“Riemann zeta function”的意思、由来-开放百科全书

The Riemann zeta function or Euler–Riemann zeta function, {{math|ζ(s)}}, is a function of a complex variable s that analytically continues the sum of the Dirichlet series

which converges when the real part of {{mvar|s}} is greater than 1. More general representations of {{math|ζ(s)}} for all {{mvar|s}} are given below. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.

As a function of a real variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis, which was not available at the time. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers.^[2]

The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, {{math|ζ(2)}}, provides a solution to the Basel problem. In 1979 Apéry proved the irrationality of {{math|ζ(3)}}. The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet {{mvar|L}}-functions and {{mvar|L}}-functions, are known.

Definition

The Riemann zeta function {{math|ζ(s)}} is a function of a complex variable {{math|s {{=}} σ + it}}. (The notation {{mvar|s}}, {{mvar|σ}}, and {{mvar|t}} is used traditionally in the study of the zeta function, following Riemann.)

The following infinite series converges for all complex numbers {{mvar|s}} with real part greater than 1, and defines {{math|ζ(s)}} in this case:

The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of {{mvar|s}} is greater than {{sfrac|1|2}}.

Universality

The critical strip of the Riemann zeta function has the remarkable property of universality. This zeta-function universality states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable. The first proof of universality was provided by Sergei Mikhailovitch Voronin in 1975.^[13] More recent work has included effective versions of Voronin's theorem^[14] and extending it to Dirichlet L-functions.^[15]^[16]

Estimates of the maximum of the modulus of the zeta function

Let the functions {{math|F(T;H)}} and {{math|G(s₀;Δ)}} be defined by the equalities

Here {{mvar|T}} is a sufficiently large positive number, {{math|0 < H ≪ ln ln T}}, {{math|s₀ {{=}} σ₀ + iT}}, {{math|{{sfrac|1|2}} ≤ σ₀ ≤ 1}}, {{math|0 < Δ < {{sfrac|1|3}}}}. Estimating the values {{mvar|F}} and {{mvar|G}} from below shows, how large (in modulus) values {{math|ζ(s)}} can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip {{math|0 ≤ Re(s) ≤ 1}}.

The case {{math|H ≫ ln ln T}} was studied by Kanakanahalli Ramachandra; the case {{math|Δ > c}}, where {{math|c}} is a sufficiently large constant, is trivial.

Anatolii Karatsuba proved,^[17]^[18] in particular, that if the values {{mvar|H}} and {{math|Δ}} exceed certain sufficiently small constants, then the estimates

The argument of the Riemann zeta function

is called the argument of the Riemann zeta function. Here {{math|arg ζ({{sfrac|1|2}} + it)}} is the increment of an arbitrary continuous branch of {{math|arg ζ(s)}} along the broken line joining the points {{math|2}}, {{math|2 + it}} and {{math|{{sfrac|1|2}} + it}}.

There are some theorems on properties of the function {{math|S(t)}}. Among those results^[19]^[20] are the mean value theorems for {{math|S(t)}} and its first integral

on intervals of the real line, and also the theorem claiming that every interval {{math|(T, T + H]}} for

points where the function {{math|S(t)}} changes sign. Earlier similar results were obtained by Atle Selberg for the case

Representations

Dirichlet series

An extension of the area of convergence can be obtained by rearranging the original series.^[21] The series

converges even for {{math|Re(s) > −1}}. In this way, the area of convergence can be extended to {{math|Re(s) > −k}} for any negative integer {{math|−k}}.

Mellin-type integrals

in the region where the integral is defined. There are various expressions for the zeta-function as Mellin transform-like integrals. If the real part of {{mvar|s}} is greater than one, we have

where {{math|Γ}} denotes the gamma function. By modifying the contour, Riemann showed that

Starting with the integral formula

one can show^[22] by substitution and iterated differentation for natural

using the notation of umbral calculus where each power

is to be replaced by

, so e.g. for

we have

while for

this becomes

We can also find expressions which relate to prime numbers and the prime number theorem. If {{math|π(x)}} is the prime-counting function, then

A similar Mellin transform involves the Riemann prime-counting function {{math|J(x)}}, which counts prime powers {{math|pⁿ}} with a weight of {{math|{{sfrac|1|n}}}}, so that

These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's prime-counting function is easier to work with, and {{math|π(x)}} can be recovered from it by Möbius inversion.

