sarah, Jemaima, Patrina, Diana and Donna for being so nice! Finally, I would like to express my sincere thanks to Prof. John Friedlander as well as my PhD supervisory committee members, Prof. Steven Kudla, and Prof. James Arthur, and my external examiner Prof. Kanaan Soundararajan for their valuable comments and suggestions. iv

5 Contents Introduction. History L-functions Work on L /L Statement of results Construction of Q σ Summary of Chapters Operations on Dirichlet series 7 2. Power Series Operations on Power series Tests of convergence Dirichlet series Operations on Dirichlet series Infinite Product and Euler Product Infinite Product Tests of convergence for infinite products Euler Product Known results on the distribution of L /L 4 3. History v

6 3.2 L-functions and L /L Zero-free region for L(s, χ) Recent progress Connection to the distribution of L /L(, χ) Ihara s study of the distribution of values of L /L Omega Theorems for L /L(, χ D ) Introduction Unconditional bounds Choice of Parameters Application of the Explicit Formula The sum over zeros Proof of the Theorem Conditional bounds Proof of Theorem Moments of L /L(, χ D ) Exceptional Zeros The main term The sum over zeros On Distribution functions Introduction Distribution Functions Fourier transforms Properties of Fourier Transforms Some properties of M r Convergence of infinite convolutions vi

7 6 Distribution of values of L /L(σ, χ D ) Introduction Admissible functions Statement of results Preliminary results Proof of Theorem Toward a Density Function Construction of M σ (x) Construction of Q σ (x) Fourier transform of distributions and connection with Fourier transform of functions Average of L /L(σ, χ D ), σ > / The main Lemma The main term Estimation of the sum over zeros under GRH Estimation of the sum over zeros, unconditionally vii

8 Notation O we say f(x) = O(g(x)) if f(x) cg(x), as x for some constant c, where g(x) 0. f(x) g(x) if f(x) = O(g(x)). o ζ L f(x) = o(g(x) means f(x) 0 as x. g(x) denotes the Riemann zeta function ζ(s) = n= denotes L-functions defined by L(s, χ) = n= with Rs >, and χ a complex or real character.. n s χ(n), s here is a complex number n s K denotes mainly an algebraic number field of finite degree over the field of rational numbers Q. ζ K is the Dedekind zeta function ζ K (s) = P ( N(P s )), P runs over prime ideals of K, and Rs >. ρ is used for non-trivial zeros of ζ, L-functions, or ζ K. ( ṁ ) denotes the Jacobi symbol. γ is Euler s constant, except when stated otherwise(denotes sometimes the imaginary part of the zeros ρ). γ K GRH Λ p, q φ D is the Euler-Kronecker constant. stands for the Generalized Riemann Hypothesis. is the von Mangoldt function. will always denote prime numbers, except when stated otherwise. is Euler s totient function. will always represent a fundamental discriminant. or represent a restricted sum over a set of integers. N(σ, T, χ) is the number of zeroes ρ = β + it of L(s, χ) such that β σ and t T. log k stands for log(log k ), where log := log and k 2. Ω, Omega we say that f(x) is Omega of g(x) and we write f(x) = Ω(g(x)) if f(x) g(x) for arbitrarily large values of x. viii

9 δ is the usual Dirac-delta function. N(Y ) this is equal to #{Fundamental discriminants D; D Y } 6/π 2 Y. π(x) this represents #{Prime p; p x} x/ log x by the Prime Number Theorem. N(χ, σ 0, T ) this represents #{ρ = σ + it; L(ρ, χ) = 0, σ σ 0 and t T }. f Y (n) 0<βD Y is equal to D Y χ D (n) = 6 π 2 χ D (n) = 3 π 2 + O(Y /2 ), for β = ±. + O(Y /2 ) and sometimes this represents L p (R d ) represents the set of all functions such that R d f(x) p dx <. L (R d ) denotes the set of functions that are bounded almost everywhere (i.e bounded up to a set of measure zero). ix

10 Chapter Introduction In this chapter we present the contents of this thesis.. History.. L-functions A classical result of Chowla [3] states that for infinitely many fundamental discriminants D we have L(, χ D ) ( + o())e γ log log D where χ D is the quadratic Dirichlet character of conductor D. Before Chowla, Littlewood [23], showed that under GRH ( ) ζ(2) 2 + o() e γ log 2 D L(, χ D) (2 + o())e γ log 2 D and that for infinitely many fundamental discriminants D, we have L(, χ D ) ( + o())e γ log 2 D and for infinitely many fundamental discriminants D, we have L(, χ D ) ( + o())ζ(2). e γ log 2 D

11 Chapter. Introduction 2 After Chowla, several authors refined these results. In particular, Granville and Soundararajan [6] studied the distribution of values of L(, χ D ), and provided a stronger version of Chowla s Omega Theorem. Conditionally, they proved that ([6], Theorem 5a) assuming GRH there are infinitely many primes q such that L(, and infinitely many primes q L(, ( ). ) e γ (log q 2 q + log 3 q log(2 log 2) ɛ) ( ). ) ζ(2) q e (log γ 2 q + log 3 q log(2 log 2) ɛ). Unconditionally, they proved ([6], Theorem 5b) that ( ). L(, ) e γ (log d 2 x + log 3 x log 4 x 0) holds for x 0 fundamental discriminants d x. Here log x = log x, and log k+ x = log(log k x). Recently, R. Vaughan [33] and H. Montgomery and Vaughan [24], have pursued a probabilistic model, which led them to formulate the following conjectures. (i)if we let P (x) denote the proportion of fundamental discriminants D x such that L(, χ D ) e γ log 2 D, then exp( C log x/ log 2 x) < P (x) < exp( c log x/ log 2 x) for appropriate constants 0 < c < C <. A similar estimate should hold for the proportion of D such that L(, χ D ) ζ(2)/(e γ log 2 D ). (ii)if we let P 2 (x) denote the proportion of fundamental discriminants D x s.t L(, χ D ) e γ (log 2 D + log 3 D ), then x θ < P 2 (x) < x Θ for some 0 < θ < Θ <. A similar estimate should hold for the proportion of D such that L(, χ D ) ζ(2)/(e γ (log 2 D + log 3 D )).

