Generalized Riemann hypothesis

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)

Main article: Dirichlet L-function

Generalized Riemann hypothesis asserts that all nontrivial zeros of Dirichlet L-function ${\textstyle L(\chi ,s)}$ for primitive Dirichlet character ${\displaystyle \chi }$ have real part ${\textstyle {\frac {1}{2}}}$.

The generalized Riemann hypothesis for Dirichlet L-functions was probably formulated for the first time by Adolf Piltz in 1884. ^[1] 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 ${\textstyle \chi$ :\mathbb {Z} \rightarrow \mathbb {C} } of modulus q is arithmetic function that is:

• completely multiplicative: ${\textstyle \chi (a\cdot b)=\chi (a)\cdot \chi (b)}$
• periodic: ${\textstyle \chi (n+q)=\chi (n)}$
• ${\textstyle \chi (n)=0}$ if and only if ${\textstyle \gcd(n,k)>1}$.

If such character ${\textstyle \chi }$, we define the corresponding Dirichlet L-function by:

${\displaystyle L(\chi ,s)=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}}$

For every complex number s such that Re s > 1 this series is absolutely convergent. By analytic continuation, this function can be extended to meromorphic function on complex plane having only possible pole in ${\textstyle s=1}$, when character is principal (have only 1 as value for numbers coprime to k). For nonprincipal character, series is conditionally convergent for ${\textstyle \operatorname {Re} (s)>0}$ and analytic continuation is entire function.

We say that Dirichlet character ${\textstyle \chi }$ is inprimitive if it is induced by another Dirichlet ${\textstyle \chi ^{\star }}$ character of lesser modulus:

${\displaystyle \chi (n)={\begin{cases}\chi ^{\star }(n),&\mathrm {if} \gcd(n,q)=1\\0,&\mathrm {if} \gcd(n,q)\neq 1\end{cases}}}$

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 ${\textstyle \chi }$ by function of character ${\textstyle \chi ^{\star }}$ that induces it:

${\displaystyle L(s,\chi )=L(s,\chi ^{\star })\prod _{p\,|\,q}\left(1-{\frac {\chi ^{\star }(p)}{p^{s}}}\right)}$

From factors in this equation we have infinitely many zeros on line: ${\textstyle \operatorname {Re} (s)=0}$. For primitive Dirichlet character L-function satisfies certain functional equation which allows us to define trivial zeros of ${\textstyle L(s,\chi )}$ as zeros corresponding to poles of gamma function in this equation:

• If ${\textstyle \chi (-1)=1}$, then all trivial zeros are simple zeros in negative even numbers. If ${\textstyle L(s,\chi )\neq \zeta (s)}$ it also includes 0.
• If ${\textstyle \chi (-1)=-1}$ 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: ${\textstyle 0<\operatorname {Re} (s)<1}$ and are symmetric with respect to critical line ${\textstyle \operatorname {Re} (s)={\tfrac {1}{2}}}$. 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 ${\textstyle \chi (n)=1}$ yields the ordinary Riemann hypothesis.
• More effective version of Dirichlet's theorem on arithmetic progressions: Let ${\textstyle \pi (x,a,d)}$ where a and d are coprime denote the number of prime numbers in arithmetic progression ${\textstyle n\cdot d+a}$ which are less than or equal to x. If the generalized Riemann hypothesis is true, then for every ε > 0:

${\displaystyle \pi (x,a,d)={\frac {1}{\varphi (d)}}\int _{2}^{x}{\frac {1}{\ln t}}\,dt+O(x^{1/2+\varepsilon })\quad {\mbox{ as }}\ x\to \infty ,}$

where ${\displaystyle \varphi }$ is Euler's totient function and ${\displaystyle O}$ is the Big O notation. This is a considerable strengthening of the prime number theorem.