Theta functions

However, this integral only converges if the real part of {{mvar|s}} is greater than 1, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all {{mvar|s}} except 0 and 1:

Laurent series

The Riemann zeta function is meromorphic with a single pole of order one at {{math|s {{=}} 1}}. It can therefore be expanded as a Laurent series about {{math|s {{=}} 1}}; the series development is then

The constants {{math|γ_n}} here are called the Stieltjes constants and can be defined by the limit

Integral

For all {{math|s ∈ ℂ}}, {{math|s ≠ 1}} the integral relation (cf. Abel–Plana formula)

Rising factorial

Another series development using the rising factorial valid for the entire complex plane is{{Citation needed|date=October 2015}}

This can be used recursively to extend the Dirichlet series definition to all complex numbers.

The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the Gauss–Kuzmin–Wirsing operator acting on {{math|x^{s − 1}}}; that context gives rise to a series expansion in terms of the falling factorial.^[24]

Hadamard product

On the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansion

where the product is over the non-trivial zeros {{mvar|ρ}} of {{math|ζ}} and the letter {{mvar|γ}} again denotes the Euler–Mascheroni constant. A simpler infinite product expansion is

This form clearly displays the simple pole at {{math|s {{=}} 1}}, the trivial zeros at −2, −4, … due to the gamma function term in the denominator, and the non-trivial zeros at {{math|s {{=}} ρ}}. (To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i.e. the factors for a pair of zeros of the form {{mvar|ρ}} and {{math|1 − ρ}} should be combined.)

Globally convergent series

A globally convergent series for the zeta function, valid for all complex numbers {{mvar|s}} except {{math|s {{=}} 1 + {{sfrac|2πi|ln 2}}n}} for some integer {{mvar|n}}, was conjectured by Konrad Knopp^[25] and proven by Helmut Hasse in 1930^[26] (cf. Euler summation):

The series only appeared in an appendix to Hasse's paper, and did not become generally known until it was discussed by Jonathan Sondow in 1994.^[25]

has found that this latter series was actually first published by Joseph Ser in 1926.^[28] New proofs for both of these results were offered by Demetrios Kanoussis in 2017.^[29] Other similar globally convergent series include

where {{math|H_n}} are the harmonic numbers,

are the Stirling numbers of the first kind,

is the Pochhammer symbol, {{math|G_n}} are the Gregory coefficients, {{math|G{{su|b=n|p=(k)}}}} are the Gregory coefficients of higher order, {{math|C_n}} are the Cauchy numbers of the second kind ({{math|C₁ {{=}} 1/2}}, {{math|C₂ {{=}} 5/12}}, {{math|C₃ {{=}} 3/8}},...), and {{math|ψ_n(a)}}

Series representation at positive integers via the primorial

Here {{math|p_n#}} is the primorial sequence and {{math|J_k}} is Jordan's totient function.^[31]

Series representation by the incomplete poly-Bernoulli numbers

The function {{mvar|ζ}} can be represented, for {{math|Re(s) > 1}}, by the infinite series

where {{math|k ∈ {−1, 0}|}}, {{math|W_k}} is the {{mvar|k}}th branch of the Lambert {{mvar|W}}-function, and {{math|B{{su|b=n, ≥2|p=(μ)}}}} is an incomplete poly-Bernoulli number.^[32]

The Mellin transform of the Engel map

The function :

is iterated to find the coeffecients appearing in Engel expansions.

The Mellin transform of the map

is related to the Riemann zeta function by the formula

Numerical algorithms

For

, the Riemann zeta function has for fixed

and for all

the following representation in terms of three absolutely and uniformly converging series,^[33]

where for positive integer

one has to take the limit value

. The derivatives of

can be calculated by differentiating the above series termwise. From this follows an algorithm which allows to compute, to arbitrary precision,

and its derivatives using at most

summands for any

, with explicit error bounds. For

, these are as follows:

For a given argument

with

and

one can approximate

to any accuracy

by summing the first series to

and neglecting

, if one chooses

as the next higher integer of the unique solution of

in the unknown

, and from this

. For

one can neglect

altogether. Under the mild condition

one needs at most

summands. Hence this algorithm is essentially as fast as the Riemann-Siegel formula. Similar algorithms are possible for Dirichlet L-functions.^[33]

Applications

The zeta function occurs in applied statistics (see Zipf's law and Zipf–Mandelbrot law).

