RH

The Riemann Hypothesis The American Institute of Mathematics This is a hard–copy version of a web page available through http://www.aimath.org Input on this material is welcomed and can be sent to workshops@aimath.org Version: Thu Jun 17 05:56:54 2004 This document is a preliminary version of a planned comprehensive resource on the Riemann Hypothesis. Suggestions and contributions are welcome and can be sent to rh(at)aimath.org 1 2 Table of Contents A. What is an L-function? . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Terminology and basic properties a. Functional equation b. Euler product c. ξ and Z functions d. Critical line and critical strip e. Trivial zeros f. Zero counting functions 2. Arithmetic L-functions a. The Riemann zeta function b. Dirichlet L-functions c. Dedekind zeta functions d. GL(2) L-functions e. Higher rank L-functions 3. The Selberg class a. Dirichlet series b. Analytic Continuation c. Functional Equation d. Euler Product e. Ramanujan Hypothesis f. Selberg Conjectures 4. Analogues of zeta-functions a. Dynamical zeta-functions b. Spectral zeta functions B. Riemann Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . 14 1. Riemann Hypotheses for global L-functions a. The Riemann Hypothesis b. The Generalized Riemann Hypothesis c. The Extended Riemann Hypothesis d. The Grand Riemann Hypothesis 2. Other statements about the zeros of L-functions a. Quasi Riemann Hypothesis b. 100 percent hypothesis c. The Density Hypothesis d. Zeros on the σ = 1 line e. Landau-Siegel zeros f. The vertical distribution of zeros 3. The Lindelof hypothesis and breaking convexity 4. Perspectives on RH a. Analytic number theory b. Physics c. Probability d. Fractal geometry C. Equivalences to RH . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1. Primes 3 a. The error term in the PNT b. More accurate estimates 2. Arithmetic functions a. Averages b. Large values 3. The Farey series a. Mikolas functions b. Amoroso’s criterion 4. Weil’s positivity criterion a. Bombieri’s refinement b. Li’s criterion 5. Complex function theory a. Speiser’s criterion b. Logarithmic integrals c. An inequality for the logarithmic derivative of xi 6. Function spaces a. The Beurling-Nyman Criterion b. Salem’s criterion 7. Other analytic estimates a. M. Riesz series b. Hardy-Littlewood series c. Polya’s integral criterion d. Newman’s criterion 8. Grommer inequalities 9. Redheffer’s matrix 10. Dynamical systems D. Attacks on the Riemann Hypothesis . . . . . . . . . . . . . . . . . . 25 1. Alain Connes’s approach a. The dynamical system problem studied by Bost–Connes b. The C*-algebra of Bost–Connes 2. Iwaniec’ approach a. families of rank 2 elliptic curves 3. Unsuccessful attacks on the Riemann Hypothesis a. Zeros of Dirichlet polynomials b. de Branges’ positivity condition E. Zeta Gallery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 F. Anecdotes about the Riemann Hypothesis . . . . . . . . . . . . . . . . 30 4 Chapter A: What is an L-function? The Riemann Hypothesis is an assertion about the zeros of the Riemann ζ-function. Generalizations of the ζ-function have been discovered, for which the analogue of the Rie- mann Hypothesis is also conjectured. These generalizations of the ζ-function are known as “zeta-functions” or “L-functions.” In this section we describe attempts to determine the collection of functions that deserve to be called L-functions. A.1 Terminology and basic properties For a more detailed discussion, see the articles on the Selberg class11 and on automor- phic L-functions12. A.1.a Functional equation of an L-function. The Riemann ζ-function4 has functional equation ξ(s) = pi− s 2Γ( s 2 )ζ(s) = ξ(1− s). (1) Dirichlet L-functions9 satisfy the functional equation ξ(s, χ) = pi− s 2Γ( s 2 + a)L(s, χ) = ∗ ∗ ξ(1− s, χ), (2) where a = 0 if χ is even and a = 1 if χ is odd, and ∗ ∗ ∗ ∗ ∗. The Dedekind zeta function66 of a number field K satisfies