Low-lying zeros of GL(2) L-functions

Transcription

1 Low-lying zeros of GL(2) L-functions Steven J Miller Williams College Steven.J.Miller@williams.edu Group, Lie and Number Theory Seminar University of Michigan, October 22, 2012

2 Introduction

3 Why study zeros of L-functions? Infinitude of primes, primes in arithmetic progression. Chebyshev s bias: π 3,4 (x) π 1,4 (x) most of the time. Birch and Swinnerton-Dyer conjecture. Goldfeld, Gross-Zagier: bound for h(d) from L-functions with many central point zeros. Even better estimates for h(d) if a positive percentage of zeros of ζ(s) are at most 1/2 ǫ of the average spacing to the next zero.

4 Distribution of zeros ζ(s) 0 for Re(s) = 1: π(x), π a,q (x). GRH: error terms. GSH: Chebyshev s bias. Analytic rank, adjacent spacings: h(d).

5 5 Introduction Classical RMT Intro to L-Functions Results Dirichlet L-functions Goals Determine correct scale and statistics to study zeros of L-functions. See similar behavior in different systems (random matrix theory). Discuss the tools and techniques needed to prove the results. State new GL(2) results, sketch proofs in simple families. In Part II: Discuss new ideas for GL(2) families.

6 Fundamental Problem: Spacing Between Events General Formulation: Studying system, observe values at t 1, t 2, t 3,... Question: What rules govern the spacings between the t i? Examples: Spacings b/w Energy Levels of Nuclei. Spacings b/w Eigenvalues of Matrices. Spacings b/w Primes. Spacings b/w n k α mod 1. Spacings b/w Zeros of L-functions.

7 7 Introduction Classical RMT Intro to L-Functions Results Dirichlet L-functions Sketch of proofs In studying many statistics, often three key steps: 1 Determine correct scale for events. 2 Develop an explicit formula relating what we want to study to something we understand. 3 Use an averaging formula to analyze the quantities above. It is not always trivial to figure out what is the correct statistic to study!

8 Classical Random Matrix Theory

9 Origins of Random Matrix Theory Classical Mechanics: 3 Body Problem Intractable. Heavy nuclei (Uranium: 200+ protons / neutrons) worse! Get some info by shooting high-energy neutrons into nucleus, see what comes out. Fundamental Equation: Hψ n = E n ψ n H : matrix, entries depend on system E n : energy levels ψ n : energy eigenfunctions

10 Origins of Random Matrix Theory Statistical Mechanics: for each configuration, calculate quantity (say pressure). Average over all configurations most configurations close to system average. Nuclear physics: choose matrix at random, calculate eigenvalues, average over matrices (real Symmetric A = A T, complex Hermitian A T = A).

11 Classical Random Matrix Ensembles A = a 11 a 12 a 13 a 1N a 12 a 22 a 23 a 2N a 1N a 2N a 3N a NN = AT, a ij = a ji Fix p, define Prob(A) = p(a ij ). 1 i j N This means Prob(A : a ij [α ij,β ij ]) = 1 i j N βij Want to understand eigenvalues of A. x ij =α ij p(x ij )dx ij.

12 Eigenvalue Distribution δ(x x 0 ) is a unit point mass at x 0 : f(x)δ(x x0 )dx = f(x 0 ).

13 Eigenvalue Distribution δ(x x 0 ) is a unit point mass at x 0 : f(x)δ(x x0 )dx = f(x 0 ). To each A, attach a probability measure: µ A,N (x) = 1 N N δ i=1 ( x λ ) i(a) 2 N

14 Eigenvalue Distribution δ(x x 0 ) is a unit point mass at x 0 : f(x)δ(x x0 )dx = f(x 0 ). To each A, attach a probability measure: b a µ A,N (x) = 1 N µ A,N (x)dx = # N δ i=1 ( x λ ) i(a) 2 N { λ i : λ i(a) 2 N [a, b] } N

