Riemann zeta function
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The Riemann zeta function or Euler–Riemann zeta function, denoted by the
The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics.
The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ(2), provides a solution to the Basel problem. In 1979 Roger Apéry proved the irrationality of ζ(3). The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known.
Definition
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Ueber_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Gr%C3%B6sse.pdf/page1-170px-Ueber_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Gr%C3%B6sse.pdf.jpg)
The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it, where σ and t are real numbers. (The notation s, σ, and t is used traditionally in the study of the zeta function, following Riemann.) When Re(s) = σ > 1, the function can be written as a converging summation or as an integral:
where
is the gamma function. The Riemann zeta function is defined for other complex values via analytic continuation of the function defined for σ > 1.
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
Euler's product formula
In 1737, the connection between the zeta function and prime numbers was discovered by 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 Πp p/p − 1) implies that there are infinitely many primes.[5] Since the logarithm of p/p − 1 is approximately 1/p, the formula can also be used to prove the stronger result that the sum of the reciprocals of the primes is infinite. On the other hand, combining that with the sieve of Eratosthenes shows that the density of the set of primes within the set of positive integers is zero.
The Euler product formula can be used to calculate the
Riemann's functional equation
This zeta function satisfies the functional equation where Γ(s) is the
A proof of the functional equation proceeds as follows: We observe that if , then
As a result, if then with the inversion of the limiting processes justified by absolute convergence (hence the stricter requirement on ).
For convenience, let
which is a special case of the theta function. Then
By the Poisson summation formula we have
so that
Hence
This is equivalent to or
So
which is convergent for all s, so holds by analytic continuation. Furthermore, the RHS is unchanged if s is changed to 1 − s. Hence
which is the functional equation.
E. C. Titchmarsh (1986). The Theory of the Riemann Zeta-function (2nd ed.).
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 (the alternating zeta function):
Incidentally, this relation gives an equation for calculating ζ(s) in the region 0 < Re(s) < 1, i.e. where the η-series is convergent (albeit non-absolutely) in the larger half-plane s > 0 (for a more detailed survey on the history of the functional equation, see e.g. Blagouchine[6][7]).
Riemann also found a symmetric version of the functional equation applying to the xi-function: which satisfies:
(Riemann's
The factor was not well-understood at the time of Riemann, until
Zeros, the critical line, and the Riemann hypothesis
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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 important results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip , which is called the critical strip. The set is called the critical line. The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that all non-trivial zeros are on the critical line. In 1989, Conrey proved that more than 40% of the non-trivial zeros of the Riemann zeta function are on the critical line.[8]
For the Riemann zeta function on the critical line, see Z-function.
Zero |
---|
1/2 ± 14.134725 i |
1/2 ± 21.022040 i |
1/2 ± 25.010858 i |
1/2 ± 30.424876 i |
1/2 ± 32.935062 i |
1/2 ± 37.586178 i |
1/2 ± 40.918719 i |
Number of zeros in the critical strip
Let be the number of zeros of in the critical strip , whose imaginary parts are in the interval . Trudgian proved that, if , then[11]
- .
The Hardy–Littlewood conjectures
In 1914, Godfrey Harold Hardy proved that ζ (1/2 + it) has infinitely many real zeros.[12]
Hardy and John Edensor Littlewood formulated two conjectures on the density and distance between the zeros of ζ (1/2 + it) on intervals of large positive real numbers. In the following, N(T) is the total number of real zeros and N0(T) the total number of zeros of odd order of the function ζ (1/2 + it) lying in the interval (0, T].
- For any ε > 0, there exists a T0(ε) > 0 such that when
- For any ε > 0, there exists a T0(ε) > 0 and cε > 0 such that the inequality
These two conjectures opened up new directions in the investigation of the Riemann zeta function.
Zero-free region
The location of the Riemann zeta function's zeros is of great importance in number theory. The prime number theorem is equivalent to the fact that there are no zeros of the zeta function on the Re(s) = 1 line.[13] A better result[14] that follows from an effective form of Vinogradov's mean-value theorem is that ζ (σ + it) ≠ 0 whenever and |t| ≥ 3.