the functional equation ξK(s) = (√|dK | 2r2pin/2 )s Γ(s/2)r1Γ(s)r2ζK(s)) = ξK(1− s). (3) Here r1 and 2r2 are the number of real and complex conjugate embeddings K ⊂ C, dK is the discriminant, and n = [K,Q] is the degree of K/Q. L-functions associated with a newform13 f ∈ Sk(Γ0(N) satisfy the functional equation ξ(s, f) = ( pi N )−s Γ ( s+ k − 1 2 ) L(s, f) = εξ(1− s, f), (4) where a = 0 if χ is even and a = 1 if χ is odd. 11page 11, The Selberg class 12page 28, Iwaniec’ approach 4page 7, The Riemann zeta function 9page 8, Dirichlet L-functions 66page 8, Dedekind zeta functions 13page 8, Dirichlet series associated with holomorphic cusp forms 5 L-functions associated with a Maass newform14 with eigenvalue λ = 1 4 + R2 on Γ0(N) satisfy the functional equation ξ(s, f) = ( N pi )s Γ ( s+ iR + a 2 ) Γ ( s− iR + a 2 ) L(s, f) = εξ(1− s, f), (5) where a = 0 if f is even and a = 1 if f is odd. GL(r) L-functions44 satisfy functional equations of the form Φ(s) := ( N pir )s/2 r∏ j=1 Γ ( s+ rj 2 ) F (s) = εΦ(1− s). [This section needs a bit of work] A.1.b Euler product. An Euler product is a representation of an L-function as a convergent infinite product over the primes p, where each factor (called the “local factor at p”) is a Dirichlet series supported only at the powers of p. The Riemann ζ-function4 has Euler product ζ(s) = ∏ p ( 1− p−s)−1 . A Dirichlet L-function9 has Euler product L(s, χ) = ∏ p ( 1− χ(p)p−s)−1 . The Dedekind zeta function66 of a number field K has Euler product ζK(s) = ∏ p ( 1−Np−s)−1 , where the product is over the prime ideals of OK . An L-functions associated with a newform13 f ∈ Sk(Γ0(N)) or a Maass newform14 f(z) on Γ0(N) has Euler product L(s, f) = ∏ p|N ( 1− app−s )−1∏ p-N ( 1− app−s + χ(p)p−2s+1 )−1 . GL(r) L-functions44 have Euler products where almost all of the local factors are (reciprocals of) polynomials in p−s of degree r. 14page 11, Dirichlet series associated with Maass forms 44page 11, Higher rank L-functions 4page 7, The Riemann zeta function 9page 8, Dirichlet L-functions 66page 8, Dedekind zeta functions 13page 8, Dirichlet series associated with holomorphic cusp forms 14page 11, Dirichlet series associated with Maass forms 44page 11, Higher rank L-functions 6 A.1.c ξ and Z functions. The functional equation69 can be written in a form which is more symmetric: ξ(s) := 1 2 s(s− 1)pi 12 sΓ(s/2)ζ(s) = ξ(1− s). Here ξ(s) is known as the Riemann ξ-function. It is an entire function of order 1, and all of its zeros lie in the critical strip. The ξ-function associated to a general L-function is similar, except that the factor 1 2 s(s− 1) is omitted, since its only purpose was to cancel the pole at s = 1. The Ξ function just involves a change of variables: Ξ(z) = ξ( 1 2 + iz). The functional equation now asserts that Ξ(z) = Ξ(−z). The Hardy Z-function is defined as follows. Let ϑ = ϑ(t) = 1 2 arg(χ( 1 2 + it)), and define Z(t) = eiϑζ( 1 2 + it) = χ( 1 2 + it)− 1 2 ζ( 1 2 + it). Then Z(t) is real for real t, and |Z(t)| = |ζ( 1 2 + it). Plots of Z(t) are a nice way to picture the ζ-function on the critical line. Z(t) is called RiemannSiegelZ[t] in Mathematica. A.1.d Critical line and critical strip. The critical line is the line of symmetry in the functional equation69 of the L-function. In the usual normalization the functional equation associates s to 1− s, so the critical line is σ = 1 2 . In the usual normalization the Dirichlet series and the Euler product converge abso- lutely for The functional equation maps σ > 1 to σ < 0. The remaining region, 0 < σ < 1 is known as the critical strip. By the Euler product there are no zeros in σ > 1, and by the functional equation there are only trivial zeros in σ < 0. So all of the nontrivial zeros are in the critical strip, and the Riemann Hypothesis asserts that the nontrivial zeros are actually on the critical line. A.1.e Trivial zeros. The trivial zeros of the ζ-function are at s = −2, −4, −6, .... The trivial zeros correspond to the poles of the associated Γ-factor. A.1.f Zero counting functions. Below we present the standard notation for the functions which count zeros of the zeta-function. Zeros of the zeta-function in the critical strip are denoted ρ = β + iγ. It is common to list the zeros with γ > 0 in order of increasing imaginary part as ρ1 = β1+iγ1, ρ2 = β2 + iγ2,.... Here zeros are repeated according to their multiplicity. We have the zero counting function N(T ) = #{ρ = β + iγ : 0 < γ ≤ T}. 