12 Chapter. Introduction 3 (iii)for any ɛ > 0, there are only finitely many D such that L(, χ D ) e γ (log 2 D + ( + ɛ) log 3 D ) or such that L(, χ D ) ζ(2) e γ (log 2 D + ( + ɛ) log 3 D ). The first of these conjectures was established by Granville and Soundararajan, and the second almost follows from GRH. A more general work on the distribution of extreme values of L-functions at the edge of the critical strip was done recently by Y. Lamzouri in [2], [22]...2 Work on L /L Ihara [] has initiated the study of the distribution of values of the logarithmic derivatives of L-functions L /L, and introduced what is called the Euler-Kronecker constant γ K of a global field K. This constant is related to L /L, and in the case of a quadratic field Q( D), Ihara-Murty-Shimura [7] proved that under GRH L L (, χ D) 2 log log D + O(). They also computed the moments of L /L(, χ), where χ runs over all non-principal multiplicative characters χ mod m. They proved lim m X m (L (, χ)/l(, χ)) a L (, χ)/l(, χ) b = C a,b χ where X m denotes the set of all non principal multiplicative characters χ mod m, and the sum is over such set. Here C a,b is a well defined constant..2 Statement of results In this thesis we prove three main results. The first result is on Omega theorems for L /L(, χ D ), where D is a fundamental discriminant, and χ D is the real character attached

13 Chapter. Introduction 4 to D. These theorems describe large values of L /L, and their influence on the behaviour of the Euler-Kronecker constant. We prove unconditionally that L L (, χ D) log log D + O() holds for infinitely many fundamental discriminants D. Furthermore, we prove that for infinitely many fundamental discriminants D, we have L L (, χ D) log log D + O(). We also prove under GRH, that for infinitely many primes q mod 4, we have L L (, χ q) log log q + log log log q + O() and for infinitely many primes q mod 4, we have L L (, χ q) log log q log log log q + O(). In the second result, we prove that the moments of L /L are constant. In other words, for each non-negative integer k, there is a constant C k so that 0<βD Y ) k ( L L (, χ D) C k Y. Here Y >, β = ±, and the asterisk on the sum indicates that we are summing over fundamental discriminants D. The Omega results, stated precisely in Chapter 4, justify the following conjecture. Conjecture. For β = ± L max x βd 2x L (, χ D) = log 2 x + log 3 x + O() and min x βd 2x L L (, χ D) = log 2 x log 3 x + O().

14 Chapter. Introduction 5 The max and min are being taken over fundamental discriminants D. The third result which is on the distribution of values of L /L(σ, χ D ) shows that, assuming GRH, for each σ > 2, there is a density function Q σ such that #{D fundamental discriminants, such that D Y, and α L /L(σ, χ D ) β} 6 π 2 2π Y β α Q σ (x)dx..2. Construction of Q σ Write exp( iz 2 Here ζ(s) is the Riemann zeta function. ζ ζ (s)) = n= λ z (n) n s We note that this is well defined, since in Chapter 2 of the thesis, we prove that if f(s) is a Dirichlet series, then so is exp(f(s)). Let us set Then, we prove that lim Y M σ (a) := N(Y ) D Y n= λ 2a (n 2 ) p n ( + p ) n 2σ. ) exp (ia L L (σ, χ D) = M σ (a). Here N(Y ) := #{D fundamental discriminants, such that D Y }. We also prove that M σ (a) = O σ (exp( C σ a /σ )) for some positive constant C σ. We then define its Fourier inverse to be the density function we are looking for Q σ (a) = M σ (a). We further compute the average over fundamental discriminants D, of L /L(σ, χ D ), under GRH, and we show that it is a constant dependent only on σ. We also compute this average unconditionally, for certain values of σ.

15 Chapter. Introduction 6.3 Summary of Chapters Chapter 2 presents some basic but useful results on Dirichlet series that we will need later. We cover the differences as well as similarities, and provide some lemmas on possible operations on them. Chapter 3 covers known results on the distribution of L /L(s, χ), s being a complex number, and χ a complex character. It is noteworthy that most of the work has been initiated mainly by Ihara in [], Ihara-Matsumoto [6], and Ihara-Murty-Shimura [7]. We also cover new results from different authors. Chapter 4 is based on a recent paper by M. Mourtada and V. Kumar Murty [27], it covers all the Omega results presented above. Chapter 5 summarizes most results from a work of B.Jessen and A.Wintner [8], on distribution functions that will be needed in Chapter 6. This last chapter gives the actual distribution results on L /L(σ, χ D ).

16 Chapter 2 Operations on Dirichlet series In this chapter, we present some useful facts about Dirichlet series. We present a comparison between power series and Dirichlet series, and study some operations on Dirichlet series. This will be in particular helpful to understand the type of obstacles we are facing when we follow methods used to study the distribution of L-functions and try to apply these to the case of L /L. We finish the chapter by reviewing some properties of infinite products and Euler products. 2. Power Series A complex power series is a series of the form f(z) = n a n z n. Here the coefficients a n are complex, and the variable z is a complex variable as well. When the series converges for all z < R and diverges for all z > R, we call R the radius of convergence of the series. 7

17 Chapter 2. Operations on Dirichlet series Operations on Power series We recall some elementary properties of power series: For two power series f(z) = n a nz n and g(z) = n b nz n and c a complex number,. f(z) + g(z) = n (a n + b n )z n is again a power series. 2. f(z)g(z) is a power series of the form n ( n k= a kb n k )z n. 3. cf(z) is a power series of the form n (ca n)z n. 4. If a 0 0 then f(z) has an inverse power series. 5. If R is the radius of convergence of f(z) then, for all z < R, f(z) is differentiable and its derivative is again a power series. Similarly, f(z) is integrable and its antiderivative is also a power series that is convergent for all z < R Tests of convergence There are many ways to test the convergence of a series. However we will present here tests that might be useful to proofs related to this thesis. Theorem. (Dirichlet s test) If a n is a sequence of real numbers and b n a sequence of complex numbers satisfying. a n a n+ 2. lim a n = 0 as n 3. N n= b n M for every integer N > 0 then the series n a nb n is convergent. Here M is some constant.