In 2015, Mossinghoff and Trudgian proved[15] that zeta has no zeros in the region
for |t| ≥ 2. This is the largest known zero-free region in the critical strip for .
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.
Other results
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
In the critical strip, the zero with smallest non-negative imaginary part is 1/2 + 14.13472514...i (OEIS: A058303). The fact that
for all complex s ≠ 1 implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line Re(s) = 1/2.
It is also known that no zeros lie on the line with real part 1.
Specific values
For any positive even integer 2n, where B2n is the 2n-th Bernoulli number. For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraic K-theory of the integers; see Special values of L-functions.
For nonpositive integers, one has for n ≥ 0 (using the convention that B1 = 1/2). In particular, ζ vanishes at the negative even integers because Bm = 0 for all odd m other than 1. These are the so-called "trivial zeros" of the zeta function.
Via analytic continuation, one can show that This gives a pretext for assigning a finite value to the divergent series 1 + 2 + 3 + 4 + ⋯, which has been used in certain contexts (Ramanujan summation) such as string theory.[16] Analogously, the particular value can be viewed as assigning a finite result to the divergent series 1 + 1 + 1 + 1 + ⋯.
The value is employed in calculating kinetic boundary layer problems of linear kinetic equations.[17][18]
Although diverges, its Cauchy principal value exists and is equal to the
The demonstration of the particular value is known as the
Taking the limit through the real numbers, one obtains . But at
Various properties
For sums involving the zeta function at integer and half-integer values, see rational zeta series.
Reciprocal
The 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 greater than 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.
Universality
The 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. The first proof of universality was provided by Sergei Mikhailovitch Voronin in 1975.[21] More recent work has included effective versions of Voronin's theorem[22] and extending it to Dirichlet L-functions.[23][24]
Estimates of the maximum of the modulus of the zeta function
Let the functions F(T;H) and G(s0;Δ) be defined by the equalities
Here T is a sufficiently large positive number, 0 < H ≪ log log T, s0 = σ0 + iT, 1/2 ≤ σ0 ≤ 1, 0 < Δ < 1/3. 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 0 ≤ Re(s) ≤ 1.
The case H ≫ log log T was studied by Kanakanahalli Ramachandra; the case Δ > c, where c is a sufficiently large constant, is trivial.
in particular, that if the values H and Δ exceed certain sufficiently small constants, then the estimateshold, where c1 and c2 are certain absolute constants.
The argument of the Riemann zeta function
The function
is called the
There are some theorems on properties of the function S(t). Among those results
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
Representations
Dirichlet series[29]
An extension of the area of convergence can be obtained by rearranging the original series. The series
converges for Re(s) > 0, while
converge even for Re(s) > −1. In this way, the area of convergence can be extended to Re(s) > −k for any negative integer −k.
The recurrence connection is clearly visible from the expression valid for Re(s) > −2 enabling further expansion by integration by parts.
Mellin-type integrals
The Mellin transform of a function f(x) is defined as[30]
in the region where the integral is defined. There are various expressions for the zeta function as Mellin transform-like integrals. If the real part of s is greater than one, we have
- and ,
where Γ denotes the gamma function. By modifying the contour, Riemann showed that
for all s (where H denotes the Hankel contour).
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 function J(x), which counts prime powers pn with a weight of 1/n, so that
Now
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.
Theta functions
The Riemann zeta function can be given by a Mellin transform[31]
in terms of Jacobi's theta function
However, this integral only converges if the real part of s is greater than 1, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all s except 0 and 1:
Laurent series
The Riemann zeta function is
The constants γn here are called the Stieltjes constants and can be defined by the limit
The constant term γ0 is the
Integral
For all s ∈ C, s ≠ 1, the integral relation (cf. Abel–Plana formula)
holds true, which may be used for a numerical evaluation of the zeta function.
Rising factorial
Another series development using the
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
Hadamard product
On the basis of
where the product is over the non-trivial zeros ρ of ζ and the letter γ again denotes the
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. (To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i.e. the factors for a pair of zeros of the form ρ and 1 − ρ should be combined.)