15 Eigenvalue Distribution δ(x x 0 ) is a unit point mass at x 0 : f(x)δ(x x0 )dx = f(x 0 ). To each A, attach a probability measure: b a µ A,N (x) = 1 N µ A,N (x)dx = k th moment = # N δ i=1 ( x λ ) i(a) 2 N { λ i : λ i(a) 2 N [a, b] } N N i=1 λ i(a) k = Trace(Ak ) 2 k N k k N. k 2 +1

16 Wigner s Semi-Circle Law Not most general case, gives flavor. Wigner s Semi-Circle Law N N real symmetric matrices, entries i.i.d.r.v. from a fixed p(x) with mean 0, variance 1, and other moments finite. Then for almost all A, as N µ A,N (x) { 2 π 1 x 2 if x 1 0 otherwise.

17 SKETCH OF PROOF: Eigenvalue Trace Lemma Want to understand the eigenvalues of A, but it is the matrix elements that are chosen randomly and independently. Eigenvalue Trace Lemma Let A be an N N matrix with eigenvalues λ i (A). Then Trace(A k ) = N λ i (A) k, n=1 where Trace(A k ) = N N a i1 i 2 a i2 i 3 a in i 1. i 1 =1 i k =1

18 SKETCH OF PROOF: Correct Scale Trace(A 2 ) = By the Central Limit Theorem: N λ i (A) 2. i=1 N N Trace(A 2 ) = a ij a ji = N λ i (A) 2 N 2 i=1 i=1 j=1 N N a 2 ij N 2 i=1 j=1 Gives NAve(λ i (A) 2 ) N 2 or Ave(λ i (A)) N.

19 SKETCH OF PROOF: Averaging Formula Recall k-th moment of µ A,N (x) is Trace(A k )/2 k N k/2+1. Average k-th moment is Trace(A k ) p(a 2 k N k/2+1 ij )da ij. i j Proof by method of moments: Two steps Show average of k-th moments converge to moments of semi-circle as N ; Control variance (show it tends to zero as N ).

20 SKETCH OF PROOF: Averaging Formula for Second Moment Substituting into expansion gives N 2 N i=1 N a 2 ji p(a 11 )da 11 p(a NN )da NN j=1 Integration factors as a ij = a 2 ijp(a ij )da ij (k,l) (i,j) k<l a kl = p(a kl )da kl = 1. Higher moments involve more advanced combinatorics (Catalan numbers).

21 SKETCH OF PROOF: Averaging Formula for Higher Moments Higher moments involve more advanced combinatorics (Catalan numbers). 1 2 k N k/2+1 N i 1 =1 N a i1 i 2 a ik i 1 p(a ij )da ij. i k =1 Main term a il i l+1 s matched in pairs, not all matchings contribute equally (if did have Gaussian, see in Real Symmetric Palindromic Toeplitz matrices; interesting results for circulant ensembles (joint with Gene Kopp, Murat Kologlu). i j

22 Numerical examples Distribution of eigenvalues Gaussian, N=400, 500 matrices Matrices: Gaussian p(x) = 1 2π e x2 /2

23 Numerical examples 2500 The eigenvalues of the Cauchy distribution are NOT semicirular Cauchy Distribution: p(x) = 1 π(1+x 2 )

24 GOE Conjecture GOE Conjecture: As N, the probability density of the spacing b/w consecutive normalized eigenvalues approaches a limit independent of p. Until recently only known if p is a Gaussian. GOE(x) π 2 xe πx2 /4.

25 Numerical Experiment: Uniform Distribution Let p(x) = 1 2 for x x 104 The local spacings of the central 3/5 of the eigenvalues of x300 uniform matrices, normalized in batches of : uniform on [ 1, 1]

26 Cauchy Distribution Let p(x) = 1 π(1+x 2 ) The local spacings of the central 3/5 of the eigenvalues of x100 Cauchy matrices, normalized in batches of : Cauchy

27 Cauchy Distribution Let p(x) = 1 π(1+x 2 ). 3.5 x 104 The local spacings of the central 3/5 of the eigenvalues of x300 Cauchy matrices, normalized in batches of : Cauchy

28 Random Graphs Degree of a vertex = number of edges leaving the vertex. Adjacency matrix: a ij = number edges b/w Vertex i and Vertex j. A = These are Real Symmetric Matrices.