18 Chapter 2. Operations on Dirichlet series Dirichlet series A Dirichlet series is a series of the form f(s) = n a n n s ; s C. Dirichlet series enjoy most properties listed above for power series, except that the domain of convergence is a half plane rather than a disc. Within wedge shaped regions in the half plane of convergence, the convergence is uniform, and so can differentiate and integrate term by term, in particular, f (s) = n a n log n n s. Proposition. ([3], p.29) The region of absolute convergence of a Dirichlet series is a half-plane Operations on Dirichlet series Lemma. Let f(s) = a n n= n s series. Proof. Formally, we have exp(f(s)) = k= k! ( be a Dirichlet series, then exp(f(s)) is a formal Dirichlet n= a n n s )k = n= n s k= k! ( a n... a nk ). n=n...n k We have to show now that for each n, T n := k= ( k! n=n...n k a n... a nk ) converges. Let M n be the number of ways one can write n = r c r, where the c r are non trivial divisors, then for all k d(n) =: d n, the number of ways one can write n = n... n k is k! M n (k d n)!. Indeed, for each such writing of n as a product of non-trivial divisors, one can deduce a writing of n as a product n... n k, for k large enough, where the empty places are filled up with. This shows that our series T n M n N n k=d n+ (a ) k (k d n )!

19 Chapter 2. Operations on Dirichlet series 0 Here N n := max{ r a c r ; n = r c r the c r being non trivial divisors}. Clearly such series converges. Hence the lemma is proved. Lemma 2. Let f(s) = a n n= n s the following are equivalent: be a Dirichlet series convergent in some half plane. Then. f(s) has a formal Dirichlet inverse. 2. a log(f(s)) has a formal Dirichlet series. Proof. ( 2) Let g(s) = b n n= n s coefficients one can show that a b =, then a 0. be such that f(s)g(s) =. Then by comparison of (2 ) Suppose a 0, then one can construct the inverse as follows: b = a, b n = a d n a n d<n d b d. Thus g(s) = (f(s)) exists as a Dirichlet series, at least formally. ( 3) Suppose f(s) has a Dirichlet inverse g(s) = n= b n, then f /f(s) has a Dirichlet n s series, this shows that d(log(f(s)))/ds has a Dirichlet series f /f(s) = c n n= n s implies that log(f(s)) = c n log n n + constant s n 2, this where the constant can be considered as a term of the series, so we conclude that log(f(s)) itself has a Dirichlet series expansion. (3 ) Suppose now that log(f(s)) has a Dirichlet series, then log(/f(s)) = log(f(s)) has a Dirichlet series, this implies according to Lemma, that exp(log(/f(s))) has a Dirichlet series, in other terms /f(s) has a Dirichlet series. This finishes the proof of the Lemma. Remark. We know from the above lemmas, that ζ (s) = n= log n n s does not have a Dirichlet inverse because its constant term is 0, thus log( ζ (s)) does not have a Dirichlet

20 Chapter 2. Operations on Dirichlet series series expansion and neither does log( ζ (s)/ζ(s)). This will prevent us from regarding the complex moments of ζ (s)/ζ(s) as Dirichlet series, since these are defined as being ( ( )) ζ (s) exp z log, z C. ζ(s) 2.3 Infinite Product and Euler Product 2.3. Infinite Product An infinite product is an expression of the form [3] P = ( + a n ) n= where the a n are supposed to be different from -. This infinite product is said to be convergent if the partial product n P n := ( + a m ) m= has a limit as n. Otherwise the product is said to be divergent. Definition. The product ( + a n ) is said to be absolutely convergent if the product ( + an ) is convergent. Proposition 2. If a n 0 for all values of n, the product ( + a n ) and the series a n converge or diverge together. Corollary. ( + a n ) converges absolutely if and only if a n is convergent (in other terms a n is absolutely convergent). Definition 2. The infinite product P = ( + u n (z)) n=

21 Chapter 2. Operations on Dirichlet series 2 where the factors are functions of the complex variable z, is uniformly convergent if the partial product n P n (z) := ( + u m (z)) m= is convergent uniformly, in a certain region of values of z, to a limit. Proposition 3. The product ( + u n (z)) is uniformly convergent in any region where the series u n (z) is uniformly convergent Tests of convergence for infinite products We state in this subsection some tests for convergence of infinite products, that are relevant to the present thesis. Proposition 4. ([30], p.4) If a n <, then the infinite product ( + a n ) n= converges. Moreover, the product converges to 0 if and only if one of its factors is Euler Product An Euler product is an expansion of a Dirichlet series into an infinite product over the prime numbers. Leonhard Euler was the first to write such a product for the Riemann Zeta function. Definition 3. An arithmetic function a : N C is said to be multiplicative, if for every two coprime integers m and n, we have a(mn) = a(m)a(n). It is called totally multiplicative if this is true for any integers m and n.

22 Chapter 2. Operations on Dirichlet series 3 Definition 4. Let a be an arithmetic multiplicative function, then the Dirichlet series has the following product expansion where for each prime p n= a(n) n s P (p, s) p P (p, s) = k=0 a(p k ) p ks. Such an expansion is called the Euler product of the Dirichlet series a(n)/n s. Remark 2. If a is totally multiplicative, then P (p, s) = a(p)p s this is true because in such case the sum of P (p, s) is a geometric series. For example, when a(n) = for all n, then we get the Euler product of the Riemann zeta function, and when a(n) = χ(n) for some multiplicative character χ then this is the product of the corresponding L-function L(s, χ).

