A friend here

And I have the answer, PC.

the answer is the square root of 5, times by the amount of booze you consumed last night, minus the the hours of sleep divided by the core temp of your freezer...

It's currently only half the question, she must have dozed off while posting...

Pepperoni please, thin and crispy.

Cheers

DefinitionThe Riemann zeta function ζ(s) is a function of a complex variable s = σ + it (here, s, σ and t are traditional notations associated to the study of the ζ-function). The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case:

The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series.

Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to real s > 1.[2]

The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for s such that σ > 1 and diverges for all other values of s. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠ 1. For s = 1 the series is the harmonic series which diverges to +∞, and

Thus the Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.

[edit] Specific values

Riemann zeta function for real s > 1For any positive even number 2n,

where B2n is a Bernoulli number; for negative integers, one has

for n ≥ 1, so in particular ζ vanishes at the negative even integers because Bm = 0 for all odd m other than 1. No such simple expression is known for odd positive integers.

The values of the zeta function obtained from integral arguments are called zeta constants. The following are the most commonly used values of the Riemann zeta function.

this is the harmonic series.

(sequence A078434 in OEIS)

this is employed in calculating the critical temperature for a Bose–Einstein condensate in a box with periodic boundary conditions, and for spin wave physics in magnetic systems.

(sequence A013661 in OEIS)

the demonstration of this equality is known as the Basel problem. The reciprocal of this sum answers the question: What is the probability that two numbers selected at random are relatively prime?[3]

(sequence A002117 in OEIS)

this is called Apéry's constant.

(sequence A0013662 in OEIS)

Stefan–Boltzmann law and Wien approximation in physics.

[edit] Euler product formulaThe connection between the zeta function and prime numbers was discovered by Leonhard Euler, who proved the identity

where, by definition, the left hand side is ζ(s) and the infinite product on the right hand side extends over all prime numbers p (such expressions are called Euler products):

Both sides of the Euler product formula converge for Re(s) > 1. The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s = 1, diverges, Euler's formula (which becomes ) implies that there are infinitely many primes.[4]

The Euler product formula can be used to calculate the asymptotic probability that s randomly selected integers are set-wise coprime. Intuitively, the probability that any single number is divisible by a prime (or any integer), p is 1/p. Hence the probability that s numbers are all divisible by this prime is 1/ps, and the probability that at least one of them is not is 1 − 1/ps. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors n and m if and only if it is divisible by nm, an event which occurs with probability 1/(nm).) Thus the asymptotic probability that s numbers are coprime is given by a product over all primes,

(More work is required to derive this result formally.)[5]

[edit] The functional equationThe Riemann zeta function satisfies the functional equation

where Γ(s) is the gamma function, which is an equality of meromorphic functions valid on the whole complex plane. This equation relates values of the Riemann zeta function at the points s and 1 − s. The functional equation (owing to the properties of sin) implies that ζ(s) has a simple zero at each even negative integer s = − 2n — these are the trivial zeros of ζ(s).[6]

The functional equation was established by Riemann in his 1859 paper On the Number of Primes Less Than a Given Magnitude and used to construct the analytic continuation in the first place. An equivalent relationship had been conjectured by Euler over a hundred years earlier, in 1749, for the Dirichlet eta function (alternating zeta function)

Incidentally, this relation is interesting also because it actually exhibits ζ(s) as a Dirichlet series (of the η-function) which is convergent (albeit non-absolutely) in the larger half-plane σ > 0 (not just σ > 1), up to an elementary factor.

Riemann also found a symmetric version of the functional equation, given by first defining

The functional equation is then given by

(Riemann defined a similar but different function which he called ξ(t).)

[edit] Zeros, the critical line, and the Riemann hypothesisMain article: Riemann hypothesis

Apart from the trivial zeros, the Riemann zeta function doesn't have any zero on the right of σ=1 and on the left of σ=0 (neither the zeros can lie too close to those lines). Furthermore, the non-trivial zeros are symmetric about the real axis and the line σ=1/2 and, according to the Riemann Hypothesis, they all lie in the line σ=1/2.

This image shows a plot of the Riemann zeta function along the critical line for real values of t running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin.The functional equation shows that the Riemann zeta function has zeros at −2, −4, ... . These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin(πs/2) being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip {s ∈ C: 0 < Re(s) < 1}, which is called the critical strip. The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has Re(s) = 1/2. In the theory of the Riemann zeta function, the set {s ∈ C: Re(s) = 1/2} is called the critical line. For the Riemann zeta function on the critical line, see Z-function.

[edit] The Hardy–Littlewood conjecturesIn 1914 Godfrey Harold Hardy proved that has infinitely many real zeros.

Let N(T) be the total number of real zeros, N0(T) be the total number of zeros of odd order of the function , lying on the interval (0,T].

The next two conjectures of Hardy and John Edensor Littlewood on the distance between real zeros of and on the density of zeros of on the intervals (T,T + H] for sufficiently great T > 0, and with as less as possible value of a > 0, where is an arbitrarily small number, open two new directions in the investigation of the Riemann zeta function:

1. for any there exists such that for and the interval (T,T + H] contains a zero of odd order of the function .

