Generalized Riemann hypothesis
Template:Short description The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these L-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the algebraic function field case (not the number field case).
Global L-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothesis (ERH) and when it is formulated for Dirichlet L-functions, it is known as the generalised Riemann hypothesis (GRH). Another approach to generalization of Riemann hypothesis was given by Atle Selberg and his introduction of class of function satisfying certain properties rather than specific functions, nowadays known as Selberg class. These three statements will be discussed in more detail below. (Many mathematicians use the label generalized Riemann hypothesis to cover the extension of the Riemann hypothesis to all global L-functions, not just the special case of Dirichlet L-functions.)
Generalized Riemann hypothesis (GRH)
Generalized Riemann hypothesis asserts that all nontrivial zeros of Dirichlet L-function <math display=inline>L(\chi,s)</math> for primitive Dirichlet character <math>\chi</math> have real part <math display=inline>\frac{1}{2}</math>.
The generalized Riemann hypothesis for Dirichlet L-functions was probably formulated for the first time by Adolf Piltz in 1884.<ref>Template:Cite book</ref> It is important to assume primitivity of character since for nonprimitive characters L-functions have infinitely many zeros off this line and don't satisfy functional equation that is used to distinguish between trivial and nontrivial zeros.
Background
A Dirichlet character <math display=inline>\chi:\mathbb{Z}\rightarrow \mathbb{C}</math> of modulus q is arithmetic function that is:
- completely multiplicative: <math display=inline>\chi(a\cdot b)=\chi(a)\cdot\chi(b)</math>
- periodic: <math display=inline>\chi(n+q)=\chi(n)</math>
- <math display=inline>\chi(n)=0</math> if and only if <math display=inline>\gcd(n, q) > 1</math>.
If such character <math display=inline>\chi</math>, we define the corresponding Dirichlet L-function by:
- <math>
L(\chi,s) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s} </math> For every complex number s such that Template:Nowrap this series is absolutely convergent. By analytic continuation, this function can be extended to meromorphic function on complex plane having only possible pole in <math display=inline>s=1</math>, when character is principal (have only 1 as value for numbers coprime to k). For nonprincipal character, series is conditionally convergent for <math display=inline>\operatorname{Re}(s)>0</math> and analytic continuation is entire function.
We say that Dirichlet character <math display=inline>\chi</math> is inprimitive if it is induced by another Dirichlet <math display=inline>\chi^\star</math> character of lesser modulus:
- <math>
\chi(n) =
\begin{cases}
\chi^\star(n), & \mathrm{if} \gcd(n,q) = 1 \\
0, & \mathrm{if} \gcd(n,q) \ne 1
\end{cases}
</math> Otherwise we say that character is primitive. Generally most statements for Dirichlet L-functions are easier to express for versions with primitive characters. Using Euler product of Dirichlet L-functions we can express L-function of imprimitive character <math display=inline>\chi</math> by function of character <math display=inline>\chi^\star</math> that induces it:
- <math>
L(s,\chi) = L(s,\chi^\star) \prod_{p \,|\, q}\left(1 - \frac{\chi^\star(p)}{p^s} \right)
</math> From factors in this equation we have infinitely many zeros on line: <math display=inline>\operatorname{Re}(s)=0</math>. For primitive Dirichlet character L-function satisfies certain functional equation which allows us to define trivial zeros of <math display=inline>L(s,\chi)</math> as zeros corresponding to poles of gamma function in this equation:
- If <math display=inline>\chi(-1)=1</math>, then all trivial zeros are simple zeros in negative even numbers. If <math display=inline>L(s,\chi)\neq \zeta(s)</math> it also includes 0.
- If <math display=inline>\chi(-1)=-1</math> then all trivial zeros are simple zeros in negative odd numbers.
Any other zeros are called nontrivial zeros. Functional equation guarantees that nontrivial zeros lies in critical strip: <math display=inline>0 < \operatorname{Re}(s) < 1</math> and are symmetric with respect to critical line <math display=inline>\operatorname{Re}(s) = \tfrac{1}{2}</math>. Generalized Riemann Hypothesis says that all nontrivial zeros lies exactly on this line.
Consequences of GRH
Like the original Riemann hypothesis, GRH has far reaching consequences about the distribution of prime numbers:
- Taking trivial character <math display=inline>\chi(n)=1</math> yields the ordinary Riemann hypothesis.
- More effective version of Dirichlet's theorem on arithmetic progressions: Let <math display=inline>\pi(x,a,d)</math> where a and d are coprime denote the number of prime numbers in arithmetic progression <math display=inline>n\cdot d + a</math> which are less than or equal to x. If the generalized Riemann hypothesis is true, then for every Template:Nowrap:
- <math>\pi(x,a,d) = \frac{1}{\varphi(d)} \int_2^x \frac{1}{\ln t}\,dt + O(x^{1/2+\varepsilon})\quad\mbox{ as } \ x\to\infty,</math>
- where <math>\varphi</math> is Euler's totient function and <math>O</math> is the Big O notation. This is a considerable strengthening of the prime number theorem.