23 Chapter 3 Known results on the distribution of L /L In this chapter we cover in detail what has been done on the distribution of values of L /L and the techniques used. We also present some conjectures and the progress that has been made on them so far. 3. History Historically, L- functions have been studied in detail in many aspects, ranging from their functional equation, to zero-free regions, to special values, and so on. The study of the distribution of values of L-functions has gained a comparable attention as well, from the work of Selberg, to Chowla s great Omega Theorem, to Montgomery and Vaughan, and recently to the work of Granville-Soundararajan, as well as many other authors. Recently, Y.Ihara [, 0], has initiated a study of the distribution of values of logarithmic derivatives of L-functions, L /L. Those differ from L-functions in being meromorphic functions rather than holomorphic functions. The logarithmic derivatives of L-functions, L /L, are important in many applications, and are expected to be related to periods of Abelian varieties. 4

24 Chapter 3. Known results on the distribution of L /L L-functions and L /L Dirichlet L-functions are given by L(s, χ) = n= χ(n), Rs >. ns Here χ is a multiplicative character. The functional equation of L-functions has been initially given by Hurwitz in 882 [4], and next by de La vallée Poussin in 896. The latter s method was a generalization of what Riemann did in his memoir on the zeta function. The equation can be presented as follows. If we let ξ(s, χ) = ( π q ) 2 (s+a) Γ( s + a )L(s, χ) 2 where a = 0 (resp. a = ) if χ is even (resp. odd) then the mentioned functional equation is ξ( s, χ) = ia q 2 ξ(s, χ). τ(χ) where for any character χ to the modulus q, the Gaussian sum τ(χ) is defined by q τ(χ) = χ(m)e q (m) m= where e q (m) := exp( 2πim q ). This functional equation is valid for all complex numbers s. From the functional equation, we can easily see that L-functions have zeros coming from the poles of the Gamma function Γ( s+a ). These zeros are called the trivial zeros. All 2 other zeros lie in the critical strip defined by 0 < Rs <. From the above functional equation, and given the fact that the Γ function has poles at 0, -, -2, -3,... we conclude that for a = 0, L-functions have the following trivial zeros s = 0, 2, 4, 6,... and for a =, the trivial zeros of L-functions are s =, 3, 5,...

25 Chapter 3. Known results on the distribution of L /L 6 All other zeros lie in the critical strip, and these are called non-trivial zeros. In addition to their functional equation, L-functions enjoy the property of having an Euler product that is valid for all complex s, with Rs L(s, χ) = p ( χ(p) p s ). The logarithmic derivatives of L-functions do not have, however, a functional equation, nor an Euler product, and given the fact that they have poles at the zeros of L(s, χ), this makes it more difficult to work with them. Despite these facts, one can work on L /L(s, χ) either outside the critical strip, for Rs > to get unconditional results, or inside the critical strip, for 0 Rs, with the assumption of the Generalized Riemann Hypothesis GRH. From this we can see that expanding the zero-free region for L-functions, would help us getting unconditional results on the distribution of values of L /L(s, χ), within that zero-free region. By differentiating the logarithm of the Euler product of L(s, χ), we get for Rs >. Here L (s, χ) = Λ(n)χ(n)n s L n= log p if n = p α, p prime Λ(n) = 0 elsewhere. Taking logarithm of both sides of the functional equation for L-functions, we get log ξ( s, χ) = log( ia q 2 ) + log ξ(s, χ). τ(χ) Differentiating this equality with respect to s, we get ξ /ξ( s, χ) = ξ /ξ(s, χ). On the other hand, we quote the following equation from [4], p. 83 that is useful in many occurrences. L L (s, χ) = 2 log q π Γ 2 Γ (s + a 2 ) + B(χ) + ρ ( s ρ + ) ρ

26 Chapter 3. Known results on the distribution of L /L 7 where a depends on the character χ, and has been defined above. The sum over ρ represents the sum over all non-trivial zeros of L(s, χ), in other terms zeros within the critical strip. As for the number B(χ), it is given by and since we get B(χ) = ξ ξ (0, χ) = ξ (, χ) ξ B(χ) = B(χ) RB(χ) = R ρ. 3.3 Zero-free region for L(s, χ) First of all, it is noteworthy that L(, χ) 0 thus is natural to think of a zero-free region of L(s, χ), around s =. The focus will be on the region within the critical strip, since for Rs >, we already know that there are only the trivial zeros of L(s, χ). Many theorems have been proved in this direction, like the following theorem, which is due partly to Gronwall and partly to Titchmarch [4]. Theorem 2. ([4], p.93) There exists a positive absolute constant c with the following property. If χ is a complex character modulo q, then L(s, χ), s = σ + it, has no zero in the region defined by σ c if t, log q t c if t. log q If χ is a real non principal character, the only possible zero of L(s, χ) in this region is a single (simple) real zero.

27 Chapter 3. Known results on the distribution of L /L 8 We present now a theorem of Landau, which ensures that if there exist values of q for which L(s, χ), where χ is a real primitive character mod q, has a real zero β with β > c, then such values of q are rare. log q Theorem 3. (Landau, [4]) If χ and χ 2 are two distinct real primitive characters to the moduli q, q 2 respectively, and if the corresponding L-functions have real zeros β, β 2 respectively, then min(β, β 2 ) < C log(q q 2 ) where C is some positive absolute constant. The possibility that q = q 2 is not excluded. Another important theorem, which is due to Page states the following Theorem 4. (Page Theorem, [4], p.95) If c is a suitable positive constant, then there is at most one real primitive character χ to a modulus q z for which L(s, χ) has a real zero β satisfying β > c log z. We state now two forms of Siegel s Theorem. Theorem 5. (Siegel, [4], p. 26) For any ɛ > 0, there exists a positive number C (ɛ) such that, if χ is a real primitive character to the modulus q, then L(, χ) > C (ɛ)q ɛ. Theorem 6. (Siegel, [4], p. 26) For any ɛ > 0, there exists a positive number C 2 (ɛ) such that, if χ is any real non principal character, with modulus q, then L(s, χ) 0 for s > C 2 (ɛ)q ɛ. 3.4 Recent progress In a paper [] dedicated to V.Drinfeld, Ihara introduced his definition of what is called the Euler-Kronecker constant γ K of a global field K which generalizes Euler s constant γ