69page 4, Functional equation of an L-function 69page 4, Functional equation of an L-function 7 In other words, N(T ) counts the number of zeros in the critical strip, up to height T . By the functional equation and the argument principle, N(T ) = 1 2pi T log ( T 2pie ) + 7 8 + S(T ) +O(1/T ), where S(T ) = 1 pi arg ζ ( 1 2 + it ) , with the argument obtained by continuous variation along the straight lines from 2 to 2+ iT to 1 2 +iT . Von Mangoldt proved that S(T ) = O(log T ), so we have a fairly precise estimate of the number of zeros of the zeta-function with height less than T . Note that Von Mangoldt’s estimate implies that a zero at height T has multiplicity O(log T ). That is still the best known result on the multiplicity of zeros. It is widely believed that all of the zeros are simple. A number of related zero counting functions have been introduced. The two most common ones are: N0(T ) = #{ρ = 1 2 + iγ : 0 < γ ≤ T}, which counts zeros on the critical line up to height T . The Riemann Hypothesis is equivalent to the assertion N(T ) = N0(T ) for all T . Selberg proved that N0(T ) À N(T ). At present the best result of this kind is due to Conrey [90g:11120], who proved that N0(T ) ≥ 0.40219N(T ) if T is sufficiently large. And, N(σ, T ) = #{ρ = β + iγ : β > σ and 0 < γ ≤ T}, which counts the number of zeros in the critical strip up to height T , to the right of the σ-line. Riemann Hypothesis is equivalent to the assertion N( 1 2 , T ) = 0 for all T . For more information on N(σ, T ), see the article on the density hypothesis25. A.2 Arithmetic L-functions Loosely speaking, arithmetic L-functions are those Dirichlet series with appropriate functional equations and Euler products which should satisfy a Riemann Hypothesis. Sel- berg has given specific requirements11 which seem likely to make this definition precise. Arithmetic L-functions arise in many situations: from the representation theory of groups associated with number fields, from automorphic forms on arithmetic groups acting on sym- metric spaces, and from the harmonic analysis on these spaces. A.2.a The Riemann zeta function. The Riemann zeta-function is defined by ζ(s) = ∞∑ n=1 1 ns = ∏ p (1− p−s)−1, where s = σ + it, the product is over the primes, and the series and product converge absolutely for σ > 1. 25page 16, The Density Hypothesis 11page 11, The Selberg class 8 The use of s = σ + it as a complex variable in the theory of the Riemann ζ-function has been standard since Riemann’s original paper. A.2.b Dirichlet L-functions. Dirichlet L-functions are Dirichlet series of the form L(s, χ) = ∞∑ n=1 χ(n) ns where χ is a primitive Dirichlet character to a modulus q. Equivalently, χ is a function from the natural numbers to the complex numbers, which is periodic with period q (i.e. χ(n + q) = χ(n) for all n ≥ 1), completely multiplicative (i.e. χ(mn) = χ(m)χ(n) for all natural numbers m and n), which vanishes at natural numbers which have a factor > 1 in common with q and which do not satisfy χ(m + q1) = χ(m) for all numbers m, q1 with q1 < q and (m, q1) = 1. The last condition gives the primitivity. Also, we do not consider the function which is identically 0 to be a character. If the modulus q = 1 then χ(n) = 1 for all n and L(s, χ) = ζ(s). If q > 1, then the series converges for all s with σ > 0 and converges absolutely for σ > 0. L(s, χ) has an Euler product L(s, χ) = ∏ p ( 1− χ)p ps )−1 . A.2.c Dedekind zeta functions. Let K be a number field (ie, a finite extension of the rationals Q), with ring of integers OK . The Dedekind zeta function of K is given by ζK(s) = ∑ a (Na)−s, for σ > 1, where the sum is over all integral ideals of OK , and Na is the norm of a. A.2.d GL(2) L-functions. We call GL(2) L-functions those Dirichlet series with functional equations and Euler products each of whose factors is the reciprocal of a degree two poly- nomial in p−s. These are associated with (i.e. their coefficients are the Fourier coefficients) of cusp forms on congruence subgroups of SL(2, Z) which are eigenfunctions of the Hecke operators and of the Atkin-Lehner operators (newforms). A.2.d.i Dirichlet series associated with holomorphic cusp forms. Level one modular forms. A cusp form of weight k for the full modular group is a holomorphic function f on the upper half-plane which satisfies f ( az + b cz + d ) = (cz + d)kf(z) for all integers a, b, c, d with ad − bc = 1 and also has the property that limy→∞ f(iy) = 0. Cusp forms for the whole modular group exist only for even integers k = 12 and k ≥ 16. The cusp forms of a given weight k of this form make a complex vector space Sk of dimension [k/12] if k 6= 2 mod 12 and of dimension [k/12]− 1 if k = 2 mod 12. Each such vector space has a special basisHk of Hecke eigenforms which consist of functions f(z) = ∑∞ n=1 af (n)e(nz) for which af (m)af (n) = ∑ d|(m,n) dk−1af (mn/d 2). 9 The Fourier coefficients af (n) are real algebraic integers of degree equal to the dimension of the vector space = #Hk. Thus, when k = 12, 16, 18, 20, 22, 26 the spaces are one dimensional and the coefficients are ordinary integers. The L-function associated with a Hecke form f of weight k is given by Lf (s) = ∞∑ n=1 af (n)/n (k−1)/2ns = ∏ p ( 1− af (p)/p (k−1)/2 ps + 1 p2s )−1 . By Deligne’s theorem af (p)/p (k−1)/2 = 2 cos θf (p) for a real θf (p). It is conjectured (Sato- Tate) that for each f the {θf (p) : p prime} is uniformly distributed on [0, pi) with respect to the measure 2 pi sin2 θdθ. We write cos θf (p) = αf (p) + αf (p) where αf (p) = e iθf (p); then Lf (s) = ∏ p ( 1− αf (p) ps )−1( 1− αf (p) ps )−1 . The functional equation satisfied by Lf (s) is ξf (s) = (2pi) −sΓ(s+ (k − 1)/2)Lf (s) = (−1)k/2ξf (1− s). Higher level forms. Let Γ0(q) denote the group of matrices ( a b c d ) with integers a, b, c, d satisfying ad− bc = 1 and q | c. This group is called the Hecke congruence group. A function f holomorphic on the upper half plane satisfying f ( az + b cz + d ) = (cz + d)kf(z) for all matrices in Γ0(q) and limy→∞ f(iy) = 0 is called a cusp form for Γ0(q); the space of these is a finite dimensional vector space Sk(q). The space Sk above is the same as Sk(1). Again, these spaces are empty unless k is an even integer. If k is an even integer, then dimSk(q) = (k − 1) 12 ν(q) + ([ k 4 ] − k − 1 4 ) ν2(q) + ([ k 3 ] − k − 1 3 ) ν3(q)− ν∞(q) 2 where ν(q) is the index of the subgroup Γ0(q) in the full modular group Γ0(1): ν(q) = q ∏ p|q ( 1 + 1 p ) ; ν∞(q) is the number of cusps of Γ0(q): ν∞(q) = ∑ d|q φ((d, q/d)); ν2(q) is the number of inequivalent elliptic points of order 2: ν2(q) = { 0 if 4 | q∏ p|q(1 + χ−4(p)) otherwise and ν3(q) is the number of inequivalent elliptic points of order 3: ν3(q) = { 0 if 9 | q∏ p|q(1 + χ−3(p)) otherwise. 