Zeta function regularization is used as one possible means of regularization of divergent series and divergent integrals in quantum field theory. In one notable example, the Riemann

zeta-function shows up explicitly in one method of calculating the Casimir effect. The zeta function is also useful for the analysis of dynamical systems.^[34]

Infinite series

The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.^[35]

There are a number of related zeta functions that can be considered to be generalizations of the Riemann zeta function. These include the Hurwitz zeta function

(the convergent series representation was given by Helmut Hasse in 1930,^[36]{{inconsistent citations}} cf. Hurwitz zeta function), which coincides with the Riemann zeta function when {{math|q {{=}} 1}} (note that the lower limit of summation in the Hurwitz zeta function is 0, not 1), the Dirichlet {{mvar|L}}-functions and the Dedekind zeta-function. For other related functions see the articles zeta function and {{mvar|L}}-function.

which coincides with the Riemann zeta function when {{math|z {{=}} 1}} and {{math|q {{=}} 1}} (note that the lower limit of summation in the Lerch transcendent is 0, not 1).

The Clausen function {{math|Cl_s(θ)}} that can be chosen as the real or imaginary part of {{math|Li_s(e{{isup|iθ}})}}.

One can analytically continue these functions to the {{mvar|n}}-dimensional complex space. The special values taken by these functions at positive integer arguments are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics.

Fractional derivative

In the case of the Riemann zeta function, a difficulty is represented by the fractional differentiation in the complex plane. The Ortigueira generalization of the classical Caputo fractional derivative solves this problem. The

-order fractional derivative of the Riemann zeta function is given by ^[37]