28 Chapter 3. Known results on the distribution of L /L 9 in many ways. Here γ = γ Q = lim ( + n log n) = n Ihara considered this constant more as an invariant of the field K and defined it as follows. For a number field K, define the Dedekind zeta function ζ K (s), Rs >, by the Dirichlet series ζ K (s) = I (NI) s where the sum ranges over integral ideals of K, and NI denotes the norm of the integral ideal I. It has an analytic continuation for all s with simple pole at s =. Write the expansion of the Dedekind zeta function near s = as ζ K (s) = c (s ) + c 0 + O(s ). Then set γ K = c 0 c. In the same paper [], Ihara proved the following explicit formula for γ K γ K = 2 ρ ρ(ρ ) 2 log d + r 2 (γ + log 4π) + r 2(γ + log 2π) where d = d K is the discriminant of K, and r, r 2 are respectively the number of real, (resp. imaginary) places of K(in other terms, the number of real, resp. complex embeddings). The sum ρ being taken over non-trivial zeros of ζ K(s). Ihara proved the following upper bound for γ K. Let α K = 2 log d K. Theorem 7. (Theorem, []) Under GRH, we have γ K < ( ) αk + (2 log α K + Φ K (α 2 α K K))

29 Chapter 3. Known results on the distribution of L /L 20 provided that the degree or ( ) αk + (2 log α K + ) α K [K : Q] > 2 [K : Q] = 2 and d K > 8. Here Φ K is the prime counting function given by Φ K (x) = x N(P ) k x ( ) x log N(P ) N(P ) k for x >, and P runs over the non-archimedean primes of K. He then deduced that under GRH γ K 2 log log d. He further provided the following unconditional lower bound. If we let then β K = { r 2 (γ + log 4π) + r 2(γ + log 2π)} Proposition 5. (Proposition 3, []) In the number field case, γ K > α K β K. This can be read as γ K > log d Ihara further proved that the above lower bound can be improved under some conditions. Theorem 8. (Theorem 3, []) Let K be an extension of Q of degree N >. Put α K = α K /(N )

30 Chapter 3. Known results on the distribution of L /L 2 = log d N and assume α K >. Then under GRH α K + α K (γ K + ) > 2(N )(log α K + ). From this theorem, Ihara deduced that when α K and N is fixed or grows slowly enough ( ) log d γ K > 2(N ) log. N In the same paper [], Ihara s main tool was the explicit formula for the prime counting function Φ K (x), for which he proved that when x is large enough, we have lim (log x Φ K(x)) = γ K +. x An important special case of Ihara s general work, was when K is either the cyclotomic field Q(µ m ) or its maximal real subfield Q(µ m ) +. He conjectured that γ Q(µm) > 0. This has been made more precise in conclusions Ihara presented in his paper [0], drawn from numerical computations. Let K m := Q(µ m ) and K + m := Q(µ m ) +. Conjecture.([0],Conjecture ) (i)γ Km and γ K + m are positive. (ii)there exist positive constants c, c 2, c +, c + 2, all 2, such that for any ɛ > 0, (c ɛ) log m < γ Km + < (c 2 + ɛ) log m, and (c + ɛ) log m < γ K + m + < (c ɛ) log m,

31 Chapter 3. Known results on the distribution of L /L 22 hold for all sufficiently large m. (iii) When m is restricted to primes, one can choose c = /2, c + = and c 2 = c + 2 = 3/2. Furthermore, Ihara [0] proved that assuming GRH for K m, and for each m consider the weighted average of cos(γ log m), c(m) := ρ m ρ /2 ρ(ρ ) / ρ ρ(ρ ) = ρ cos(γ log m) /4 + γ 2 where ρ = /2 + iγ. Now, set γ K := γ K +. Proposition 6. ([0], Proposition 3) (Under GRH) Let M be any given infinite set of prime numbers, and let m run over M. Then, γ K m / log m (resp. γ K + m/ log m ) is bounded if and only if mc(m) (resp. mc + (m) ) is so, and when these conditions are satisfied, we have γk m / log m = ( mc(m) ) + O( ) (m M) log m γ / log m = 3 K m ( mc + (m) ) + O( ) (m M) log m respectively. In particular, the above Conjecture (iii) is equivalent to that mc(m) (resp. mc + (m) ) belongs to the interval ( ɛ, ɛ) when m is sufficiently large. Tsfasman [32] proved, assuming GRH, that lim inf γ K log d K where the lim inf is taken over number fields K with d K.

32 Chapter 3. Known results on the distribution of L /L Connection to the distribution of L /L(, χ) It is noteworthy that an equivalent definition of γ K is ( ζ γ K = lim K (s) + ). s ζ K s In their paper [7], Ihara, Murty and Shimura showed that and γ Q(µm) = γ + χ χ 0 L /L(, χ) γ Q(µm) + = γ + χ χ 0 χ( )= L /L(, χ) where χ runs over all non-principal multiplicative characters χ : (Z/m) C and χ 0 being the trivial character. They also proved under GRH and unconditionally, for any ɛ > 0 γ Q(µm), γ Q(µm) + = O((log m)2 ) γ Q(µm), γ Q(µm) + = O(mɛ ). More strongly, Badzyan [2] showed that under GRH, we have γ Q(µm) = O(log m log log m). Recently, Murty [26] has proved unconditionally that Ihara s conjecture is true on average. Theorem 9. ([26], Theorem.) We have γ q π (Q) log Q 2 Q<q Q where π (Q) denotes the number of primes in the interval ( Q, Q] and the sum is over 2 primes q in this interval. Fouvry [5] has refined this to an asymptotic formula.

