Time-dependent density functional theory: Difference between revisions

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This article gives some specific values of the [[Riemann zeta function]], including values at integer arguments, and some series involving them.
The author's title is Christy Brookins. Office supervising is where her primary income comes from. What I love doing is soccer but I don't have the time recently. For a while I've been in Mississippi but now I'm contemplating other choices.<br><br>Here is my web blog ... best psychic ([http://165.132.39.93/xe/visitors/372912 http://165.132.39.93])
 
==The Riemann zeta function at 0 and 1==
At [[Zero (complex analysis)|zero]], one has
:<math>\zeta(0)= -B_1=-\tfrac{1}{2}.\!</math>
 
At 1 there is a [[Pole (complex analysis)|pole]], so ζ(1) is not defined but the left and right limits are:
:<math>\lim_{\epsilon\to 0^{\pm}}\zeta(1+\epsilon) = \pm\infty</math>
and because it is a pole of 1st order its principal value exists and is γ.
 
==Positive integers==
===Even positive integers===
For the even positive integers, one has the relationship to the [[Bernoulli numbers]]:
 
:<math>\zeta(2n) = (-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!} \!</math>
 
for ''n'' ∈ '''N'''. The first few values are given by:
 
:<math>\zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6} = 1.6449\dots\!</math> ({{OEIS2C|A013661}})
:::(the demonstration of this equality is known as the [[Basel problem]])
:<math>\zeta(4) = 1 + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90} = 1.0823\dots\!</math> ({{OEIS2C|A013662}})
:::(the [[Stefan–Boltzmann law]] and [[Wien approximation]] in physics)
:<math>\zeta(6) = 1 + \frac{1}{2^6} + \frac{1}{3^6} + \cdots = \frac{\pi^6}{945} = 1.0173...\dots\!</math> ({{OEIS2C|A013664}})
:<math>\zeta(8) = 1 + \frac{1}{2^8} + \frac{1}{3^8} + \cdots = \frac{\pi^8}{9450} = 1.00407... \dots\!</math> ({{OEIS2C|A013666}})
:<math>\zeta(10) = 1 + \frac{1}{2^{10}} + \frac{1}{3^{10}} + \cdots = \frac{\pi^{10}}{93555} = 1.000994...\dots\!</math> ({{OEIS2C|A013668}})
:<math>\zeta(12) = 1 + \frac{1}{2^{12}} + \frac{1}{3^{12}} + \cdots = \frac{691\pi^{12}}{638512875} = 1.000246\dots\!</math> ({{OEIS2C|A013670}})
:<math>\zeta(14) = 1 + \frac{1}{2^{14}} + \frac{1}{3^{14}} + \cdots = \frac{2\pi^{14}}{18243225} = 1.0000612\dots\!</math> ({{OEIS2C|A013672}}).
 
The relationship between zeta at the positive even integers and the Bernoulli numbers may be written as
 
:<math>A_n \zeta(n) = B_n \pi^n\,\!</math>
 
where ''A<sub>n</sub>'' and ''B<sub>n</sub>'' are integers for all even ''n''. These are given by the integer sequences  {{OEIS2C|id=A046988}} and {{OEIS2C|id=A002432}} in [[OEIS]].  Some of these values are reproduced below:
 
{| class="wikitable"
|+ coefficients
|-
! n
! A
! B
|-
| 2
| 6
| 1
|-
| 4
| 90
| 1
|-
| 6
| 945
| 1
|-
| 8
| 9450
| 1
|-
| 10
| 93555
| 1
|-
| 12
| 638512875
| 691
|-
| 14
| 18243225
| 2
|-
| 16
| 325641566250
| 3617
|-
| 18
| 38979295480125
| 43867
|-
| 20
| 1531329465290625
| 174611
|-
| 22
| 13447856940643125
| 155366
|-
| 24
| 201919571963756521875
| 236364091
|-
| 26
| 11094481976030578125
| 1315862
|-
| 28
| 564653660170076273671875
| 6785560294
|-
| 30
| 5660878804669082674070015625
| 6892673020804
|-
| 32
| 62490220571022341207266406250
| 7709321041217
|-
| 34
| 12130454581433748587292890625
| 151628697551
|}
 
If we let η<sub>''n''</sub> be the coefficient ''B''/''A'' as above,
:<math>\zeta(2n) = \sum_{\ell=1}^{\infty}\frac{1}{\ell^{2n}}=\eta_n\pi^{2n},</math>
then we find recursively,
 
:<math>\begin{align}
\eta_1 &= 1/6; \\
\eta_n &= \sum_{\ell=1}^{n-1}(-1)^{\ell-1}\frac{\eta_{n-\ell}}{(2\ell+1)!}+(-1)^{n+1}\frac{n}{(2n+1)!}.
\end{align}</math>
 
This recurrence relation may be derived from that for the [[Bernoulli number]]s.
 