29 McKay s Law (Kesten Measure) with d = 3 Density of Eigenvalues for d-regular graphs f(x) = { d 2π(d 2 x 2 ) 4(d 1) x 2 x 2 d 1 0 otherwise.

30 McKay s Law (Kesten Measure) with d = 6 Fat Thin: fat enough to average, thin enough to get something different than semi-circle (though as d recover semi-circle).

31 3-Regular Graph with 2000 Vertices: Comparison with the GOE Spacings between eigenvalues of 3-regular graphs and the GOE:

32 Introduction to L-Functions

33 Riemann Zeta Function ζ(s) = n=1 1 n s = p prime ( 1 1 p s ) 1, Re(s) > 1. Functional Equation: ( s ξ(s) = Γ π 2) s 2 ζ(s) = ξ(1 s). Riemann Hypothesis (RH): All non-trivial zeros have Re(s) = 1 2 ; can write zeros as 1 2 +iγ. Observation: Spacings b/w zeros appear same as b/w eigenvalues of Complex Hermitian matrices A T = A.

34 General L-functions L(s, f) = n=1 a f (n) n s = p prime L p (s, f) 1, Re(s) > 1. Functional Equation: Λ(s, f) = Λ (s, f)l(s, f) = Λ(1 s, f). Generalized Riemann Hypothesis (RH): All non-trivial zeros have Re(s) = 1 2 ; can write zeros as 1 2 +iγ. Observation: Spacings b/w zeros appear same as b/w eigenvalues of Complex Hermitian matrices A T = A.

35 Zeros of ζ(s) vs GUE 70 million spacings b/w adjacent zeros of ζ(s), starting at the 10 20th zero (from Odlyzko).

36 Explicit Formula (Contour Integration) ζ (s) ζ(s) = d ds logζ(s) = d ds log p ( 1 p s ) 1

37 Explicit Formula (Contour Integration) ζ (s) ζ(s) = d ds logζ(s) = d ds log p = d log ( 1 p s) ds p ( 1 p s ) 1 = p log p p s 1 p s = p log p p s + Good(s).

38 Explicit Formula (Contour Integration) ζ (s) ζ(s) = d ds logζ(s) = d ds log p = d log ( 1 p s) ds = p p log p p s 1 p s = p log p p s ( 1 p s ) 1 + Good(s). Contour Integration: ζ (s) x s ζ(s) s ds vs log p p ( ) s x ds p s.

39 Explicit Formula (Contour Integration) ζ (s) ζ(s) = d ds logζ(s) = d ds log p = d log ( 1 p s) ds = p p log p p s 1 p s = p log p p s ( 1 p s ) 1 + Good(s). Contour Integration: ζ (s) ζ(s) φ(s)ds vs p log p φ(s)p s ds.

40 Explicit Formula (Contour Integration) ζ (s) ζ(s) = d ds logζ(s) = d ds log p = d log ( 1 p s) ds = p p log p p s 1 p s = p log p p s ( 1 p s ) 1 + Good(s). Contour Integration (see Fourier Transform arising): ζ (s) ζ(s) φ(s)ds vs log p φ(s)e σ log p e it log p ds. Knowledge of zeros gives info on coefficients. p

41 Explicit Formula: Examples Riemann Zeta Function: Let ρ denote the sum over the zeros of ζ(s) in the critical strip, g an even Schwartz function of compact support and φ(r) = g(u)eiru du. Then ( ) i φ(γ ρ ) = 2φ 2 log p g(k log p) 2 pk/2 ρ p k=1 + 1 ( 1 + Γ ( iy + 5 ) ) 2 4 π iy 1 2 Γ( iy + 5)) 1 2 logπ φ(y) dy. 2 4