33 Chapter 3. Known results on the distribution of L /L Ihara s study of the distribution of values of L /L In his paper [0], Ihara constructed and studied a function M σ (z) on C parametrized by σ > /2, together with its Fourier transform M σ (z). This function is closely related to the density measure for the distribution of values on C of the logarithmic derivatives of L-functions L(s, χ), where s is fixed, Rs = σ, and χ runs over an infinite family of Dirichlet characters. Let P be any non-empty finite set of non-archimedean prime divisors of K, T P := P P C and let g σ,p : T P C be defined by g σ,p (t P ) = P P g σ,p (t P ) where g σ,p (t P ) = t P log N(P) t P N(P) σ and t P = (t P ) P P. Ihara [0] proved the following Theorem 0. ([0], Theorem ) Let σ > 0. There exists a unique function M σ,p (z) of z C, which is a hyperfunction (Schwartz distribution) when P =, that satisfies M σ,p (ω)φ(ω) dω = Φ(g σ,p (t P ))d t P C T P for any continuous function Φ(ω) on C, where dω = (2π) dxdy (ω = x + yi), and d t P is the normalized Haar measure on T P. It is compactly supported, and satisfies M σ,p 0, M σ,p (ω) dω = C Let P y := {P P ; N(P) y} and D (a,b) := a+b z a z b. Theorem. ([0], Theorem 2) Let σ > /2, P = P y, and let y. Then. M σ,p (z) converges uniformly to a non-negative real valued C -function M σ (z). 2. Each D (a,b) M σ,p (z) converges uniformly to D (a,b) M σ (z) (starting with P sufficiently large).

34 Chapter 3. Known results on the distribution of L /L For any n, z n M σ (z) belongs to L The function M σ (z) is not identically zero; in fact M σ (z) dz =. It satisfies M σ (z) = M σ (z) = M σ (z). C

35 Chapter 4 Omega Theorems for L /L(, χ D ) In this chapter we present our results on Omega theorems for L /L. This is based on our paper [25]. 4. Introduction A classical result of Chowla [3] states that for infinitely many fundamental discriminants D we have L(, χ D ) ( + o())e γ log log D where χ D is the quadratic Dirichlet character of conductor D. In other words, L(, χ D ) = Ω(log log D ). Before Chowla, Littlewood [23] proved, assuming GRH, that L(, χ) (2 + o())e γ log log m holds for all sufficiently large conductors m, χ being a character of conductor m. After Chowla, several authors refined this result. In particular, Granville and Soundararajan [6] studied the distribution of values of L(, χ D ), and provided a stronger version of Chowla s Omega Theorem. Conditionally, they proved the following 26

36 Chapter 4. Omega Theorems for L /L(, χ D ) 27 Theorem 2. ([6], Theorem 5a)Assume GRH. For any ɛ > 0, and all large x, there are x 2 primes q x such that L(, ( ). ) e γ (log q 2 q + log 3 q log(2 log 2) ɛ) and x 2 primes q such that L(, Unconditionally, they proved ( ). ) ζ(2) q e (log γ 2 q + log 3 q log(2 log 2) ɛ). Theorem 3. ([6], Theorem 5b) For large x there are at least x 0 square-free integers d x such that ( ). L(, ) e γ (log d 2 x + log 3 x log 4 x 0). Granville and Soundararajan [6] further proved the following important theorem about the distribution of L(, χ D ). Let us set Φ x (τ) := ( d x L(,χ d )>e γ τ ) ( / d x ) and Ψ x (τ) := ( d x L(,χ d ) π2 6e γ τ )( d x ). Where stands for the sum over fundamental discriminants. Theorem 4. ([6], Theorem 4) Let x be large and let log 2 x A e be a real number. Uniformly in the range τ R (x) log 2 (x) we have ( Φ x (τ) = exp eτ C ( + O( τ A + ) τ )) and the same asymptotic holds for Ψ x (τ). Here we may take R (x) = log 2 x + log 4 x 20 unconditionally, and R (x) = log 2 x + log 3 x 20 if the GRH is true.

37 Chapter 4. Omega Theorems for L /L(, χ D ) 28 In this chapter, we prove analogous results for the logarithmic derivatives L L (, χ D), and investigate the growth of the logarithmic derivatives of real Dirichlet L-functions. This work was motivated by the lack of literature on the subject, despite the fact that a lot of work has been done on the study of the value distribution theory of L-functions (for example the work of Selberg and others). Ihara began a systematic study of the value distribution of the logarithmic derivatives of Dirichlet L-functions [] and has obtained many interesting results. In particular, he introduced the Euler-Kronecker constant γ K of a number field K. It generalizes the Euler constant γ and if K = Q( D), then Thus, our result restated gives γ K = γ + L L (, χ D). γ K = Ω(log log D ). When K = Q, this is just Euler s constant, and when K = Q(ζ m ) a cyclotomic field, recent result has been obtained by Murty [26], concerning the average of γ K. Our result is in fact consistent with a previous conditional result by Ihara, Murty, and Shimura in [7]. They proved the following theorem assuming the Generalized Riemann Hypothesis. Theorem 5. ([7], Theorem 3) Let χ be a non-principal primitive Dirichlet character of a number field K with conductor f χ. Then L (, χ)/l(, χ) < 2(log log ( ) log d χ + ) γk dk + log log d χ + O. log d χ Here, d χ = d K N(f χ ) and γ K = γ K +, γ K being the Euler-Kronecker invariant of K. The above theorem implies that under GRH L L (, χ D) 2 log log D + O(). Set log x = log x, and log k+ x = log(log k x). Then we prove unconditionally that

38 Chapter 4. Omega Theorems for L /L(, χ D ) 29 Theorem 6. There are infinitely many fundamental discriminants D (both positive and negative) such that L L (, χ D) log 2 D + O(). Furthermore, for x large enough, there are x 0 fundamental discriminants 0 < D x such that L L (, χ D) log 2 D + O(). Remark 3. In fact, the bound obtained in the proof of the second part of Theorem 6 is log 2 x + O() which is stronger than what is stated. The key point in proving Theorem 6 is the use of the explicit formula from [7], p.26: L ( ) y ρ log y (, χ) = Φ(y) L y ρ( ρ) + O y ρ where y >, the sum is taken over all non-trivial zeros ρ of L(s, χ), and (4..) Φ(y) = ( y y m )Λ(m)χ(m). m<y As a motivation for where this formula comes from, we recall that L L (s, χ) = m= Λ(m)χ(m) m s, Rs >. One would expect that m<y Λ(m)χ(m)/m s is an approximation for L /L(s, χ). We may smoothen it by integrating When s = this is equal to Φ(y). y y m<t Λ(m)χ(m) dt. m s The first part of Theorem 6 is proved by averaging over a certain special set of integers. The second part of Theorem 6 together with the following stronger conditional result are motivated by a result of Granville-Soundararajan on L(, χ D ) [6].

