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# Riemann zeta function

# 1 Definition

The *Riemann zeta function* is defined to be the complex valued
function given by the series

$\zeta(s):=\sum_{{n=1}}^{\infty}\frac{1}{n^{s}},$ | (1) |

which is valid (in fact, absolutely convergent) for all complex numbers $s$ with $\operatorname{Re}(s)>1$. We list here some of the key properties [1] of the zeta function.

1. For all $s$ with $\operatorname{Re}(s)>1$, the zeta function satisfies the

*Euler product formula*$\zeta(s)=\prod_{{p}}\frac{1}{1-p^{{-s}}},$ (2) where the product is taken over all positive integer primes $p$, and converges uniformly in a neighborhood of $s$.

2. The zeta function has a meromorphic continuation to the entire complex plane with a simple pole at $s=1$, of residue $1$, and no other singularities.

3. The zeta function satisfies the

*functional equation*$\zeta(s)=2^{s}\pi^{{s-1}}\sin\frac{\pi s}{2}\Gamma(1-s)\zeta(1-s),$ (3) for any $s\in\mathbb{C}$ (where $\Gamma$ denotes the Gamma function).

# 2 Distribution of primes

The Euler product formula (2) given above expresses the zeta function as a product over the primes $p\in\mathbb{Z}$, and consequently provides a link between the analytic properties of the zeta function and the distribution of primes in the integers. As the simplest possible illustration of this link, we show how the properties of the zeta function given above can be used to prove that there are infinitely many primes.

If the set $S$ of primes in $\mathbb{Z}$ were finite, then the Euler product formula

$\zeta(s)=\prod_{{p\in S}}\frac{1}{1-p^{{-s}}}$ |

would be a finite product, and consequently $\lim_{{s\to 1}}\zeta(s)$ would exist and would equal

$\lim_{{s\to 1}}\zeta(s)=\prod_{{p\in S}}\frac{1}{1-p^{{-1}}}.$ |

But the existence of this limit contradicts the fact that $\zeta(s)$ has a pole at $s=1$, so the set $S$ of primes cannot be finite.

A more sophisticated analysis of the zeta function along these lines
can be used to prove both the analytic prime number theorem and
Dirichlet’s theorem on primes in arithmetic progressions^{1}^{1}In the case of arithmetic progressions, one also needs to examine the closely related Dirichlet $L$–functions in addition to the zeta function itself.. Proofs of
the prime number theorem can be found in [2]
and [5], and for proofs of Dirichlet’s theorem on primes
in arithmetic progressions the reader may look in [3]
and [7].

# 3 Zeros of the zeta function

A *nontrivial zero* of the Riemann zeta function is defined to be
a root $\zeta(s)=0$ of the zeta function with the property that $0\leq\operatorname{Re}(s)\leq 1$. Any other zero is called *trivial zero* of
the zeta function.

The reason behind the terminology is as follows. For complex numbers $s$ with real part greater than 1, the series definition (1) immediately shows that no zeros of the zeta function exist in this region. It is then an easy matter to use the functional equation (3) to find all zeros of the zeta function with real part less than 0 (it turns out they are exactly the values $-2n$, for $n$ a positive integer). However, for values of $s$ with real part between 0 and 1, the situation is quite different, since we have neither a series definition nor a functional equation to fall back upon; and indeed to this day very little is known about the behavior of the zeta function inside this critical strip of the complex plane.

It is known that the prime number theorem is equivalent to the
assertion that the zeta function has no zeros $s$ with $\operatorname{Re}(s)=0$ or
$\operatorname{Re}(s)=1$. The celebrated *Riemann hypothesis* asserts that all nontrivial zeros $s$ of the zeta function satisfy the much more precise equation $\operatorname{Re}(s)=1/2$. If true, the hypothesis would have profound
consequences on the distribution of primes in the
integers [5].

# References

- 1
Lars Ahlfors,
*Complex Analysis, Third Edition*, McGraw–Hill, Inc., 1979. - 2
Joseph Bak & Donald Newman,
*Complex Analysis, Second Edition*, Springer–Verlag, 1991. - 3
Gerald Janusz,
*Algebraic Number Fields, Second Edition*, American Mathematical Society, 1996. - 4
Serge Lang,
*Algebraic Number Theory, Second Edition*, Springer–Verlag, 1994. - 5
Stephen Patterson,
*Introduction to the Theory of the Riemann Zeta Function*, Cambridge University Press, 1988. - 6
B. Riemann,
*Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse*, http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/ - 7
Jean–Pierre Serre,
*A Course in Arithmetic*, Springer–Verlag, 1973.