Given that

is a fractional number such that

, the half-plane of convergence is

See also

Notes

1. ^{{cite web|url=http://nbviewer.ipython.org/github/empet/Math/blob/master/DomainColoring.ipynb |title=Jupyter Notebook Viewer|website=Nbviewer.ipython.org |date= |accessdate=2017-01-04}}
2. ^This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is considered by many mathematicians to be the most important unsolved problem in pure mathematics.{{cite web|last=Bombieri|first= Enrico|url=http://www.claymath.org/sites/default/files/official_problem_description.pdf|title=The Riemann Hypothesis – official problem description|publisher=Clay Mathematics Institute|accessdate=2014-08-08|format=PDF}}
3. ^{{cite book | last = Devlin | first = Keith | authorlink = Keith Devlin | title = The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time | publisher = Barnes & Noble | year = 2002 | location = New York | pages = 43–47 | isbn = 978-0-7607-8659-8}}
4. ^{{cite book | last = Polchinski | first = Joseph | authorlink = Joseph Polchinski | title = String Theory, Volume I: An Introduction to the Bosonic String | publisher = Cambridge University Press | year = 1998 | page = 22 | isbn = 978-0-521-63303-1}}
5. ^{{cite journal|first=A. J. |last=Kainz |first2=U. M. |last2=Titulaer |title=An accurate two-stream moment method for kinetic boundary layer problems of linear kinetic equations |pages=1855–1874 |journal=J. Phys. A: Math. Gen. |volume=25 |issue=7 |date=1992|bibcode=1992JPhA...25.1855K |doi=10.1088/0305-4470/25/7/026 }}
6. ^{{cite book|authorlink=C. Stanley Ogilvy|first=C. S. |last=Ogilvy |first2=J. T. |last2=Anderson |title=Excursions in Number Theory |pages=29–35 |publisher=Dover Publications |date=1988 |isbn=0-486-25778-9}}
7. ^{{cite book|first=Charles Edward |last=Sandifer |title=How Euler Did It |publisher=Mathematical Association of America |date=2007 |page=193 |isbn=978-0-88385-563-8}}
8. ^{{cite journal|first=J. E. |last=Nymann|title=On the probability that {{mvar|k}} positive integers are relatively prime|journal=Journal of Number Theory|volume=4|year=1972|pages=469–473|doi=10.1016/0022-314X(72)90038-8|issue=5|bibcode = 1972JNT.....4..469N }}
9. ^I. V. Blagouchine The history of the functional equation of the zeta-function. Seminar on the History of Mathematics, Steklov Institute of Mathematics at St. Petersburg, 1 March 2018. [https://iblagouchine.perso.centrale-marseille.fr/Blagouchine-The-history-of-the-functional-equation-of-the-zeta-function-(1-March-2018).php PDF]
10. ^[https://link.springer.com/article/10.1007/s11139-013-9528-5 I. V. Blagouchine Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results. The Ramanujan Journal, vol. 35, no. 1, pp. 21-110, 2014. Addendum: vol. 42, pp. 777–781, 2017.] [https://iblagouchine.perso.centrale-marseille.fr/Blagouchine-Malmsten-integrals-and-their-evaluation-by-contour-integration-methods-(Ramanujan-J-2014).php PDF]
11. ^{{cite journal|first=Harold G.|last=Diamond|title=Elementary methods in the study of the distribution of prime numbers|journal=Bulletin of the American Mathematical Society|volume=7|issue=3|year=1982|pages=553–89|mr=670132|doi=10.1090/S0273-0979-1982-15057-1}}
12. ^{{cite journal | last1 = Ford | first1 = K. | year = 2002 | title = Vinogradov's integral and bounds for the Riemann zeta function | url = | journal = Proc. London Math. Soc. | volume = 85 | issue = 3| pages = 565–633 | doi = 10.1112/S0024611502013655 }}
13. ^{{cite journal|last=Voronin|first=S. M.|date=1975|title=Theorem on the Universality of the Riemann Zeta Function|journal=Izv. Akad. Nauk SSSR, Ser. Matem.|volume=39|pages=475–486}} Reprinted in Math. USSR Izv. (1975) 9: 443–445.
14. ^{{ cite journal |author1=Ramūnas Garunkštis |author2=Antanas Laurinčikas |author3=Kohji Matsumoto |author4=Jörn Steuding |author5=Rasa Steuding |title=Effective uniform approximation by the Riemann zeta-function |journal=Publicacions Matemàtiques |volume=54 |date=2010 |pages=209–219 |jstor=43736941 |doi=10.1090/S0025-5718-1975-0384673-1}}
15. ^{{ cite journal |author=Bhaskar Bagchi |title=A Joint Universality Theorem for Dirichlet L-Functions |journal=Mathematische Zeitschrift |issn=0025-5874 |volume=181 |date=1982 |pages=319–334 |doi=10.1007/bf01161980}}
16. ^{{cite book |last=Steuding |first=Jörn |date=2007 |title=Value-Distribution of L-Functions |location=Berlin |publisher=Springer |page=19 |isbn=3-540-26526-0 |series=Lecture Notes in Mathematics |doi=10.1007/978-3-540-44822-8}}
17. ^{{cite journal| first=A. A.| last=Karatsuba| title= Lower bounds for the maximum modulus of {{math|ζ(s)}} in small domains of the critical strip | pages=796–798| journal= Mat. Zametki| volume=70|issue=5| year=2001}}
18. ^{{cite journal| first=A. A.| last=Karatsuba| title= Lower bounds for the maximum modulus of the Riemann zeta function on short segments of the critical line| pages=99–104| journal= Izv. Ross. Akad. Nauk, Ser. Mat.| volume=68|issue=8| year=2004}}