39 Chapter 4. Omega Theorems for L /L(, χ D ) 30 Theorem 7. Assume GRH. For x large enough, there are x 2 primes q x such that and x 2 primes q x such that L L (, χ q) log 2 x + log 3 x + O() L L (, χ q) log 2 x log 3 x + O(). In [6], Granville and Soundararajan further provided a probabilistic model for the value distribution of L(, χ D ) giving uniform results. We did investigate their method, and it does not seem so far that one can construct a similar model for the value distribution of L /L(, χ D ) following the same procedure. The main trouble comes from the fact that we cannot compute the complex powers of L /L(, χ D ), without lots of loss of generality, since ( L L (, χ D)) z := exp(z log( L L (, χ D))). However for this to be defined, we need to investigate when is L L (, χ D) 0. and this is not as direct as the fact that for all characters χ one has L(, χ) 0. In contrast with the omega results of Theorem 6 and Theorem 7, we will show unconditionally in Theorem 8 that the moments of L /L(, χ D ) are constants. Theorem 8. For Y >, k and β = ±, we have where 0<βD Y ) k ( L L (, χ D) = C k Y + O k (Y 5 C k = 3 π 2 n= Λ k (n 2 ) n 2 ( + p ). p n 6 +ɛ )

40 Chapter 4. Omega Theorems for L /L(, χ D ) 3 Here the asterisk on the sum indicates that we are summing over fundamental discriminants 0 < βd Y, and the functions Λ k will be defined in Section 4.4 of this chapter. The moments of L /L(, χ), where χ is a non-principal multiplicative character to a prime modulus m have been computed unconditionally by Ihara, Murty and Shimura in [7], where they showed that Theorem 9. ([7], Theorem 5) We have, unconditionally, X m χ X m P (a,b) (L (, χ)/l(, χ)) = ( ) a+b µ (a,b) + O(m ɛ ) for any ɛ > 0. In particular the limit formula lim m X m χ X m P (a,b) (L (, χ)/l(, χ)) = ( ) a+b µ (a,b) holds unconditionally. The same remain valid if X m is replaced by X ± m. Here X m denotes the set of all non-principal multiplicative characters χ mod m, X + m (resp. X m) the subset of X m consisting of even (resp. odd) characters, P (a,b) (z) := z a z b and µ (a,b) = n= Λ a (n)λ b (n) n 2. Notice that as we are working here with real Dirichlet characters, the moments are reduced to powers of L /L(, χ D ). We note that our method follows closely the argument in [7] pages Putting our results together, it seems evident enough that one might expect the following. Conjecture. For β = ±, and L max x βd 2x L (, χ D) = log 2 x + log 3 x + O() L min x βd 2x L (, χ D) = log 2 x log 3 x + O() where the min and max are taken over fundamental discriminants D of the appropriate sign.

41 Chapter 4. Omega Theorems for L /L(, χ D ) Unconditional bounds In this section we prove Theorem Choice of Parameters Let x be a sufficiently large positive integer, k a positive integer, η a suitably small positive real number, and let [ g = η log x ]. log log x Denote by p i the i th odd prime with p = 3. Set a = p p 2...p g. Then by the Prime Number Theorem, for any ɛ > 0, we have: a = e θ(pg) e (+ɛ)g log g ɛ x η, where θ(y) := log p. Choose a positive integer b so that p y ( ) b = for i g and b mod 8. p i Notice that the existence of such b is guaranteed by the Chinese Remainder Theorem. we may assume b < 8a. The Jacobi symbol ( n ) is defined to be zero if m is even or (m, n) >, and as a product m of Legendre symbols corresponding to the prime factors of m elsewhere: ( ) ( ) α ( ) α2 ( ) αr n n n n =... where m = q α q α qr αr. m q q 2 q r If 8an + b is square free, it is a fundamental discriminant (as b mod 4). In this case, set χ n (m) = ( ) m. 8an + b

42 Chapter 4. Omega Theorems for L /L(, χ D ) 33 For m odd, we have ( ) 8an + b χ n (m) =, m by the Jacobi Reciprocity Law. Now consider the Dirichlet series L L (s, χ n) = m= χ n (m)λ(m) m s, Rs > where L(s, χ n ) is the Dirichlet L-function corresponding to the character χ n Application of the Explicit Formula We will need the following lemma: Lemma 3. Let Φ(y) be as in (4..), then Φ(y) = p<y χ(p) log p p + O() Proof. Write Φ(y) = + 2 where 2 = log 2 y m=2 j <y ( y ( ) j 2 2 ), j 8an + b and is the sum over odd m < y. We see that 2 = O(). Now, write = +, A B where in A, m ranges over primes p < y, and in B m ranges over prime powers. We have B log p( y y p ) log p + log y j j 2 p< p 2 y p< log p y p j <y This proves the lemma. log p y.

43 Chapter 4. Omega Theorems for L /L(, χ D ) 34 Let y > p g be a real number to be chosen later, choose χ = χ n and write p<y χ n (p) log p p = + 2 where in, p ranges over primes p p g and in 2 p ranges over primes p g < p < y. ( ) Since p a and = for all p p g, the contribution of primes p p g is b p = ( y ) log p. y p p p g In turn this is equal to ([4], p.57, and (), p.) : = = = = y y { y log p p p p g } log p p p g { y(log p g + O()) Ψ(p g ) + { y (log p g + O()) Ψ(p g ) + y { y(log p g + O()) p g + O y = log p g + O(). p j p g j 2 p p g log pg j log p ( pg log p g } log p } log p )} As it follows from the prime number theorem and the fact that p g < y. Summing this over n (x, 2x], with 8an + b square free, it becomes x<n 2x = N log p g + O(N), (4.2.) where N = #{n (x, 2x], such that 8an + b is square free}. Then one can show easily that N x i.e it is bounded from above and below by a constant multiple of x ([3], Lemma 3 in Collected Papers). As for primes p g < p < y we have to estimate x<n 2x 2 = y p g<p<y ( y p ) log p x<n 2x ( 8an + b p ). (4.2.2)

