Generalized Riemann hypotheses

The Riemann hypothesis is one of the most important hypotheses in mathematics . The hypothesis is a statement about the zeros of the Riemann zeta function . Various geometric and arithmetic objects can be described by the so-called global L-functions , which are formally similar to the Riemann zeta function. We can then ask the same question about the roots of these L- functions, which gives different generalizations of the Riemann hypothesis. Many mathematicians believe in the validity of these generalizations of the Riemann hypothesis . The only case when such a hypothesis was proved occurred in the not (not in the case of the number field).

Global L -functions can be associated with elliptic curves , numerical fields (in this case they are called Dedekind zeta functions ), and Dirichlet characters (in this case they are called Dirichlet L-functions ). When the Riemann hypothesis is formulated for the Dedekind zeta-functions, it is called the extended Riemann hypothesis (RGR), and when it is formulated for the Dirichlet L- functions, it is known as the generalized Riemann hypothesis (RGR). These two statements are discussed in more detail below. Many mathematicians use the name generalized Riemann hypothesis to extend the Riemann hypothesis to all global L -functions, not only a special case of Dirichlet L- functions.

Content

The Generalized Riemann Hypothesis (OGR)

The generalized Riemann hypothesis (for Dirichlet L- functions) was apparently first formulated by in 1884 ^[1] . Like the original Riemann hypothesis, the generalized hypothesis has far-reaching consequences on the distribution of primes .

Formal statement of a hypothesis . The Dirichlet character is a completely multiplicative arithmetic function χ such that there exists a positive integer k with χ ( n + k ) = χ ( n ) for all n and χ ( n ) = 0 if gcd ( n , k )> 1. If such a character is given, we determine the corresponding Dirichlet L-function

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

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

for any complex number s with the real part> 1. Using analytic continuation, this function can be extended to a meromorphic function defined on the entire complex plane. The generalized Riemann hypothesis states that for any Dirichlet character χ and any complex number s with L (χ, s ) = 0, the following holds: if the real number s is between 0 and 1, then it is actually 1/2.

The case χ ( n ) = 1 for all n gives the usual Riemann hypothesis.

OGR Consequences

Dirichlet’s theorem states that when a and d are coprime natural numbers , then the arithmetic progression a , a + d , a +2 d , a +3 d , ... contains infinitely many primes. Let π ( x , a , d ) denote the number of progressive primes that are less than or equal to x . If the generalized Riemann hypothesis is true, then for any coprime a and d and any ε> 0

\pi (x,a,d)={\frac {1}{\varphi (d)}}\int _{2}^{x}{\frac {1}{\ln t}}\,dt+O(x^{1/2+\epsilon })

{\ displaystyle \ pi (x, a, d) = {\ frac {1} {\ varphi (d)}} \ int _ {2} ^ {x} {\ frac {1} {\ ln t}} \ , dt + O (x ^ {1/2 + \ epsilon})}

at

x\to \infty

{\ displaystyle x \ to \ infty}

,

where φ ( d ) is the Euler function , and $O$ ${\ displaystyle O}$ - “O” is big . This is a significant strengthening of the prime number distribution theorem .

If the OGR is true, then any proper subgroup of the multiplicative group $(\mathbb {Z} /n\mathbb {Z} )^{\times }$ ${\ displaystyle (\ mathbb {Z} / n \ mathbb {Z}) ^ {\ times}}$ does not contain numbers less than 2 (ln n ) ² , as well as numbers coprime to n and less than 3 (ln n ) ² ^[2] . In other words, $(\mathbb {Z} /n\mathbb {Z} )^{\times }$ ${\ displaystyle (\ mathbb {Z} / n \ mathbb {Z}) ^ {\ times}}$ generated by a set of numbers less than 2 (ln n ) ² . This fact is often used in evidence and many consequences follow from it, for example (assuming that the OGR is true):

The Miller-Rabin test is guaranteed to work for polynomial time (A test with polynomial time that does not require OGR, the Agrawal-Kayal-Saxen test , was published in 2002).
The Gelfond - Shanks algorithm is guaranteed to work in polynomial time.
The Ivanuos – Karpinski – Sachen deterministic algorithm ^[3] for expanding polynomials over finite fields with simple degree n and smooth n - 1 works in polynomial time.