The even zeta constants have the [[generating function]]:
:<math>\sum_{n=0}^\infty \zeta(2n) x^{2n} = -\frac{\pi x}{2} \cot(\pi x) = -\frac{1}{2} + \frac{\pi^2}{6} x^2 + \frac{\pi^4}{90} x^4+\frac{\pi^6}{945}x^6 + \cdots</math>
Since
:<math>\lim_{n\rightarrow\infty} \zeta(2n)=1,</math>
the formula also shows that for  <math> n\in\mathbb{N}, n\rightarrow\infty</math>,
:<math>\left|B_{2n}\right| \sim \frac{2(2n)!}{(2\pi)^{2n}}</math>.
 
===Odd positive integers===
For the first few odd natural numbers one has
 
:<math>\zeta(1) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty\!</math>
:::(the [[harmonic series (mathematics)|harmonic series]]);
:<math>\zeta(3) = 1 + \frac{1}{2^3} + \frac{1}{3^3} + \cdots = 1.20205\dots\!</math>
:::([[Apéry's constant]])
:<math>\zeta(5) = 1 + \frac{1}{2^5} + \frac{1}{3^5} + \cdots = 1.03692\dots\!</math> {{OEIS2C|A013663}}
:<math>\zeta(7) = 1 + \frac{1}{2^7} + \frac{1}{3^7} + \cdots = 1.00834\dots\!</math> {{OEIS2C|A013665}}
:<math>\zeta(9) = 1 + \frac{1}{2^9} + \frac{1}{3^9} + \cdots = 1.002008\dots\!</math> {{OEIS2C|A013667}}
 
It is known that ζ(3) is irrational ([[Apéry's theorem]]) and that infinitely many of the numbers ζ(2''n''+1) (''n'' ∈ '''N''') are irrational.<ref>{{cite journal | last1 = Rivoal | first1 = T. | year = 2000 | title = La fonction zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs | url = | journal = Comptes Rendus de l'Académie des Sciences. Série I. Mathématique | volume = 331 | issue = | pages = 267–270 |doi = 10.1016/S0764-4442(00)01624-4  |arxiv=math/0008051}}</ref> There are also results on the irrationality of values of the Riemann zeta function at the elements of certain subsets of the positive odd integers; for example, at least one of ζ(5), ζ(7), ζ(9), or ζ(11) is irrational.<ref>{{cite journal |author=W. Zudilin |title=One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational |journal=Russ. Math. Surv. |year=2001 |volume=56 |issue=4 |pages=774–776}}</ref>
 
They appear in physics, in [[Correlation function (statistical mechanics) |correlation functions]] of antiferromagnetic  [[Heisenberg model (quantum) | xxx spin chain]].<ref>{{citation|title=Quantum correlations and number theory|first1=H. E.|last1=Boos|first2=V. E.|last2=Korepin|first3=Y.|last3=Nishiyama|first4=M.|last4=Shiroishi|arxiv=cond-mat/0202346|journal=J. Phys. A|volume=35|pages=4443–4452|year=2002}}.</ref>
 
Most of the identities following below are provided by [[Simon Plouffe]]. They are notable in that they converge quite rapidly, giving almost three digits of precision per iteration, and are thus useful for high-precision calculations.
 
====''&zeta;''(5)====
Plouffe gives the following identities
 
:<math>\begin{align}
\zeta(5)&=\frac{1}{294}\pi^5 -\frac{72}{35} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} -1)}-\frac{2}{35} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} +1)}\\
\zeta(5)&=12 \sum_{n=1}^\infty \frac{1}{n^5 \sinh (\pi n)} -\frac{39}{20} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} -1)}-\frac{1}{20} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} +1)}
\end{align}</math>
 
====''&zeta;''(7)====
:<math>\zeta(7)=\frac{19}{56700}\pi^7 -2 \sum_{n=1}^\infty \frac{1}{n^7 (e^{2\pi n} -1)}\!</math>
 
Note that the sum is in the form of the [[Lambert series]].
 