42 Explicit Formula: Examples Dirichlet L-functions: Let h be an even Schwartz function and L(s,χ) = n χ(n)/ns a Dirichlet L-function from a non-trivial character χ with conductor m and zeros ρ = 1 + iγ 2 χ; if the Generalized Riemann Hypothesis is true then γ R. Then ( ) log(m/π) h γ ρ = h(y)dy 2π ρ ( ) log p log p χ(p) 2 p 2 p log(m/π)ĥ log p log(m/π)ĥ log(m/π) ( 2 log p log(m/π) p 1/2 ) χ 2 (p) p ( 1 ) + O. log m

43 Explicit Formula: Examples Cuspidal Newforms: Let F be a family of cupsidal newforms (say weight k, prime level N and possibly split by sign) L(s, f) = n λ f(n)/n s. Then 1 F φ f F γ f ( log R 2π γ f ) P(f;φ) = p N = φ(0)+ 1 2 φ(0) 1 P(f;φ) F f F ( ) log log R + O log R ( ) log p 2 log p λ f (p) φ. log R p log R

44 Measures of Spacings: n-level Correlations {α j } increasing sequence, box B R n 1. n-level correlation lim N } ( ) #{ α α j1 j,...,α 2 j α n 1 j n B, j i j k (Instead of using a box, can use a smooth test function.) N

45 Measures of Spacings: n-level Correlations {α j } increasing sequence, box B R n 1. 1 Normalized spacings of ζ(s) starting at (Odlyzko). 2 2 and 3-correlations of ζ(s) (Montgomery, Hejhal). 3 n-level correlations for all automorphic cupsidal L-functions (Rudnick-Sarnak). 4 n-level correlations for the classical compact groups (Katz-Sarnak). 5 Insensitive to any finite set of zeros.

46 Measures of Spacings: n-level Density and Families φ(x) := i φ i(x i ), φ i even Schwartz functions whose Fourier Transforms are compactly supported. n-level density D n,f (φ) = j 1,...,jn distinct ( ) ( ) φ 1 L f γ (j 1) φ n L f γ (jn) f f

47 Measures of Spacings: n-level Density and Families φ(x) := i φ i(x i ), φ i even Schwartz functions whose Fourier Transforms are compactly supported. n-level density D n,f (φ) = ( ) ( ) φ 1 L f γ (j 1) φ n L f γ (jn) j 1,...,jn distinct 1 Individual zeros contribute in limit. 2 Most of contribution is from low zeros. 3 Average over similar curves (family). f f

48 Measures of Spacings: n-level Density and Families φ(x) := i φ i(x i ), φ i even Schwartz functions whose Fourier Transforms are compactly supported. n-level density D n,f (φ) = ( ) ( ) φ 1 L f γ (j 1) φ n L f γ (jn) j 1,...,jn distinct 1 Individual zeros contribute in limit. 2 Most of contribution is from low zeros. 3 Average over similar curves (family). Katz-Sarnak Conjecture For a nice family of L-functions, the n-level density depends only on a symmetry group attached to the family. f f

49 Normalization of Zeros Local (hard, use C f ) vs Global (easier, use log C = F N 1 f F N log C f ). Hope: φ a good even test function with compact support, as F, 1 F N f F N D n,f (φ) = 1 F N f F N j 1,...,jn j i ±j k i ( ) log C f φ i 2π γ(j i) E φ(x)w n,g(f) (x)dx. Katz-Sarnak Conjecture As C f the behavior of zeros near 1/2 agrees with N limit of eigenvalues of a classical compact group.

50 1-Level Densities The Fourier Transforms for the 1-level densities are W 1,SO(even) (u) = δ 0 (u)+ 1 2 η(u) Ŵ 1,SO (u) = δ 0 (u)+ 1 2 W 1,SO(odd) (u) = δ 0 (u) 1 2 η(u)+1 Ŵ 1,Sp (u) = δ 0 (u) 1 2 η(u) Ŵ 1,U (u) = δ 0 (u) where δ 0 (u) is the Dirac Delta functional and { 1 if u < 1 1 η(u) = if u = if u > 1

51 Correspondences Similarities between L-Functions and Nuclei: Zeros Schwartz test function Support of test function Energy Levels Neutron Neutron Energy.

