In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
is a function in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, which is also important in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
. It is often denoted and is named after the mathematician
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
. When the argument is a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
greater than one, the zeta function satisfies the equation
It can therefore provide the sum of various convergent
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
, such as
Explicit or numerically efficient formulae exist for at integer arguments, all of which have real values, including this example. This article lists these formulae, together with tables of values. It also includes derivatives and some series composed of the zeta function at integer arguments.
The same equation in above also holds when is a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
whose
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
is greater than one, ensuring that the infinite sum still converges. The zeta function can then be extended to the whole of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
by
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
, except for a
simple pole
In complex analysis (a branch of mathematics), a pole is a certain type of singularity (mathematics), singularity of a complex-valued function of a complex number, complex variable. In some sense, it is the simplest type of singularity. Technical ...
at . The
complex derivative
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
exists in this more general region, making the zeta function a
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), pole ...
. The above equation no longer applies for these extended values of , for which the corresponding summation would diverge. For example, the full zeta function exists at (and is therefore finite there), but the corresponding series would be
whose
partial sum
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
s would grow indefinitely large.
The zeta function values listed below include function values at the negative even numbers (, ), for which and which make up the so-called trivial zeros. The
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane. The successful characterisation of its non-trivial zeros in the wider plane is important in number theory, because of the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
.
The Riemann zeta function at 0 and 1
At
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
, one has
At 1 there is a
pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
, so ''ζ''(1) is not finite but the left and right limits are:
Since it is a pole of first order, it has a
complex residue
Positive integers
Even positive integers
For the even positive integers
, one has the relationship to the
Bernoulli numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
:
The computation of ''ζ''(2) is known as the
Basel problem
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
. The value of ''ζ''(4) is related to the
Stefan–Boltzmann law
The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths ...
and
Wien approximation
Wien's approximation (also sometimes called Wien's law or the Wien distribution law) is a law of physics used to describe the spectrum of thermal radiation (frequently called the blackbody function). This law was first derived by Wilhelm Wien in ...
in physics. The first few values are given by:
Taking the limit
, one obtains
.
The relationship between zeta at the positive even integers and the Bernoulli numbers may be written as
where
and
are integers for all even
. These are given by the integer sequences and , respectively, in
OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
. Some of these values are reproduced below:
If we let
be the coefficient of
as above,
then we find recursively,
This recurrence relation may be derived from that for the
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s.
Also, there is another recurrence:
which can be proved, using that
The values of the zeta function at non-negative even integers have the
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
:
Since
The formula also shows that for
,
Odd positive integers
The sum of the
harmonic series is infinite.
The value is also known as
Apéry's constant
In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number
:
\begin
\zeta(3) &= \sum_^\infty \frac \\
&= \lim_ \left(\frac + \frac + \cdots + \frac\right),
\end
...
and has a role in the electron's gyromagnetic ratio.
The value also appears in
Planck's law
In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature , when there is no net flow of matter or energy between the body and its environment.
At ...
.
These and additional values are:
It is known that is irrational (
Apéry's theorem
In mathematics, Apéry's theorem is a result in number theory that states the Apéry's constant ζ(3) is irrational. That is, the number
:\zeta(3) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots = 1.2020569\ldots
cannot be written as a fract ...
) and that infinitely many of the numbers , 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 is irrational.
The positive odd integers of the zeta function appear in physics, specifically
correlation functions of antiferromagnetic
XXX spin chain.
Most of the identities following below are provided by
Simon Plouffe
Simon Plouffe (born June 11, 1956) is a mathematician who discovered the Bailey–Borwein–Plouffe formula (BBP algorithm) which permits the computation of the ''n''th binary digit of π, in 1995. His other 2022 formula allows extracting the '' ...
. 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
''ζ''(7)
Note that the sum is in the form of a
Lambert series
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form
:S(q)=\sum_^\infty a_n \frac .
It can be resumed formally by expanding the denominator:
:S(q)=\sum_^\infty a_n \sum_^\infty q^ = \sum_^\infty b_m ...
.