- Every proper subgroup of the multiplicative group <math display=inline>(\mathbb Z/n\mathbb Z)^\times</math> has set of generators less than <math display=inline>2\ln(n)^2</math>. In other words, every subgroup of multiplicative group omits a number less than <math display=inline>2\ln(n)^2</math>, as well as a number coprime to <math>n</math> less than <math display=inline>3\ln(n)^2</math>.<ref>Template:Cite journal</ref> This has many consequences in computational number theory:
- In 1976, G. Miller showed that Miller-Rabin test is guaranteed to run in polynomial time. In 2002, Manindra Agrawal, Neeraj Kayal and Nitin Saxena proved unconditionally that AKS primality test is guaranteed to run in polynomial time.
- The Shanks–Tonelli algorithm is guaranteed to run in polynomial time.
- The Ivanyos–Karpinski–Saxena deterministic algorithm<ref>Template:Cite book</ref> for factoring polynomials over finite fields with prime constant-smooth degrees is guaranteed to run in polynomial time.
- For every prime p there exists a primitive root mod p (a generator of the multiplicative group of integers modulo p) that is less than <math>O((\ln p)^6).</math><ref>Template:Cite journal</ref>
- Estimate of the character sum in the Pólya–Vinogradov inequality can be improved to <math display=inline>O\left(\sqrt{q}\log\log q\right)</math>, q being the modulus of the character.
- In 1913, Grönwall showed that the generalized Riemann hypothesis implies that Gauss's list of imaginary quadratic fields with class number 1 is complete, though Baker, Stark and Heegner later gave unconditional proofs of this without using the generalized Riemann hypothesis.
- In 1917, Hardy and Littlewood showed that the generalized Riemann hypothesis implies a conjecture of Chebyshev that <math display="block">\lim_{x\to 1^-} \sum_{p>2}(-1)^{(p+1)/2} x^p=+\infty,</math> which says that primes 3 mod 4 are more common than primes 1 mod 4 in some sense. (For related results, see Template:Slink.)
- In 1923, Hardy and Littlewood showed that the generalized Riemann hypothesis implies Goldbach weak conjecture for sufficently large odd numbers. In 1997 Deshouillers, Effinger, te Riele, and Zinoviev showed that actually 5 is sufficiently large, so GRH implies weak Goldbach conjecture. In 1937 Vinogradov gave an unconditional proof for sufficiently large odd numbers. The yet to be verified proof of Harald Helfgott improved Vinogradov's method by verifying GRH for several thousand small characters up to a certain imaginary part to prove the conjecture for all integers above 1029, integers below which have already been verified by calculation.<ref>p5. Template:Cite arXiv</ref>
- In 1934, Chowla showed that the generalized Riemann hypothesis implies that the first prime in the arithmetic progression a mod m is at most <math display=inline>Km^2\log(m)^2</math> for some fixed constant K.
- In 1967, Hooley showed that the generalized Riemann hypothesis implies Artin's conjecture on primitive roots.
- In 1973, Weinberger showed that the generalized Riemann hypothesis implies that Euler's list of idoneal numbers is complete.
- Template:Harvtxt showed that the generalized Riemann hypothesis implies that Ramanujan's integral quadratic form Template:Nowrap represents all integers that it represents locally, with exactly 18 exceptions.
- In 2021, Alexander (Alex) Dunn and Maksym Radziwill proved Patterson's conjecture on cubic Gauss sums, under the assumption of the GRH.<ref>Template:Cite web</ref><ref name="Dunn Radziwiłł 2021 Patterson ">Template:Cite arXiv</ref>
Extended Riemann hypothesis (ERH)
Suppose <math display=inline>K</math> is a number field with ring of integers <math display=inline>O_{K}</math> (this ring is the integral closure of the integers <math>\mathbb Z</math> in K). If <math display=inline>I</math> is a nonzero ideal of <math display=inline>O_{K}</math>, we denote its norm by <math display=inline>N(I)</math>. The Dedekind zeta-function of K is then defined by:
- <math>
\zeta_K(s) = \sum_{I\subseteq O_K} \frac{1}{N(I)^s} </math> for every complex number s with real part > 1. The sum extends over all non-zero ideals <math display=inline>I</math> of <math display=inline>O_{K}</math>. That function can be extended by analytic continuation to the meromorphic function on complex plane with only possible pole at <math display=inline>s=1</math> and satisfies a functional equation that gives exact location of trivial zeroes and guarantees that nontrivial zeros lie inside critical strip <math display=inline> 0 \leq Re(s) \leq 1 </math> and are symmetric with respect to critical line: <math display=inline>Re(s)=\tfrac{1}{2}</math>.
The extended Riemann hypothesis asserts that for every number field K each nontrivial zero of <math display=inline>\zeta_K</math> has real part <math display=inline>\tfrac{1}{2}</math> (and thus lies on the critical line).