52 Results

53 Some Number Theory Results Orthogonal: Iwaniec-Luo-Sarnak, Ricotta-Royer: 1-level density for holomorphic even weight k cuspidal newforms of square-free level N (SO(even) and SO(odd) if split by sign). Symplectic: Rubinstein, Gao, Levinson-Miller, and Entin, Roddity-Gershon and Rudnick: n-level densities for twists L(s,χ d ) of the zeta-function. Unitary: Fiorilli-Miller, Hughes-Rudnick: Families of Primitive Dirichlet Characters. Orthogonal: Miller, Young: One and two-parameter families of elliptic curves.

54 Recent Results with Colleagues Orthogonal: n-level densities of cuspidal newforms: Hughes-Miller: up to ( 1, 1 ). n 1 n 1 Iyer-Miller-Triantafillou: up to ( 1, 1 ). n 2 n 2 Symplectic: n-level densities of L(s,χ d ): Levinson-Miller: up to ( 2/n, 2/n) for n 7. Unitary: 1-level of L(s, χ): Fiorilli-Miller: lower order terms beyond square-root, large support conditionally.

55 55 Introduction Classical RMT Intro to L-Functions Results Dirichlet L-functions Recent Results with Colleagues Orthogonal: n-level densities of cuspidal newforms: Hughes-Miller: up to ( 1, 1 ). n 1 n 1 Iyer-Miller-Triantafillou: up to ( 1, 1 ). n 2 n 2 Symplectic: n-level densities of L(s,χ d ): Levinson-Miller: up to ( 2/n, 2/n) for n 7. Unitary: 1-level of L(s, χ): Fiorilli-Miller: lower order terms beyond square-root, large support conditionally.

56 Main Tools 1 Control of conductors: Usually monotone, gives scale to study low-lying zeros. 2 Explicit Formula: Relates sums over zeros to sums over primes. 3 Averaging Formulas: Petersson formula in Iwaniec-Luo-Sarnak, Orthogonality of characters in Fiorilli-Miller, Gao, Hughes-Rudnick, Levinson-Miller, Rubinstein.

57 57 Introduction Classical RMT Intro to L-Functions Results Dirichlet L-functions Applications of n-level density One application: bounding the order of vanishing at the central point. Average rank φ(0) φ(x)w G(F) (x)dx if φ non-negative.

58 Applications of n-level density One application: bounding the order of vanishing at the central point. Average rank φ(0) φ(x)w G(F) (x)dx if φ non-negative. Can also use to bound the percentage that vanish to order r for any r. Theorem (Miller, Hughes-Miller) Using n-level arguments, for the family of cuspidal newforms of prime level N (split or not split by sign), for any r there is a c r such that probability of at least r zeros at the central point is at most c n r n. Better results using 2-level than Iwaniec-Luo-Sarnak using the 1-level for r 5.

59 Example: Dirichlet L-functions

60 Dirichlet Characters (m prime) (Z/mZ) is cyclic of order m 1 with generator g. Let ζ m 1 = e 2πi/(m 1). The principal character χ 0 is given by { 1 (k, m) = 1 χ 0 (k) = 0 (k, m) > 1. The m 2 primitive characters are determined (by multiplicativity) by action on g. As each χ : (Z/mZ) C, for each χ there exists an l such that χ(g) = ζm 1 l. Hence for each l, 1 l m 2 we have { ζm 1 la k g a (m) χ l (k) = 0 (k, m) > 0

61 Dirichlet L-Functions Let χ be a primitive character mod m. Let m 1 c(m,χ) = χ(k)e 2πik/m. k=0 c(m,χ) is a Gauss sum of modulus m. where L(s,χ) = (1 χ(p)p s ) 1 p ( ) Λ(s,χ) = π 1 s +ǫ 2 (s+ǫ) Γ m 1 2 (s+ǫ) L(s,χ), 2 ǫ = { 0 if χ( 1) = 1 1 if χ( 1) = 1