• Every proper subgroup of the multiplicative group ${\textstyle (\mathbb {Z} /n\mathbb {Z} )^{\times }}$ has set of generators less than ${\textstyle 2\ln(n)^{2}}$. In other words, every subgroup of multiplicative group omits a number less than ${\textstyle 2\ln(n)^{2}}$, as well as a number coprime to ${\displaystyle n}$ less than ${\textstyle 3\ln(n)^{2}}$. ^[2] 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 ^[3] 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 ${\displaystyle O((\ln p)^{6}).}$ ^[4]
• Estimate of the character sum in the Pólya–Vinogradov inequality can be improved to ${\textstyle O\left({\sqrt {q}}\log \log q\right)}$, 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 $${\displaystyle \lim _{x\to 1^{-}}\sum _{p>2}(-1)^{(p+1)/2}x^{p}=+\infty ,}$$ which says that primes 3 mod 4 are more common than primes 1 mod 4 in some sense. (For related results, see Prime number theorem § Prime number race .)
• 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. ^[5]
• In 1934, Chowla showed that the generalized Riemann hypothesis implies that the first prime in the arithmetic progression a mod m is at most ${\textstyle Km^{2}\log(m)^{2}}$ 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.
• Ono & Soundararajan (1997) showed that the generalized Riemann hypothesis implies that Ramanujan's integral quadratic form x² + y² + 10z² 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. ^{[6] [7]}

Extended Riemann hypothesis (ERH)

Suppose ${\textstyle K}$ is a number field with ring of integers ${\textstyle O_{K}}$ (this ring is the integral closure of the integers ${\displaystyle \mathbb {Z} }$ in K). If ${\textstyle I}$ is a nonzero ideal of ${\textstyle O_{K}}$, we denote its norm by ${\textstyle N(I)}$. The Dedekind zeta-function of K is then defined by:

${\displaystyle \zeta _{K}(s)=\sum _{I\subseteq O_{K}}{\frac {1}{N(I)^{s}}}}$

for every complex number s with real part > 1. The sum extends over all non-zero ideals ${\textstyle I}$ of ${\textstyle O_{K}}$. That function can be extended by analytic continuation to the meromorphic function on complex plane with only possible pole at ${\textstyle s=1}$ and satisfies a functional equation that gives exact location of trivial zeroes and guarantees that nontrivial zeros lie inside critical strip ${\textstyle 0\leq Re(s)\leq 1}$ and are symmetric with respect to critical line: ${\textstyle Re(s)={\tfrac {1}{2}}}$.

The extended Riemann hypothesis asserts that for every number field K each nontrivial zero of ${\textstyle \zeta _{K}}$ have real part ${\textstyle {\tfrac {1}{2}}}$ and then lie on critical line.

Consequences of ERH

• The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be ${\displaystyle \mathbb {Q} }$, whose ring of integers is: ${\displaystyle O_{\mathbb {Q} }=\mathbb {Z} }$.
• Generalized Riemann hypothesis for Dirichlet L-functions is equivalent to ERH for K being abelian extension of rational numbers, since for abelian extensions ${\textstyle \zeta _{K}}$ is finite product of some Dirichlet L-functions depending on K. Conversely, all L-functions for character modulo n appears in product for ${\textstyle K=\mathbb {Q} (\zeta _{n})}$, where ${\textstyle \zeta _{n}}$ 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 ^[8] 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

${\displaystyle {\frac {|C|}{|G|}}{\Bigl (}\operatorname {Li} (x)+O{\bigl (}{\sqrt {x}}(n\log x+\log |\Delta |){\bigr )}{\Bigr )},}$

where the constant implied in the big-O notation is absolute, n is the degree of L over Q, and Δ its discriminant.

• Weinberger (1973) 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.
• Odlyzko (1990) 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 ${\textstyle F(s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}}$ is in Selberg class if it satisfies following properties:

• Analyticity: ${\displaystyle F(s)}$ has a meromorphic continuation to the entire complex plane, with the only possible pole (if any) in ${\textstyle s=1}$.
• Ramanujan conjecture: a₁ = 1 and ${\displaystyle a_{n}\ll _{\varepsilon }n^{\varepsilon }}$ for any ε > 0;
• Functional equation: there is a gamma factor of the form

${\displaystyle \gamma (s)=Q^{s}\prod _{i=1}^{k}\Gamma (\omega _{i}s+\mu _{i})}$

where ${\textstyle Q}$ is real and positive, ${\textstyle \Gamma }$ the gamma function, the ${\textstyle \omega _{i}}$ real and positive, and the ${\textstyle \mu _{i}}$ complex with non-negative real part, as well as a so-called root number: ${\textstyle \alpha \in \mathbb {C} ,\;|\alpha |=1}$, such that the function:

${\displaystyle \Phi (s)=\gamma (s)F(s)\,}$

satisfies:

${\displaystyle \Phi (s)=\alpha \,{\overline {\Phi (1-{\overline {s}})}};}$

• Euler product: For Re(s) > 1, F(s) can be written as a product over primes:

${\displaystyle F(s)=\prod _{p}F_{p}(s)}$

with

${\displaystyle F_{p}(s)=\exp \left(\sum _{n=1}^{\infty }{\frac {b_{p^{n}}}{p^{ns}}}\right)}$

and, for some ${\textstyle \theta <{\tfrac {1}{2}}}$,

${\displaystyle b_{p^{n}}=O(p^{n\theta }).\,}$

From analicity follows that poles of gamma factor in ${\textstyle Re(s)<1}$ must be cancelled by zeros of ${\textstyle F(s)}$, that zeros are called trivial zeros. Functional equation guarantees that all nontrivial zeros lie in critical strip ${\textstyle 0<Re(s)<1}$ and are symmetric with respect to critical line ${\textstyle Re(s)={\tfrac {1}{2}}}$.

Generalized Riemann hypothesis for Selberg class states that all nontrivial zeros of function ${\textstyle F}$ belonging to Selberg class have real part ${\textstyle {\tfrac {1}{2}}}$ and then lie on critical line.

Selberg class along with proposition of Riemann hypothesis for it was firs introduced in ( Selberg 1992 ). 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 ${\displaystyle Re(s)=6}$, function obtained by translation of ${\displaystyle {\tfrac {11}{2}}}$ is in Selberg class.

See also

• Artin's conjecture
• Artin L-function
• Dirichlet L-function
• Dedekind zeta function
• Selberg class
• Grand Riemann hypothesis

References

1. Davenport, Harold (2000). Multiplicative Number Theory. Graduate Texts in Mathematics. Vol. 74. Revised and with a preface by Hugh L. Montgomery (Third ed.). New York: Springer-Verlag. p. 124. ISBN   0-387-95097-4.
2. Bach, Eric (1990). "Explicit bounds for primality testing and related problems". Mathematics of Computation . 55 (191): 355–380. doi: 10.2307/2008811 . JSTOR   2008811.
3. Ivanyos, Gabor; Karpinski, Marek; Saxena, Nitin (2009). "Schemes for deterministic polynomial factoring". Proceedings of the 2009 international symposium on Symbolic and algebraic computation (ISAAC). pp. 191–198. arXiv: 0804.1974 . doi:10.1145/1576702.1576730. ISBN   9781605586090. S2CID   15895636.
4. Shoup, Victor (1992). "Searching for primitive roots in finite fields". Mathematics of Computation. 58 (197): 369–380. doi: 10.2307/2153041 . JSTOR   2153041.
5. p5. Helfgott, Harald (2013). "Major arcs for Goldbach's theorem". arXiv: 1305.2897 [math.NT].
6. "Caltech Mathematicians Solve 19th Century Number Riddle". California Institute of Technology. October 31, 2022.
7. Dunn, Alexander; Radziwiłł, Maksym (2021). "Bias in cubic Gauss sums: Patterson's conjecture". arXiv: 2109.07463 [math.NT].
8. Lagarias, J.C.; Odlyzko, A.M. (1977). "Effective Versions of the Chebotarev Theorem". Algebraic Number Fields: 409–464.

• Ono, Ken; Soundararajan, K. (1997), "Ramanujan's ternary quadratic form", Inventiones Mathematicae , 130 (3): 415–454, Bibcode:1997InMat.130..415O, doi:10.1007/s002220050191, S2CID   122314044
• Odlyzko, A. M. (1990), "Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results", Séminaire de Théorie des Nombres de Bordeaux, Série 2, 2 (1): 119–141, doi: 10.5802/jtnb.22 , MR   1061762
• Weinberger, Peter J. (1973), "On Euclidean rings of algebraic integers", Analytic number theory ( St. Louis Univ., 1972), Proc. Sympos. Pure Math., vol. 24, Providence, R.I.: Amer. Math. Soc., pp. 321–332, MR   0337902
• Selberg, Atle (1992), "Old and new conjectures and results about a class of Dirichlet series", Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Salerno: Univ. Salerno, pp. 367–385, MR   1220477, Zbl   0787.11037 Reprinted in Collected Papers, vol 2, Springer-Verlag, Berlin (1991)

Further reading

• "Riemann hypothesis, generalized", Encyclopedia of Mathematics , EMS Press, 2001 [1994]