Consequences of ERH
- The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be <math>\mathbb Q</math>, whose ring of integers is: <math>O_{\mathbb Q} = \mathbb Z</math>.
- Generalized Riemann hypothesis for Dirichlet L-functions is equivalent to ERH for K being abelian extension of rational numbers, since for abelian extensions <math display=inline>\zeta_K</math> is finite product of some Dirichlet L-functions depending on K. Conversely, all L-functions for character modulo n appears in product for <math display=inline>K=\mathbb{Q}(\zeta_n)</math>, where <math display=inline>\zeta_n</math> is n-th primitive root of unity.
- For general extensions, similar role to Dirichlet L-functions is played by Artin L-functions. Then, ERH is equivalent to Riemann Hypothesis for Artin L-functions.
- The ERH implies an effective version<ref>Template:Cite journal</ref> of the Chebotarev density theorem: if L/K is a finite Galois extension with Galois group G, and C a union of conjugacy classes of G, the number of unramified primes of K of norm below x with Frobenius conjugacy class in C is
- <math>\frac{|C|}{|G|}\Bigl(\operatorname{Li}(x)+O\bigl(\sqrt x(n\log x+\log|\Delta|)\bigr)\Bigr),</math>
- where the constant implied in the big-O notation is absolute, n is the degree of L over Q, and Δ its discriminant.
- Template:Harvtxt showed that ERH implies that any number field with class number 1 is either Euclidean or an imaginary quadratic number field of discriminant −19, −43, −67, or −163.
- Template:Harvtxt discussed how the ERH can be used to give sharper estimates for discriminants and class numbers of number fields.
Generalized Riemann hypothesis for Selberg class
Selberg class is defined the following way:
We say that Dirichlet series <math display=inline>F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}</math> is in Selberg class if it satisfies following properties:
- Analyticity: <math>F(s)</math> has a meromorphic continuation to the entire complex plane, with the only possible pole (if any) in <math display=inline>s=1</math>.
- Ramanujan conjecture: a1 = 1 and <math>a_n \ll_\varepsilon n^\varepsilon</math> for any ε > 0;
- Functional equation: there is a gamma factor of the form
- <math>\gamma(s)=Q^s\prod_{i=1}^k \Gamma (\omega_is+\mu_i)</math>
- where <math display=inline>Q</math> is real and positive, <math display=inline>\Gamma</math> the gamma function, the <math display=inline>\omega_i</math> real and positive, and the <math display=inline>\mu_i</math> complex with non-negative real part, as well as a so-called root number: <math display=inline>\alpha\in\mathbb C,\;|\alpha|=1</math>, such that the function:
- <math>\Phi(s) = \gamma(s) F(s)\,</math>
- satisfies:
- <math>\Phi(s)=\alpha\,\overline{\Phi(1-\overline{s})};</math>
- Euler product: For Re(s) > 1, F(s) can be written as a product over primes:
- <math>F(s)=\prod_p F_p(s)</math>
- with
- <math> F_p(s)=\exp\left(\sum_{n=1}^\infty\frac{b_{p^n}}{p^{ns}}\right)</math>
- and, for some <math display=inline>\theta < \tfrac{1}{2}</math>,
- <math>b_{p^n}=O(p^{n\theta}).\,</math>
From analyticity follows that poles of gamma factor in <math display=inline>Re(s)<1</math> must be cancelled by zeros of <math display=inline>F(s)</math>, that zeros are called trivial zeros. Functional equation guarantees that all nontrivial zeros lie in critical strip <math display=inline>0< Re(s)<1</math> and are symmetric with respect to critical line <math display=inline>Re(s)=\tfrac{1}{2}</math>.
Generalized Riemann hypothesis for Selberg class states that all nontrivial zeros of function <math display=inline>F</math> belonging to Selberg class have real part <math display=inline>\tfrac{1}{2}</math> and then lie on critical line.
Selberg class along with proposition of Riemann hypothesis for it was firs introduced in Template:Harv. Instead of considering specific functions, Selberg approach was to give axiomatic definition consisting of properties characterizing most of objects called L-functions or zeta functions and expected to satisfy counterparts or generalizations of Riemann hypothesis.
Consequences
- Artin L-functions and Dedekind zeta functions belong to Selberg class, then Riemann Hypothesis for Selberg class implies extended Riemann hypothesis.
- Nontrivial zeros for much more general L-functions than Dedekind zeta functions lie on critical lines. One example can be Ramanujan L-function related to modular form called Dedekind eta function. Despite Ramanujan L-function itself don't belong to Selberg class and its critical line is <math>Re(s)=6</math>, function obtained by translation of <math>\tfrac{11}{2}</math> is in Selberg class.
See also
- Artin's conjecture
- Artin L-function
- Dirichlet L-function
- Dedekind zeta function
- Selberg class
- Grand Riemann hypothesis
References
- Template:Citation
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- Template:Citation Reprinted in Collected Papers, vol 2, Springer-Verlag, Berlin (1991)