====''&zeta;''(2''n'' + 1)====
By defining the quantities
 
:<math>S_\pm(s) = \sum_{n=1}^\infty \frac{1}{n^s (e^{2\pi n} \pm 1)}</math>
 
a series of relationships can be given in the form
 
:<math>0=A_n \zeta(n) - B_n \pi^{n} + C_n S_-(n) + D_n S_+(n)\,</math>
 
where ''A''<sub>''n''</sub>, ''B''<sub>''n''</sub>, ''C''<sub>''n''</sub> and ''D''<sub>''n''</sub> are positive integers. Plouffe gives a table of values:
 
{| class="wikitable"
|+ coefficients
|-
! n
! A
! B
! C
! D
|-
| 3
| 180
| 7
| 360
| 0
|-
| 5
| 1470
| 5
| 3024
| 84
|-
| 7
| 56700
| 19
| 113400
| 0
|-
| 9
| 18523890
| 625
| 37122624
| 74844
|-
| 11
| 425675250
| 1453
| 851350500
| 0
|-
| 13
| 257432175
| 89
| 514926720
| 62370
|-
| 15
| 390769879500
| 13687
| 781539759000
| 0
|-
| 17
| 1904417007743250
| 6758333
| 3808863131673600
| 29116187100
|-
| 19
| 21438612514068750
| 7708537
| 42877225028137500
| 0
|-
| 21
| 1881063815762259253125
| 68529640373
| 3762129424572110592000
| 1793047592085750
|}
 
These integer constants may be expressed as sums over Bernoulli numbers, as given in (Vepstas, 2006) below.
The only fast algorithm for the calculation of Riemann's zeta function for any integer argument was found by E. A. Karatsuba.<ref>E. A. Karatsuba: Fast computation of the Riemann zeta-function ''ζ''(''s'') for integer values of the argument&nbsp;''s''. Probl. Inf. Transm. Vol.31, No.4, pp. 353–362 (1995).</ref><ref>E. A. Karatsuba: Fast computation of the Riemann zeta function for integer argument. Dokl. Math. Vol.54, No.1, p. 626 (1996).</ref><ref>E. A. Karatsuba: Fast evaluation of ''ζ''(3).  Probl. Inf. Transm. Vol.29, No.1, pp. 58–62 (1993).</ref>
 
==Negative integers==
In general, for negative integers, one has
 
:<math>\zeta(-n)=-\frac{B_{n+1}}{n+1}.</math>
 
The so-called "trivial zeros" occur at the negative even integers:
 
:<math>\zeta(-2n)=0.\,</math>
 
The first few values for negative odd integers are
 
:<math>\zeta(-1)=-\frac{1}{12}</math>
:<math>\zeta(-3)=\frac{1}{120}</math>
:<math>\zeta(-5)=-\frac{1}{252}</math>
:<math>\zeta(-7)=\frac{1}{240}.</math>
 
However, just like the [[Bernoulli numbers]], these do not stay small for increasingly negative odd values. For details on the first value, see [[1 + 2 + 3 + 4 + · · ·]].
 
So ζ(''m'') can be used as the definition of all (including those for index 0 and 1) Bernoulli numbers.
 
==Derivatives==
The derivative of the zeta function at the negative even integers is given by
 
:<math>\zeta^{\prime}(-2n) = (-1)^n \frac {(2n)!} {2 (2\pi)^{2n}} \zeta (2n+1).</math>
 
The first few values of which are
 
:<math>\zeta^{\prime}(-2) = -\frac{\zeta(3)}{4\pi^2}</math>
:<math>\zeta^{\prime}(-4) = \frac{3}{4\pi^4} \zeta(5)</math>
:<math>\zeta^{\prime}(-6) = -\frac{45}{8\pi^6} \zeta(7)</math>
:<math>\zeta^{\prime}(-8) = \frac{315}{4\pi^8} \zeta(9).</math>
 
One also has
 
:<math>\zeta^{\prime}(0) = -\frac{1}{2}\ln(2\pi)\approx -0.918938533\ldots</math> {{OEIS2C|A075700}}
 
and
 
:<math>\zeta^{\prime}(-1)=\frac{1}{12}-\ln A \approx -0.1654211437\ldots</math> {{OEIS2C|A084448}}
 
where ''A'' is the [[Glaisher–Kinkelin constant]].
 