62 Explicit Formula Let φ be an even Schwartz function with compact support ( σ, σ), let χ be a non-trivial primitive Dirichlet character of conductor m. φ (γ log( m) ) π = p φ(y)dy 2π log p ( log p log(m/π) φ log(m/π) ( p ( 1 +O log m log p log(m/π) φ ). 2 log p log(m/π) ) [χ(p)+ χ(p)]p 1 2 ) [χ 2 (p)+ χ 2 (p)]p 1

63 Expansion {χ 0 } {χ l } 1 l m 2 are all the characters mod m. Consider the family of primitive characters mod a prime m (m 2 characters): + O 1 m 2 φ(y)dy χ χ 0 p 1 m 2 χ χ 0 p ( 1 ). log m log p ( log p log(m/π) φ log(m/π) ( log p log(m/π) φ 2 log p log(m/π) ) [χ(p)+ χ(p)]p 1 2 ) [χ 2 (p)+ χ 2 (p)]p 1 Note can pass Character Sum through Test Function.

64 Character Sums { m 1 k 1(m) χ(k) = 0 otherwise. χ

65 Character Sums { m 1 k 1(m) χ(k) = 0 otherwise. χ For any prime p m χ χ 0 χ(p) = { 1+m 1 1 p 1(m) otherwise.

66 Character Sums { m 1 k 1(m) χ(k) = 0 otherwise. χ For any prime p m χ χ 0 χ(p) = { 1+m 1 1 p 1(m) otherwise. Substitute into 1 log p ( log p ) [χ(p)+ χ(p)]p m 2 log(m/π) φ 1 2 log(m/π) χ χ 0 p

67 First Sum: no contribution if σ < 2 2 m m 1 m 2 m σ p log p ( log p ) p log(m/π) φ 1 2 log(m/π) m σ p 1(m) log p ( log p ) p log(m/π) φ 1 2 log(m/π)

68 First Sum: no contribution if σ < 2 2 m m 1 m 2 1 m m σ p m σ p log p ( log p ) p log(m/π) φ 1 2 log(m/π) m σ p 1(m) p log p ( log p ) p log(m/π) φ 1 2 log(m/π) m σ p 1(m) p 1 2

69 First Sum: no contribution if σ < 2 2 m m 1 m 2 1 m m σ p m σ p log p ( log p ) p log(m/π) φ 1 2 log(m/π) m σ p 1(m) p log p ( log p ) p log(m/π) φ 1 2 log(m/π) m σ p 1(m) p m m σ k k m σ k 1(m) k m+1 k 1 2

70 First Sum: no contribution if σ < 2 2 m m 1 m 2 1 m 1 m m σ p m σ p log p ( log p ) p log(m/π) φ 1 2 log(m/π) m σ p 1(m) p m σ k 1 2 k log p ( log p ) p log(m/π) φ 1 2 log(m/π) + 1 m m σ p 1(m) m σ k p m m σ k m mσ/2. k k m σ k 1(m) k m+1 k 1 2

71 Second Sum 1 m 2 χ χ 0 [χ 2 (p)+ χ 2 (p)] = χ χ 0 ( ) Up to O we find that 1 log m p log p ( 2 log(m/π) φ log p log(m/π) { 1 m σ/2 p 1 + 2m 2 m 2 m 2 p 1 m σ/2 k 1 + m 2 k m σ/2 k 1(m) k m+1 ) χ 2 (p)+ χ 2 (p). p 2(m 2) p ±1(m) 2 p ±1(m) m σ/2 p ±1(m) k 1 + p 1 m σ/2 k 1(m) k m 1 k 1

