Time-dependent density functional theory

This article gives some specific values of the Riemann zeta function, including values at integer arguments, and some series involving them.

The Riemann zeta function at 0 and 1

At zero, one has

$\zeta (0)=-B_{1}=-{\tfrac {1}{2}}.\!$

At 1 there is a pole, so ζ(1) is not defined but the left and right limits are:

$\lim _{\epsilon \to 0^{\pm }}\zeta (1+\epsilon )=\pm \infty$

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:

$\zeta (2n)=(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}\!$

for n ∈ N. The first few values are given by:

$\zeta (2)=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}=1.6449\dots \!$ (Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.)

(the demonstration of this equality is known as the Basel problem)

$\zeta (4)=1+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}=1.0823\dots \!$ (Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.)

(the Stefan–Boltzmann law and Wien approximation in physics)

$\zeta (6)=1+{\frac {1}{2^{6}}}+{\frac {1}{3^{6}}}+\cdots ={\frac {\pi ^{6}}{945}}=1.0173...\dots \!$ (Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.)

$\zeta (8)=1+{\frac {1}{2^{8}}}+{\frac {1}{3^{8}}}+\cdots ={\frac {\pi ^{8}}{9450}}=1.00407...\dots \!$ (Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.)

$\zeta (10)=1+{\frac {1}{2^{10}}}+{\frac {1}{3^{10}}}+\cdots ={\frac {\pi ^{10}}{93555}}=1.000994...\dots \!$ (Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.)

$\zeta (12)=1+{\frac {1}{2^{12}}}+{\frac {1}{3^{12}}}+\cdots ={\frac {691\pi ^{12}}{638512875}}=1.000246\dots \!$ (Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.)

$\zeta (14)=1+{\frac {1}{2^{14}}}+{\frac {1}{3^{14}}}+\cdots ={\frac {2\pi ^{14}}{18243225}}=1.0000612\dots \!$ (Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.).

The relationship between zeta at the positive even integers and the Bernoulli numbers may be written as

$A_{n}\zeta (n)=B_{n}\pi ^{n}\,\!$

where A_n and B_n are integers for all even n. These are given by the integer sequences Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland. and Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland. in OEIS. Some of these values are reproduced below:

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_n be the coefficient B/A as above,

$\zeta (2n)=\sum _{\ell =1}^{\infty }{\frac {1}{\ell ^{2n}}}=\eta _{n}\pi ^{2n},$

then we find recursively,

${\begin{aligned}\eta _{1}&=1/6;\\\eta _{n}&=\sum _{\ell =1}^{n-1}(-1)^{\ell -1}{\frac {\eta _{n-\ell }}{(2\ell +1)!}}+(-1)^{n+1}{\frac {n}{(2n+1)!}}.\end{aligned}}$

This recurrence relation may be derived from that for the Bernoulli numbers.

The even zeta constants have the generating function:

$\sum _{n=0}^{\infty }\zeta (2n)x^{2n}=-{\frac {\pi x}{2}}\cot(\pi x)=-{\frac {1}{2}}+{\frac {\pi ^{2}}{6}}x^{2}+{\frac {\pi ^{4}}{90}}x^{4}+{\frac {\pi ^{6}}{945}}x^{6}+\cdots$

Since

$\lim _{n\rightarrow \infty }\zeta (2n)=1,$

the formula also shows that for $n\in \mathbb {N} ,n\rightarrow \infty$,

$\left|B_{2n}\right|\sim {\frac {2(2n)!}{(2\pi )^{2n}}}$.

Odd positive integers

For the first few odd natural numbers one has

$\zeta (1)=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots =\infty \!$

(the harmonic series);

$\zeta (3)=1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots =1.20205\dots \!$

(Apéry's constant)

$\zeta (5)=1+{\frac {1}{2^{5}}}+{\frac {1}{3^{5}}}+\cdots =1.03692\dots \!$ Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.

$\zeta (7)=1+{\frac {1}{2^{7}}}+{\frac {1}{3^{7}}}+\cdots =1.00834\dots \!$ Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.

$\zeta (9)=1+{\frac {1}{2^{9}}}+{\frac {1}{3^{9}}}+\cdots =1.002008\dots \!$ Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.

It is known that ζ(3) is irrational (Apéry's theorem) and that infinitely many of the numbers ζ(2n+1) (n ∈ N) are irrational. 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.

They appear in physics, in correlation functions of antiferromagnetic xxx spin chain.

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.