2. for any there exist and , such that for and the inequality is true.

[edit] The Selberg conjectureIn 1942 Atle Selberg investigated the problem of Hardy–Littlewood 2 and proved that for any there exists such and , such that for and the inequality is true.

In his turn, Selberg claim a conjecture[7] that it's possible to decrease the value of the exponent a = 0.5 for .

In 1984 Anatolii Alexeevitch Karatsuba proved[8][9][10] that for a fixed satisfying the condition , a sufficiently large T and , , the interval (T,T + H) contains at least cHlnT real zeros of the Riemann zeta function and therefore confirmed the Selberg conjecture.

The estimates of Atle Selberg and Karatsuba can not be improved in respect of the order of growth as .

In 1992 A.A. Karatsuba proved,[11] that an analog of the Selberg conjecture holds for «almost all» intervals (T,T + H], , where is an arbitrarily small fixed positive number. The Karatsuba method permits to investigate zeros of the Riemann zeta-function on «supershort» intervals of the critical line, that is, on the intervals (T,T + H], the length H of which grows slower than any, even arbitrarily small degree T. In particular, he proved that for any given numbers , satisfying the conditions almost all intervals (T,T + H] for contain at least zeros of the function . This estimate is quite close to the one that follows from the Riemann hypothesis.

[edit] Other resultsThe location of the Riemann zeta function's zeros is of great importance in the theory of numbers. From the fact that all non-trivial zeros lie in the critical strip one can deduce the prime number theorem. A better result[12] is that ζ(σ + it) ≠ 0 whenever |t| ≥ 3 and

The strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound consequences in the theory of numbers.

It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence (γn) contains the imaginary parts of all zeros in the upper half-plane in ascending order, then

The critical line theorem asserts that a positive percentage of the nontrivial zeros lies on the critical line.

In the critical strip, the zero with smallest non-negative imaginary part is 1/2 + i14.13472514... Directly from the functional equation one sees that the non-trivial zeros are symmetric about the axis Re(s) = 1/2. Furthermore, the fact that ζ(s) = ζ(s*)* for all complex s ≠ 1 (* indicating complex conjugation) implies that the zeros of the Riemann zeta function are symmetric about the real axis.

The statistics of the Riemann zeta zeros are a topic of interest to mathematicians because of their connection to big problems like the Riemann hypothesis, distribution of prime numbers, etc. Through connections with random matrix theory and quantum chaos, the appeal is even broader. The fractal structure of the Riemann zeta zero distribution has been studied using rescaled range analysis.[13] The self-similarity of the zero distribution is quite remarkable, and is characterized by a large fractal dimension of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude, and also for the zeros of other L-functions.

[edit] Various propertiesFor sums involving the zeta-function at integer and half-integer values, see rational zeta series.

[edit] ReciprocalThe reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius function μ(n):

for every complex number s with real part > 1. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series.

The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than 1/2.

[edit] UniversalityThe 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.

[edit] Estimates of the maximum of the modulus of the zeta functionLet the functions F(T;H) and G(s0;Δ) be defined by the equalities

Here T is a sufficiently large positive number, , s0 = σ0 + iT, , . Estimating the values F and G from below shows, how large (in modulus) values ζ(s) can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip .

The case was studied by Ramachandra; the case Δ > c, where c is a sufficiently large constant, is trivial.

Karatsuba proved,[14][15] in particular, that if the values H and Δ exceed certain sufficiently small constants, then the estimates

hold, where c1,c2 are certain absolute constants.

[edit] The argument of the Riemann zeta-functionThe function is called the argument of the Riemann zeta function. Here is the increment of an arbitrary continuous branch of argζ(s) along the broken line joining the points 2,2 + it and There are some theorems on properties of the function S(t). Among those results[16][17] are the mean value theorems for S(t) and its first integral on intervals of the real line, and also the theorem claiming that every interval (T,T + H] for contains at least

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

[edit] Representations[edit] Mellin transformThe Mellin transform of a function ƒ(x) is defined as

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

where Γ denotes the Gamma function. By modifying the contour Riemann showed that

for all s, where the contour C starts and ends at +∞ and circles the origin once.

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

for values with Re(s) > 1.

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

Now we have

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 π(x) can be recovered from it by Möbius inversion.

[edit] Theta functionsThe Riemann zeta function can be given formally by a divergent Mellin transform

in terms of Jacobi's theta function

However this integral does not converge for any value of s and so needs to be regularized: this gives the following expression for the zeta function:

This leads to the rapidly and globally converging series

where Γ(s,z) is the upper incomplete Gamma function.

[edit] Laurent seriesThe Riemann zeta function is meromorphic with a single pole of order one at s = 1. It can therefore be expanded as a Laurent series about s = 1; the series development then is

The constants γn here are called the Stieltjes constants and can be defined by the limit

The constant term γ0 is the Euler–Mascheroni constant.