==Series involving ''&zeta;''(''n'')==
The following sums can be derived from the generating function:
:<math>\sum_{k=2}^\infty \zeta(k) x^{k-1}=-\psi_0(1-x)-\gamma</math>
where ''ψ''<sub>0</sub> is the [[digamma function]].
 
:<math>\sum_{k=2}^\infty (\zeta(k) -1) = 1</math>
:<math>\sum_{k=1}^\infty (\zeta(2k) -1) = \frac{3}{4}</math>
:<math>\sum_{k=1}^\infty (\zeta(2k+1) -1) = \frac{1}{4}</math>
:<math>\sum_{k=2}^\infty (-1)^k(\zeta(k) -1) = \frac{1}{2}.</math>
 
Series related to the [[Euler–Mascheroni constant]] (denoted by γ) are
:<math>\sum_{k=2}^\infty (-1)^k \frac{\zeta(k)}{k} = \gamma</math>
:<math>\sum_{k=2}^\infty \frac{\zeta(k) - 1}{k} = 1 - \gamma</math>
:<math>\sum_{k=2}^\infty (-1)^k \frac{\zeta(k)-1}{k} = \ln2 + \gamma - 1</math>
 
and using the principle value
:<math> \zeta(k) = \lim_{\varepsilon \to 0} \frac{\zeta(k+\varepsilon)+\zeta(k-\varepsilon)}{2},</math>
which of course affects only the value at 1. These formulae can be stated as
 
:<math>\sum_{k=1}^\infty (-1)^k \frac{\zeta(k)}{k} = 0</math>
:<math>\sum_{k=1}^\infty \frac{\zeta(k) - 1}{k} = 0</math>
:<math>\sum_{k=1}^\infty (-1)^k \frac{\zeta(k)-1}{k} = \ln2</math>
 
and show that they depend on the principal value of&nbsp;''ζ''(1)&nbsp;=&nbsp;''γ''.
 
== Nontrivial zeros ==
{{main|Riemann hypothesis}}
 
Zeros of the Riemann zeta except negative integers are called "nontrivial zeros". See [[Andrew Odlyzko]]'s website for their tables and bibliographies.
 
==References==
{{reflist}}
* {{cite journal
|first1=E. A.
|last1=Karatsuba
|title=Fast calculation of the Riemann Zeta function zeta(s) for integer values of the argument s
|year=1995
|journal=Probl. Perdachi Inf.
|volume=31
|issue=4
|pages=69–80
|mr=1367927
|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paperid=294&option_lang=eng
}}
* [[Simon Plouffe]], "[http://www.lacim.uqam.ca/~plouffe/identities.html Identities inspired from Ramanujan Notebooks]", (1998).
* [[Simon Plouffe]], "[http://www.lacim.uqam.ca/~plouffe/inspired22.html Identities inspired by Ramanujan Notebooks part 2] [http://www.lacim.uqam.ca/~plouffe/inspired2.pdf PDF]" (2006).
* {{cite arxiv
|first1=Linas
|last1=Vepstas
|url=http://www.linas.org/math/plouffe-ram.pdf
|title=On Plouffe's Ramanujan Identities
|eprint=math.NT/0609775
|year=2006
}}
* {{cite journal
|first1=Wadim
|last1=Zudilin
|authorlink=Wadim Zudilin
|title=One of the Numbers ζ(5), ζ(7), ζ(9), ζ(11) Is Irrational
|journal=[[Russian Mathematical Surveys]]
|volume= 56
|pages=774&ndash;776
|year=2001
|doi=10.1070/RM2001v056n04ABEH000427
|mr=1861452
}} [http://wain.mi.ras.ru/PS/zeta5-11$.pdf PDF] [http://wain.mi.ras.ru/PS/zeta5-11.pdf PDF Russian] [http://wain.mi.ras.ru/PS/zeta5-11.ps.gz PS Russian]
* Nontrival zeros reference by [[Andrew Odlyzko]]:
** [http://www.dtc.umn.edu/~odlyzko/doc/zeta.html Bibliography]
** [http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html Tables]
 
[[Category:Mathematical constants]]
[[Category:Zeta and L-functions]]
[[Category:Irrational numbers]]

Latest revision as of 20:35, 19 September 2014

The author's title is Christy Brookins. Office supervising is where her primary income comes from. What I love doing is soccer but I don't have the time recently. For a while I've been in Mississippi but now I'm contemplating other choices.

Here is my web blog ... best psychic (http://165.132.39.93)

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