''ζ''(2''n'' + 1)
By defining the quantities
a series of relationships can be given in the form
where ''A''
''n'', ''B''
''n'', ''C''
''n'' and ''D''
''n'' are positive integers. Plouffe gives a table of values:
These integer constants may be expressed as sums over Bernoulli numbers, as given in (Vepstas, 2006) below.
A fast algorithm for the calculation of Riemann's zeta function for any integer argument is given by E. A. Karatsuba.
Negative integers
In general, for negative integers (and also zero), one has
The so-called "trivial zeros" occur at the negative even integers:
(
Ramanujan summation
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has pro ...
)
The first few values for negative odd integers are
However, just like the
Bernoulli numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
, these do not stay small for increasingly negative odd values. For details on the first value, see
1 + 2 + 3 + 4 + · · ·
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1 ...
.
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
The first few values of which are
One also has
where ''A'' is the
Glaisher–Kinkelin constant In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted , is a mathematical constant, related to the -function and the Barnes -function. The constant appears in a number of sums and integrals, especially those ...
. The first of these identities implies that the regularized product of the reciprocals of the positive integers is
, thus the amusing "equation"
.
From the logarithmic derivative of the functional equation,
Series involving ''ζ''(''n'')
The following sums can be derived from the generating function:
where is the
digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
:\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac.
It is the first of the polygamma functions. It is strictly increasing and strictly ...
.
Series related to the
Euler–Mascheroni constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural l ...
(denoted by ) are
and using the principal value
which of course affects only the value at 1, these formulae can be stated as
and show that they depend on the principal value of
Nontrivial zeros
Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". The
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
states that the real part of every nontrivial zero must be . In other words, all known nontrivial zeros of the Riemann zeta are of the form where ''y'' is a real number. The following table contains the decimal expansion of Im(''z'') for the first few nontrivial zeros:
Andrew Odlyzko
Andrew Michael Odlyzko (Andrzej Odłyżko) (born 23 July 1949) is a Polish-American mathematician and a former head of the University of Minnesota's Digital Technology Center and of the Minnesota Supercomputing Institute. He began his career in ...
computed the first 2 million nontrivial zeros accurate to within 4, and the first 100 zeros accurate within 1000 decimal places. See their website for the tables and bibliographies.
Ratios
Although evaluating particular values of the zeta function is difficult, often certain ratios can be found by inserting
particular values of the gamma function
The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general. Othe ...
into the functional equation
We have simple relations for half-integer arguments
Other examples follow for more complicated evaluations and relations of the gamma function. For example a consequence of the relation
is the zeta ratio relation
where AGM is the
arithmetic–geometric mean. In a similar vein, it is possible to form radical relations, such as from
:
the analogous zeta relation is
References
Further reading
*
*
Simon Plouffe
Simon Plouffe (born June 11, 1956) is a mathematician who discovered the Bailey–Borwein–Plouffe formula (BBP algorithm) which permits the computation of the ''n''th binary digit of π, in 1995. His other 2022 formula allows extracting the '' ...
,
Identities inspired from Ramanujan Notebooks", (1998).
*
Simon Plouffe
Simon Plouffe (born June 11, 1956) is a mathematician who discovered the Bailey–Borwein–Plouffe formula (BBP algorithm) which permits the computation of the ''n''th binary digit of π, in 1995. His other 2022 formula allows extracting the '' ...
,
Identities inspired by Ramanujan Notebooks part 2PDF
" (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
''Uspekhi Matematicheskikh Nauk'' (russian: Успехи математических наук) is a Russian mathematical journal, published by the Russian Academy of Sciences and Moscow Mathematical Society and translated into English as ''Russi ...
, volume= 56
, pages=774–776
, year=2001
, issue=4
, doi=10.1070/RM2001v056n04ABEH000427
, mr=1861452
, bibcode=2001RuMaS..56..774Z
, s2cid=250734661
}
PDFPDF RussianPS Russian
* Nontrival zeros reference by Andrew Odlyzko
Andrew Michael Odlyzko (Andrzej Odłyżko) (born 23 July 1949) is a Polish-American mathematician and a former head of the University of Minnesota's Digital Technology Center and of the Minnesota Supercomputing Institute. He began his career in ...
:
*
Bibliography
*
Mathematical constants
Zeta and L-functions
Irrational numbers