If the OGR is true, then for any prime p there exists a primitive root modulo p (the generator of the multiplicative group of integers modulo p ) is smaller $O((\ln p)^{6}).\,$ ${\ displaystyle O ((\ ln p) ^ {6}). \,}$ ^[4] .

The weak Goldbach hypothesis also follows from the generalized Riemann hypothesis. Harald Helfgott's proof of this hypothesis confirms the OGR for several thousand small characters, which made it possible to prove the hypothesis for all integers (odd) numbers greater than 10 ²⁹ . For integers below this boundary, the hypothesis was verified by direct enumeration ^[5] .

Assuming that the OGR is true, the estimate of the sum of characters in can be improved to $O\left({\sqrt {q}}\log \log q\right)$ ${\ displaystyle O \ left ({\ sqrt {q}} \ log \ log q \ right)}$ , where q is the character modulus.

Extended Riemann Hypothesis (RGR)

Let K be a number field (a finite-dimensional extension of the field of rational numbers Q ) with a ring of integers O _K (this ring is the whole closure of the integers Z in K ). If a is an ideal of the ring O _K other than the zero ideal, we denote its by Na . The Dedekind zeta function over K is then defined as

\zeta _{K}(s)=\sum _{a}{\frac {1}{(Na)^{s}}}

{\ displaystyle \ zeta _ {K} (s) = \ sum _ {a} {\ frac {1} {(Na) ^ {s}}}}

for any complex number s with the real part> 1.

The Dedekind zeta function satisfies the functional equation and can be extended by analytic continuation to the entire complex plane. The resulting function encodes important information about the number field K. The extended Riemann hypothesis states that for any number field K and any complex number s for which ζ _K ( s ) = 0, it holds: if the real part of s lies between 0 and 1, then it is actually 1 / 2.

The initial Riemann hypothesis follows from the extended hypothesis if we take the number field Q with the ring of integers Z.

The RGR implies an effective version ^[6] : if L / K is a finite Galois extension with a Galois group G , and C is the union of the cosets G , the number of ideals K with norm below x with the Frobenius adjacency class in C is

{\frac {|C|}{|G|}}{\Bigl (}\mathrm {li} (x)+O{\bigl (}{\sqrt {x}}(n\log x+\log |\Delta |){\bigr )}{\Bigr )},

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

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

Notes

↑ Davenport, 2000 , p. 124.
↑ Bach, 1990 , p. 355-380.
↑ Ivanyos, Karpinski, Saxena, 2009 , p. 191–198.
↑ Shoup, 1992 , p. 369-380.
↑ Helfgott, 2013 .
↑ Lagarias, Odlyzko, 1977 , p. 409-464.

Literature

Lagarias JC, Odlyzko AM Effective Versions of the Chebotarev Theorem // Algebraic Number Fields. - 1977. - S. 409-464 .
Eric Bach. Explicit bounds for primality testing and related problems // Mathematics of Computation . - 1990. - T. 55 , no. 191 . - S. 355-380 . - DOI : 10.2307 / 2008811 .
Gabor Ivanyos, Marek Karpinski, Nitin Saxena. Schemes for Deterministic Polynomial Factoring // Proc. ISAAC - 2009. - S. 191–198 . - ISBN 9781605586090 . - DOI : 10.1145 / 1576702.1576730 .
Helfgott HA Major arcs for Goldbach's theorem . - 2013 .-- arXiv : 1305.2897v3 .
Victor Shoup. Searching for primitive roots in finite fields // Mathematics of Computation. - 1992. - Vol. 58. - Vol. 197 . - S. 369-380 . - DOI : 10.2307 / 2153041 .
Harold Davenport. Multiplicative number theory. - Third edition, Revised and with a preface by Hugh L. Montgomery . - New York: Springer-Verlag, 2000 .-- T. 74 .-- C. xiv + 177. - (Graduate Texts in Mathematics). - ISBN 0-387-95097-4 .