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11M06*no label found*

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## Attached Articles

values of the Riemann zeta function in terms of Bernoulli numbers by Mathprof

value of the Riemann zeta function at $s=0$ by Wkbj79

critical strip by Wkbj79

convergence of Riemann zeta series by pahio

Riemann zeta function has no zeros on $\Re s=0,1$ by rm50

value of Riemann zeta function at $s = 4$ by pahio

Euler product formula by pahio

Lindel\"of hypothesis by pahio

dilogarithm function by pahio

## Corrections

classification by Wkbj79 ✓

capitalization by Mathprof ✘

capitalization by Mathprof ✘

related by pahio ✓

## Comments

## Addition to References...

Dear Djao,

I feel you must add to the refrences the standard expositions/monographs on the zeta functions, e.g.

1. Titchmarsh, EC, "The Theory of the Riemann Zeta Function",

revised by DR Heath-Brown, Oxford Univ. Press, 1986.

2. Edwards, HM, "The Riemann Zeta Function",?,?...

(please find out publisher and date)

3. Karatsuba,AA and Voronin, SM (translated from Russian in English

by Neal Koblitz), "The Riemann Zeta-Function", DeGruyter Expositions in Mathematics No. 5. 1992.

4. Karatsuba AA, Complex Analysis in Number Theory, CRC Press, 1995.

5. Ivic, A

6.

7....

Regards and Best Wishes

Manoj.

## Proof

I have proven RH in general case. It holds for any L-fonction.

Are you interested to read this paper? The article is written in french. My work is available in pdf format on my site http://henri.voici.org. There are two preprints (they are currently under evaluation by "Le Journal de ThÃ©orie des Nombres de Bordeaux".

I am interested in having your opinion.

Henri

## inappropriate forum

Let me remind you that this article is part of the "encyclopedia" section of planetmath. An encyclopedia, by definition of encyclopedia, is not a suitable forum to use for reporting on current research. The purpose of an encyclopedia is to deliver a refined presentation of well known established facts in a manner suitable for use as a reference. It follows then that any new result, no matter what the result is, should not be reported or presented in any scientific encyclopedia until it has established itself in the refereed literature for a minimum of several years.

Your proof, whatever its merits may otherwise be, does not at this time even come close to satisfying the standards for consideration in an encyclopedia context.

## Is the functional equation correct?

I might be totally off-base here since I'm only a math-hobbyist, but it seems to me that if the zeta functional equation in this article is correct, Zeta(s) is a product of sin((pi s)/2)) and some other factors. However, as far as I know, Zeta(2) converges to a non-negative number (pi^2/6 I think??), but sin((pi 2)/2) = sin(pi) = 0

Can anyone enlighten me?

## Re: Is the functional equation correct?

Oh nevermind. I'm sorry. This is probably the wrong place to ask questions like this since the formula is most certainly correct and I'm really asking for an explanation.

## Re: Is the functional equation correct?

We may both be off base discussing it here but I think I can answer your question. At the positive integer s=2 where the sin factor goes to zero, the Gamma function factor is evaluated at the negative integer 1-s=-1, where it is infinite. The functional equation thus has to be interpreted as a limit at that point, and the limit is a finite nonzero number. The same thing happens for other positive even integers. But negative even integers (see the section on "Zeros of the zeta function") do behave the way you expected!

## Re: Is the functional equation correct?

So then does that paragraph in the entry fulfill the request (in the Requests section) for the functional equation?

## Re: Is the functional equation correct?

There is a request for a proof of the functional equation. Stating the functional equation and clarifying what it means when some of the factors become infinite does not amount to a proof of the functional equation.

## Re: Is the functional equation correct?

> I might be totally off-base here since I'm only a

> math-hobbyist, but it seems to me that if the zeta

> functional equation in this article is correct, Zeta(s) is a

> product of sin((pi s)/2)) and some other factors. However,

> as far as I know, Zeta(2) converges to a non-negative number

> (pi^2/6 I think??), but sin((pi 2)/2) = sin(pi) =

>

> Can anyone enlighten me?

sum_{n=1}^infinity 1/n^2 = pi^2 / 6 . This can be obtained from the sum of odd ... But this IS the case if the average value of the function is zero. ... integral_0^{2pi} f(u) du = integral_0^0 f(u) du = 0 . If you think of this one term ... x_p = (4/pi) omega_n^2 (sin(t)/(omega_n^2 - 1) + sin(3t)/(omega_n^2 - 9) ...

They can be manipulated but not evaluated. Functions, on the other hand, are a little better for ... L := [[0,0],[1,3],[2,2],[6,3],[1,1]]; pointplot(L); How boring. ... I think somebody had some fun implementing plotting options in Maple. ... First, name the plots: P1 := plot([sin(t),cos(2*t+Pi/2),t=0..10*Pi]): P2 ...

Hope this Helps!

Martin Musatov