10 It is clear from this formula that the dimension of Sk(q) grows approximately linearly with q and k. For the spaces Sk(q) the issue of primitive forms and imprimitive forms arise, much as the situation with characters. In fact, one should think of the Fourier coefficients of cusp forms as being a generalization of characters. They are not periodic, but they act as harmonic detectors, much as characters do, through their orthogonality relations (below). Imprimitive cusp forms arise in two ways. Firstly , if f(z) ∈ Sk(q), then f(z) ∈ Sk(dq) for any integer d > 1. Secondly, if f(z) ∈ Sk(q), then f(dz) ∈ Sk(Γ0(dq)) for any d > 1. The dimension of the subspace of primitive forms is given by dimSnewk (q) = ∑ d|q µ2(d) dimSk(q/d) where µ2(n) is the multiplicative function defined for prime powers by µ2(p e) = −2 if e = 1, = 1 if e = 2 , and = 0 if e > 2. The subspace of newforms has a Hecke basis Hk(q) consisting of primitive forms, or newforms, or Hecke forms. These can be identified as those f which have a Fourier series f(z) = ∞∑ n=1 af (n)e(nz) where the af (n) have the property that the associated L-function has an Euler product Lf (s) = ∞∑ n=1 af (n)/n (k−1)/2 ns = ∏ p-q ( 1− af (p)/p (k−1)/2 ps + 1 p2s )−1∏ p|q ( 1− af (p)/p (k−1)/2 ps )−1 . The functional equation satisfied by Lf (s) is ξf (s) = (2pi/ √ q)−sΓ(s+ (k − 1)/2)Lf (s) = ±(−1)k/2ξf (1− s). A.2.d.i.A Examples. See the website1 for many specific examples. Ramanujan’s tau-function defined implicitly by x ∞∏ n=1 (1− xn)24 = ∞∑ n=1 τ(n)xn also yields the simplest cusp form. The associated Fourier series ∆(z) := ∑∞ n=1 τ(n) exp(2piinz) satisfies ∆ ( az + b cz + d ) = (cz + d)12∆(z) for all integers a, b, c, d with ad− bc = 1 which means that it is a cusp form of weight 12 for the full modular group. The unique cusp forms of weights 16, 18, 20, 22, and 26 for the full modular group can be given explicitly in terms of (the Eisenstein series) E4(z) = 1 + 240 ∞∑ n=1 σ3(n)e(nz) 1http://www.math.okstate.edu/∼loriw/degree2/degree2hm/degree2hm.html 11 and E6(z) = 1− 504 ∞∑ n=1 σ5(n)e(nz) where σr(n) is the sum of the rth powers of the positive divisors of n: σr(n) = ∑ d|n dr. Then, ∆(z)E4(z) gives the unique Hecke form of weight 16; ∆(z)E6(z) gives the unique Hecke form of weight 18; ∆(z)E4(z) 2 is the Hecke form of weight 20; ∆(z)E4(z)E6(z) is the Hecke form of weight 22; and ∆(z)E4(z) 2E6(z) is the Hecke form of weight 26. The two Hecke forms of weight 24 are given by ∆(z)E4(z) 3 + x∆(z)2 where x = −156± 12√144169. An example is the L-function associated to an elliptic curve E : y2 = x3 + Ax + B where A,B are integers. The associated L-function, called the Hasse-Weil L-function, is LE(s) = ∞∑ n=1 a(n)/n1/2 ns = ∏ p-N ( 1− a(p)/p 1/2 ps + 1 p2s )−1∏ p|N ( 1− a(p)/p 1/2 ps )−1 where N is the conductor of the curve. The coefficients an are constructed easily from ap for prime p; in turn the ap are given by ap = p−Np where Np is the number of solutions of E when considered modulo p. The work of Wiles and others proved that these L-functions are associated to modular forms of weight 2. A.2.d.j Dirichlet series associated with Maass forms. A.2.e Higher rank L-functions. A.3 The Selberg class Selberg [94f:11085] has given an elegant a set of axioms which presumably describes exactly the set of arithmetic L-functions. He also made two deep conjectures82 about these L-functions which have far reaching consequences. The collection of Dirichlet series satisfying Selberg’s axioms is called “The Selberg Class.” This set has many nice properties. For example, it is closed under products. The elements which cannot be written as a nontrivial product are called “primitive,” and every member can be factored uniquely into a product of primitive elements. In some cases it is useful to slightly relax the axioms so that the set is closed under the operation F (s) 7→ F (s+ iy) for real y. Some of the important problems concerning the Selberg Class are: 1. Show that the members of the Selberg Class are arithmetic L-functions. 2. Prove a prime number theorem26 for members of the Selberg class. 82page 12, Selberg Conjectures 26page 16, Zeros on the σ = 1 line 12 See Perelli and Kaczorowski [MR 2001g:11141], Conrey and Ghosh [95f:11064] and Chapter 7 of Murty and Murty [98h:11106] for more details and some additional consequences of Selberg’s conjectures. A.3.a Axiom 1: Dirichlet series. For <(s) > 1, F (s) = ∞∑ n=1 an ns . A.3.b Axiom 2: Analytic Continuation. F (s) extends to a meromor