72 Dirichlet Characters: m Square-free Fix an r and let m 1,...,m r be distinct odd primes. m = m 1 m 2 m r M 1 = (m 1 1)(m 2 1) (m r 1) = φ(m) M 2 = (m 1 2)(m 2 2) (m r 2). M 2 is the number of primitive characters mod m, each of conductor m. A general primitive character mod m is given by χ(u) = χ l1 (u)χ l2 (u) χ lr (u). Let F = {χ : χ = χ l1 χ l2 χ lr }. 1 log p ( log p M 2 log(m/π) φ p 1 log p M 2 log(m/π) φ p ( log(m/π) ) p 1 2 [χ(p)+ χ(p)] χ F 2 log p ) p 1 log(m/π) χ F[χ 2 (p)+ χ 2 (p)]

73 Characters Sums m i 2 l i =1 χ li (p) = { m i 1 1 p 1(m i ) 1 otherwise. Define Then χ(p) = χ F = δ mi (p, 1) = m 1 2 l 1 =1 r i=1 m r 2 m i 2 l i =1 { 1 p 1(m i ) 0 otherwise. χ l1 (p) χ lr (p) l r=1 χ li (p) = ( ) r 1+(m i 1)δ mi (p, 1). i=1

74 Expansion Preliminaries k(s) is an s-tuple (k 1, k 2,..., k s ) with k 1 < k 2 < < k s. This is just a subset of (1, 2,...,r), 2 r possible choices for k(s). s δ k(s) (p, 1) = δ mki (p, 1). i=1 If s = 0 we define δ k(0) (p, 1) = 1 p. Then = r ( i=1 ) 1+(m i 1)δ mi (p, 1) r ( 1) r s δ k(s) (p, 1) s=0 k(s) s (m ki 1). i=1

75 75 Introduction Classical RMT Intro to L-Functions Results Dirichlet L-functions First Sum m σ p 1 2 p 1 ( 1+ M 2 r δ k(s) (p, 1) s=1 k(s) s ) (m ki 1). i=1 As m/m 2 3 r, s = 0 sum contributes S 1,0 = 1 M 2 mσ hence negligible for σ < 2. p p r m 1 2 σ 1,

76 First Sum m σ p 1 2 p Now we study S 1,k(s) = 1 M 2 1 M 2 1 M 2 1 ( 1+ M 2 r δ k(s) (p, 1) s=1 k(s) s ) (m ki 1). i=1 s m σ (m ki 1) p 1 2 δk(s) (p, 1) i=1 s (m ki 1) i=1 p m σ n 1(m k(s) ) n 1 2 s 1 m σ (m ki 1) s i=1 (m n r m 1 2 σ 1. k i ) i=1 n

77 First Sum There are 2 r choices, yielding S 1 6 r m 1 2 σ 1, which is negligible as m goes to infinity for fixed r if σ < 2. Cannot let r go to infinity. If m is the product of the first r primes, r log m = log p k k=1 = p r log p r 77 Therefore 6 r m log 6 m 1.79.

78 Second Sum Expansions and Bounds m i 2 l i =1 χ 2 l i (p) = { m i 1 1 p ±1(m i ) 1 otherwise = = χ 2 (p) = χ F r i=1 m i 2 l i =1 χ 2 l i (p) m 1 2 l 1 =1 χ 2 l 1 (p) χ 2 l r (p) m r 2 l r=1 r ( ) 1+(m i 1)δ mi (p, 1)+(m i 1)δ mi (p, 1). i=1

79 Second Sum Expansions and Bounds Handle similarly as before. Say p 1 mod m k1,..., m ka p 1 mod m ka+1,...,m kb How small can p be? +1 congruences imply p m k1 m ka congruences imply p m ka+1 m kb 1. Since the product of these two lower bounds is greater than b i=1 (m k i 1), at least one must be greater than ( b ) 1 2 i=1 (m k i 1). There are 3 r pairs, yielding r Second Sum = S 2,k(s),j(s) 9 r m 1 2. s=0 k(s) j(s)

80 Summary Agrees with Unitary for σ < 2 for square-free m [N, 2N] from: Theorem m square-free odd integer with r = r(m) factors; m = r i=1 m i; M 2 = r i=1 (m i 2). Then family F m of primitive characters mod m has First Sum 1 M 2 2 r m 1 2 σ Second Sum 1 M 2 3 r m 1 2.