44 Chapter 4. Omega Theorems for L /L(, χ D ) 35 We have m b(8a) z m z 2 m sqfree ( ) m = p φ(8a) χ mod 8a z m z 2 χ(b)χ(m) ( ) m µ(d) (4.2.3) p d 2 m µ being the Mobius function, and φ the Euler s totient function. Writing m = d 2 u in the sum, we find that the right hand side of (4.2.3) is d z 2 (d,8ap)= φ(8a) χ mod 8a χ(d 2 )µ(d)χ(b) ( ) u χ(u). z p d 2 u z 2 d 2 The inner sum is 8ap log (8ap), ( by the Polya Vinogradov Theorem ([4], p.35), which assumes that χ(.) χ is a character mod 8a, and (8a, p) =. Remark 4. Notice that 8a = 8 ( p i, and p > p g, therefore χ(.) p i p ( ) g different conductors. Thus χ(.).. p. p. p ), where ), as they have Hence the entire quantity is z 2 8ap log (8ap), If z = 8ax + b, and z 2 = 6ax + b, this can be rewritten as a px log (8ap). Hence, inserting this into (4.2.2), we find that the contribution of primes p g < p < y is

45 Chapter 4. Omega Theorems for L /L(, χ D ) 36 ( ) y y p (log p)a px log (8ap) (4.2.4) p g<p<y a { } log p x(log (xy)) p (4.2.5) p p g<p<y a xy log (xy). (4.2.6) Hence, putting together (4.2.), and (4.2.6), we get x<n 2x Φ n (y) = N log p g + O(N) + O(a xy log (xy)). (4.2.7) Here Φ n is defined as in 4.., with χ = χ n The sum over zeros Next, we consider the contribution of the sum over zeros, we need to estimate x<n 2x { y ρ y ρ( ρ) + O ρ ( log y y )}. If we truncate the sum over ρ = β + iγ at γ T, we introduce an error of ([7], 5.4.6, p.269) x<n 2x where q = 8an + b. As q ax, this is { log (qt ) + T } (log q)2 y x T log (axt ) + x y (log (ax))2. The contribution of non-exceptional zeros with β 5 6 is ([7], 5.4.4) (log (axt )) 2 x x<n 2x y 6 (log (axt ))2. y 6

46 Chapter 4. Omega Theorems for L /L(, χ D ) 37 The term exceptional zeros refers to those exceptions to Page theorem ([4], p.95), concerning the set of positive integers q z, where z 3: If c is a suitable positive constant, there is at most one real primitive character χ to a modulus q z, for which L(s, χ) has a real zero β satisfying β > c log z. If such a character exists, it is called exceptional, and so is called the zero β. For the zeros with β > 5 6 and γ T, we separate exceptional and non-exceptional zeros. For all non-exceptional zeros, we use Jutila s zero-density estimate [9], Theorem 2(see below), which implies that for σ 5, 6 x<n 2x N(σ, T, χ n ) (axt ) 3 4 +ɛ, where the sum is over fundamental discriminants, and N(σ, T, χ) := #{ρ = β + iγ a non-trivial zero of L(s, χ) such that β σ and γ T }. Then is y β = γ T β y σ dn(σ, T, χ n ) = y σ N(σ, T, χ n ) N(σ, T, χ n )y σ log y dσ. Summing this over n (x, 2x] such that 8an + b is square free, and noticing that b mod 4, hence 8an + b is a fundamental discriminant. We get x<n 2x y β y (axt ) 4 +ɛ + (axt ) 3 4 +ɛ y σ log ydσ (4.2.8) γ T β 5 6 y 5 6 (axt ) 3 4 +ɛ + (axt ) 3 4 +ɛ y (4.2.9) y(axt ) 3 4 +ɛ. (4.2.0) 5 6

47 Chapter 4. Omega Theorems for L /L(, χ D ) 38 We use this to estimate x<n 2x γ T β 5 6 y ρ ρ( ρ), (4.2.) where the asterisk indicates that we sum only over non-exceptional zeros, again we need only to consider β 5. Thus we have 6 ρ( ρ) β( β) + γ 2 and 2 < β < c log (axt ) ([4],Page Theorem, p.95). Hence, the contribution of zeros with γ 2 to (4.2.) is by (4.2.0) : y(log (axt ))(ax) 3 4 +ɛ and the contribution of zeros with j < γ j + is Summing this over j, this is j 2 y(axj) 3 4 +ɛ. Hence y(ax) 3 4 +ɛ. x<n 2x y γ T β 5 6 y ρ ρ( ρ) (ax) 3 4 +ɛ log (axt ). Theorem 20. (Jutila s zero-density estimate [9], Theorem 2) Let N(α, T, χ) be the number of zeros (ρ = σ + it) of the function L(s, χ) in the rectangle α σ, t T.

48 Chapter 4. Omega Theorems for L /L(, χ D ) 39 Then for /2 α <, T, ɛ > 0 we have D X N(α, T, χ D ) ɛ (XT ) (7 6α)/(6 4α)+ɛ. The asterisk on the sum means that we are summing over fundamental discriminants D X Proof of the Theorem From the identities (4..), (4.2.7), and the calculations of Subsections and 4.2.3, we deduce that : x<n 2x ( L L (, χ n) + y y β 0 ) ( ) N log y = O β 0 ( β 0 ) y ( ) +O (ax) 3 4 +ɛ log (axt ) N log p g + O(N) + O(a (log (axt ))2 xy log (xy)) + x. Now, N x, and log p g log(g log g). Moreover, recall that y 6 We choose y so that a = e p pg log p e (+ɛ)pg x η. a xy log (xy) = o(n log log x) = o(x log log x) and (log (axt )) 2 = o(y 6 ). Clearly, any y = x α with 0 < α < satisfies this. With such a choice, we get x<n 2x where, β 0 > log (8an+b) ( L L (, χ y β 0 ) n) + = N log p g + O(x) (y )β 0 ( β 0 ) ([4], p. 95). Notice that the contribution of the possible Landau-Siegel zero in this identity is positive, so we can remove it as it points in the right direction. Then there exists x < n 2x so that for D = 8an + b L L (, χ n) log p g + O() log 2 D + O().