19. ^{{cite journal |first=A. A. |last=Karatsuba |title=Density theorem and the behavior of the argument of the Riemann zeta function |pages=448–449 |journal=Mat. Zametki |issue=60 |year=1996}}
20. ^{{cite journal |first=A. A. |last=Karatsuba |title=On the function {{math|S(t)}}| pages=27–56| journal= Izv. Ross. Akad. Nauk, Ser. Mat. |volume=60 |issue=5 |year=1996}}
21. ^{{cite book|first=Konrad|last=Knopp|title=Theory of Functions|date=1945|pages=51–55}}
22. ^{{cite web|url=http://math.stackexchange.com/questions/117529/evaluating-the-definite-integral-int-0-infty-frac-mathrmex-left-mathr?rq=1 |title=Evaluating the definite integral...|website=math.stackexchange.com}}
23. ^{{Cite book |first=Jürgen |last=Neukirch |title=Algebraic number theory |publisher=Springer |date=1999 |page=422 |isbn=3-540-65399-6}}
24. ^{{cite web|url=http://linas.org/math/poch-zeta.pdf |format=PDF |title=A series representation for the Riemann Zeta derived from the Gauss-Kuzmin-Wirsing Operator |website=Linas.org |accessdate=2017-01-04}}
25. ^{{cite journal|first = Jonathan|last = Sondow|title = Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series|journal = Proceedings of the American Mathematical Society|year = 1994|volume = 120|issue = 2|pages = 421–424|url = http://www.ams.org/journals/proc/1994-120-02/S0002-9939-1994-1172954-7/S0002-9939-1994-1172954-7.pdf|doi = 10.1090/S0002-9939-1994-1172954-7|doi-access = free}}
26. ^{{cite journal | last = Blagouchine | first = Iaroslav V. | arxiv = 1501.00740 | doi = 10.1016/j.jnt.2015.06.012 | journal = Journal of Number Theory | pages = 365–396 | title = Expansions of generalized Euler's constants into the series of polynomials in {{pi}}⁻² and into the formal enveloping series with rational coefficients only | volume = 158 | year = 2016}}
27. ^¹²{{cite journal | last = Blagouchine | first = Iaroslav V. | arxiv = 1606.02044 | url = http://math.colgate.edu/~integers/vol18a.html | journal = Integers (Electronic Journal of Combinatorial Number Theory) | pages = 1–45 | title = Three Notes on Ser's and Hasse's Representations for the Zeta-functions | volume = 18A | year = 2018| bibcode = 2016arXiv160602044B}}
28. ^{{cite journal|first = Joseph|last = Ser|authorlink = Joseph Ser|title = Sur une expression de la fonction ζ(s) de Riemann|trans-title = Upon an expression for Riemann's ζ function|year = 1926|journal = Comptes rendus hebdomadaires des séances de l'Académie des Sciences|volume = 182|pages = 1075–1077|language = French}}
29. ^{{cite journal|title = A New Proof of H. Hasse's Global Expression for the Riemann's Zeta Function|year = 2017|first = Demetrios P.|last = Kanoussis|url = https://www.researchgate.net/publication/317823796_A_New_Proof_of_HHasse%27s_Global_Expression_for_the_Riemann%27s_Zeta_Function}}
30. ^{{cite book|first = Peter|last = Borwein|authorlink = Peter Borwein|url = http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf|chapter = An Efficient Algorithm for the Riemann Zeta Function|series = Conference Proceedings, Canadian Mathematical Society|year = 2000|title = Constructive, Experimental, and Nonlinear Analysis|volume = 27|pages = 29–34|isbn = 978-0-8218-2167-1|editor-first = Michel A.|editor-last = Théra|publisher = American Mathematical Society, on behalf of the Canadian Mathematical Society|location = Providence, RI}}
31. ^{{cite journal|first=István|last=Mező|title=The primorial and the Riemann zeta function|journal= The American Mathematical Monthly|year=2013|volume=120|issue=4|page=321}}
32. ^{{cite journal|first1=Takao|last1=Komatsu|first2=István|last2=Mező|title=Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers|journal=Publicationes Mathematicae Debrecen|year=2016|volume=88|issue=3–4|pages=357–368|doi=10.5486/pmd.2016.7361|url=https://arxiv.org/pdf/1510.05799|arxiv=1510.05799}}
33. ^¹{{cite arxiv|last=Fischer|first=Kurt|date=2017-03-04|title=The Zetafast algorithm for computing zeta functions|eprint=1703.01414}}
34. ^{{cite web|url=http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/spinchains.htm |title=Work on spin-chains by A. Knauf, et. al |website=Empslocal.ex.ac.uk |date= |accessdate=2017-01-04}}
35. ^Most of the formulas in this section are from § 4 of J. M. Borwein et al. (2000)
36. ^¹²{{Cite journal |first=Helmut |last=Hasse |authorlink=Helmut Hasse |title=Ein Summierungsverfahren für die Riemannsche {{mvar|ζ}}-Reihe |trans-title=A summation method for the Riemann ζ series |year=1930 |journal=Mathematische Zeitschrift |volume=32 |issue=1 |pages=458–464 |doi=10.1007/BF01194645 |language=German}}
37. ^{{cite book | last1 = Guariglia | first1 = E. | year = 2015 | title=Fractional derivative of the Riemann zeta function | publisher= In: Fractional Dynamics (Cattani, C., Srivastava, H., and Yang, X. Y.). De Gruyter|pages=357–368 | doi = 10.1515/9783110472097-022 }}