ζ(5)

Plouffe gives the following identities

${\begin{aligned}\zeta (5)&={\frac {1}{294}}\pi ^{5}-{\frac {72}{35}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}-1)}}-{\frac {2}{35}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}+1)}}\\\zeta (5)&=12\sum _{n=1}^{\infty }{\frac {1}{n^{5}\sinh(\pi n)}}-{\frac {39}{20}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}-1)}}-{\frac {1}{20}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}+1)}}\end{aligned}}$

ζ(7)

$\zeta (7)={\frac {19}{56700}}\pi ^{7}-2\sum _{n=1}^{\infty }{\frac {1}{n^{7}(e^{2\pi n}-1)}}\!$

Note that the sum is in the form of the Lambert series.

ζ(2n + 1)

By defining the quantities

$S_{\pm }(s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}(e^{2\pi n}\pm 1)}}$

a series of relationships can be given in the form

$0=A_{n}\zeta (n)-B_{n}\pi ^{n}+C_{n}S_{-}(n)+D_{n}S_{+}(n)\,$

where A_n, B_n, C_n and D_n are positive integers. Plouffe gives a table of values:

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.

Negative integers

In general, for negative integers, one has

$\zeta (-n)=-{\frac {B_{n+1}}{n+1}}.$

The so-called "trivial zeros" occur at the negative even integers:

$\zeta (-2n)=0.\,$

The first few values for negative odd integers are

$\zeta (-1)=-{\frac {1}{12}}$

$\zeta (-3)={\frac {1}{120}}$

$\zeta (-5)=-{\frac {1}{252}}$

$\zeta (-7)={\frac {1}{240}}.$

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

$\zeta ^{\prime }(-2n)=(-1)^{n}{\frac {(2n)!}{2(2\pi )^{2n}}}\zeta (2n+1).$

The first few values of which are

$\zeta ^{\prime }(-2)=-{\frac {\zeta (3)}{4\pi ^{2}}}$

$\zeta ^{\prime }(-4)={\frac {3}{4\pi ^{4}}}\zeta (5)$

$\zeta ^{\prime }(-6)=-{\frac {45}{8\pi ^{6}}}\zeta (7)$

$\zeta ^{\prime }(-8)={\frac {315}{4\pi ^{8}}}\zeta (9).$

One also has

$\zeta ^{\prime }(0)=-{\frac {1}{2}}\ln(2\pi )\approx -0.918938533\ldots$ Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.

and

$\zeta ^{\prime }(-1)={\frac {1}{12}}-\ln A\approx -0.1654211437\ldots$ Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.

where A is the Glaisher–Kinkelin constant.

Series involving ζ(n)

The following sums can be derived from the generating function:

$\sum _{k=2}^{\infty }\zeta (k)x^{k-1}=-\psi _{0}(1-x)-\gamma$

where ψ₀ is the digamma function.

$\sum _{k=2}^{\infty }(\zeta (k)-1)=1$

$\sum _{k=1}^{\infty }(\zeta (2k)-1)={\frac {3}{4}}$

$\sum _{k=1}^{\infty }(\zeta (2k+1)-1)={\frac {1}{4}}$

$\sum _{k=2}^{\infty }(-1)^{k}(\zeta (k)-1)={\frac {1}{2}}.$

Series related to the Euler–Mascheroni constant (denoted by γ) are

$\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}=\gamma$

$\sum _{k=2}^{\infty }{\frac {\zeta (k)-1}{k}}=1-\gamma$

$\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}=\ln 2+\gamma -1$

and using the principle value

$\zeta (k)=\lim _{\varepsilon \to 0}{\frac {\zeta (k+\varepsilon )+\zeta (k-\varepsilon )}{2}},$

which of course affects only the value at 1. These formulae can be stated as

$\sum _{k=1}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}=0$

$\sum _{k=1}^{\infty }{\frac {\zeta (k)-1}{k}}=0$

$\sum _{k=1}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}=\ln 2$

and show that they depend on the principal value of ζ(1) = γ.

Nontrivial zeros

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

Zeros of the Riemann zeta except negative integers are called "nontrivial zeros". See Andrew Odlyzko's website for their tables and bibliographies.

References

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  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• Simon Plouffe, "Identities inspired from Ramanujan Notebooks", (1998).
• Simon Plouffe, "Identities inspired by Ramanujan Notebooks part 2 PDF" (2006).
• Template:Cite arxiv
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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang PDF PDF Russian PS Russian
• Nontrival zeros reference by Andrew Odlyzko:
  • Bibliography
  • Tables