[edit] IntegralFor all the integral relation

holds true, which may be used for a numerical evaluation of the Zeta-function.[18] A similar expression is obtained for the Lerch's zeta function:

[edit] Rising factorialAnother series development using the rising factorial valid for the entire complex plane is

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 xs−1; that context gives rise to a series expansion in terms of the falling factorial.

[edit] Hadamard productFor other uses, see Matrix multiplication.

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

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

This form clearly displays the simple pole at s = 1, the trivial zeros at −2, −4, ... due to the gamma function term in the denominator, and the non-trivial zeros at s = ρ.

[edit] Logarithmic derivative on the critical strip

where is the density of zeros of ζ on the critical strip 0 < Re(s) < 1 (δ is the Dirac delta distribution, and the sum is over the nontrivial zeros ρ of ζ).

[edit] Globally convergent seriesA globally convergent series for the zeta function, valid for all complex numbers s except s = 1 + 2πin/log(2) for some integer n, was conjectured by Konrad Knopp and proved by Helmut Hasse in 1930 (cf. Euler summation):

The series only appeared in an Appendix to Hasse's paper, and did not become generally known until it was rediscovered more than 60 years later (see Sondow, 1994).

Peter Borwein has shown a very rapidly convergent series suitable for high precision numerical calculations. The algorithm, making use of Chebyshev polynomials, is described in the article on the Dirichlet eta function.

[edit] ApplicationsThe 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 in quantum field theory. In one notable example, the Riemann zeta-function shows up explicitly in the calculation of the Casimir effect.

[edit] GeneralizationsThere 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 [19], cf. Hurwitz zeta function), which coincides with the Riemann zeta function when q = 1 (note that the lower limit of summation in the Hurwitz zeta function is 0, not 1), the Dirichlet L-functions and the Dedekind zeta-function. For other related functions see the articles Zeta function and L-function.

The polylogarithm is given by

which coincides with the Riemann zeta function when z = 1.

The Lerch transcendent is given by

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

The Clausen function Cls(θ) that can be chosen as the real or imaginary part of Lis(e iθ).

The multiple zeta functions are defined by

One can analytically continue these functions to the n-dimensional complex space. The special values of these functions are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics.

[edit] See alsoGeneralized Riemann hypothesis

Riemann–Siegel theta function

Prime zeta function

1 + 2 + 3 + 4 + ···

DefinitionThe Riemann zeta function ζ(s) is a function of a complex variable s = σ + it (here, s, σ and t are traditional notations associated to the study of the ζ-function). The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case:

The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series.

...

Can you post the equations please? This was really interesting but impossible to follow if you don't actually show the integrals and infinite series making up the post.

Luckily, I am very familiar with the gamma function, so I was almost able to understand that. But you did a poor job with the rest of it.

It's a good thing that I did a Google search and found the page you quoted, or the rest of the TV members would be as disappointed as I was.

http://en.wikipedia.org/wiki/Riemann_zeta_function

Too hard tonight, mate. I'll work on it tomorrow.

Thanks for your concern

(smart arse person)

Am I the only one that has no idea what greb and BB are nattering about?

Am I the only one that has no idea what greb and BB are nattering about?

Probably.

Patsy's still sleeping it off.

Too hard tonight, mate. I'll work on it tomorrow.

Thanks for your concern

(smart arse person)

As the person that presented Prof Butcher (Head of Math, University of Auckland, 1990) with a proof of my formula that showed that the Pascal Triangle could extend to negative numbers and become a rectangle (dealing with geometric series), I realised that the post was simply a (deservedly) tongue-in-cheek response.

True story.

Magic Eight Ball

Am I the only one that has no idea what greb and BB are nattering about?

i am lost, and i think GregB might be taking this thread a bit to seroisly

Where's Zzaa when you need him?

SC

Pepperoni please, thin and crispy.

Cheers

Not too crispy. And a simple mozzarella and basil would be the choice.

Just what I thought.......

Am I the only one that has no idea what greb and BB are nattering about?

i am lost, and i think GregB might be taking this thread a bit to seroisly

As if it were possible to take math and physics too seriously. Next thing you'll be saying is that none of you read Complex Analysis books before going to bed. Whatever Patsycat had in mind, I'm sure a holomorphic function is the answer.

Just what I thought.......

Sorry, that's wrong.

You forgot to carry the one.

If you divide it by 2 you get;

You want to be careful making holomorphic statements or you'll have Endure down here carrying on....

You want to be careful making holomorphic statements or you'll have Endure down here carrying on....

He'd never last.

Just what I thought.......

Sorry, that's wrong.

You forgot to carry the one.

dam_n it. I wondered why it crashed.

If you divide it by 2 you get;

Ahhhh...so the answer must have been 6.28318 (rounded down)

I hope Patsy's ok....I hope it's just that her modem crashed at that moment, and nothing more serious.

Blinky - look what you've done!

You want to be careful making holomorphic statements or you'll have Endure down here carrying on....

i thought that said homophobic statements......

I like the second half, Humph. It makes quite a lot of sense when you close your eyes.

If it's difficult you can always work it out using your finger ;-)

555 Sometimes I worry about you, Patsy.

If it's difficult you can always work it out using your finger ;-)

We are talking about computation, not constipation.