Globally convergent series
A globally convergent series for the zeta function, valid for all complex numbers s except s = 1 + 2πi/ln 2n for some integer n, was conjectured by Konrad Knopp in 1926 [34] and proven by Helmut Hasse in 1930[35] (cf. Euler summation):
The series appeared in an appendix to Hasse's paper, and was published for the second time by Jonathan Sondow in 1994.[36]
Hasse also proved the globally converging series
in the same publication.[35] Research by Iaroslav Blagouchine[37][34] has found that a similar, equivalent series was published by Joseph Ser in 1926.[38]
In 1997 K. Maślanka gave another globally convergent (except s = 1) series for the Riemann zeta function:
where real coefficients are given by:
Here are the Bernoulli numbers and denotes the Pochhammer symbol.[39][40]
Note that this representation of the zeta function is essentially an interpolation with nodes, where the nodes are points , i.e. exactly those where the zeta values are precisely known, as Euler showed. An elegant and very short proof of this representation of the zeta function, based on Carlson's theorem, was presented by Philippe Flajolet in 2006.[41]
The asymptotic behavior of the coefficients is rather curious: for growing values, we observe regular oscillations with a nearly exponentially decreasing amplitude and slowly decreasing frequency (roughly as ). Using the saddle point method, we can show that
where stands for:
(see [42] for details).
On the basis of this representation, in 2003 Luis Báez-Duarte provided a new criterion for the Riemann hypothesis.[43][44][45] Namely, if we define the coefficients as
then the Riemann hypothesis is equivalent to
Rapidly convergent series
Series representation at positive integers via the primorial
Here pn# is the primorial sequence and Jk is Jordan's totient function.[47]
Series representation by the incomplete poly-Bernoulli numbers
The function ζ can be represented, for Re(s) > 1, by the infinite series
where k ∈ {−1, 0}, Wk is the kth branch of the Lambert W-function, and B(μ)
n, ≥2 is an incomplete poly-Bernoulli number.[48]
The Mellin transform of the Engel map
The function is iterated to find the coefficients appearing in Engel expansions.[49]
The Mellin transform of the map is related to the Riemann zeta function by the formula
Thue-Morse sequence
Certain linear combinations of Dirichlet series whose coefficients are terms of the
where is the term of the Thue-Morse sequence. In fact, for all with real part greater than , we have
Numerical algorithms
A classical algorithm, in use prior to about 1930, proceeds by applying the
where, letting denote the indicated Bernoulli number,
and the error satisfies
with σ = Re(s).[51]
A modern numerical algorithm is the Odlyzko–Schönhage algorithm.
Applications
The zeta function occurs in applied statistics (see Zipf's law and Zipf–Mandelbrot law).
Musical tuning
In the theory of musical tunings, the zeta function can be used to find equal divisions of the octave (EDOs) that closely approximate the intervals of the harmonic series. For increasing values of , the value of
peaks near integers that correspond to such EDOs.[53] Examples include popular choices such as 12, 19, and 53.[54]
Infinite series
The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.[55]
In fact the even and odd terms give the two sums
and
Parametrized versions of the above sums are given by
and
with and where and are the polygamma function and Euler's constant, respectively, as well as
all of which are continuous at . Other sums include
where Im denotes the
There are yet more formulas in the article Harmonic number.
Generalizations
There are a number of related
(the convergent series representation was given by
The polylogarithm is given by
which coincides with the Riemann zeta function when z = 1. The Clausen function Cls(θ) can be chosen as the real or imaginary part of Lis(eiθ).
The
which coincides with the Riemann zeta function when z = 1 and q = 1 (the lower limit of summation in the Lerch transcendent is 0, not 1).
The
One can analytically continue these functions to the n-dimensional complex space. The special values taken by these functions at positive integer arguments are called
See also
- 1 + 2 + 3 + 4 + ···
- Arithmetic zeta function
- Generalized Riemann hypothesis
- Lehmer pair
- Prime zeta function
- Riemann Xi function
- Renormalization
- Riemann–Siegel theta function
- ZetaGrid
Notes
- ^ "Jupyter Notebook Viewer". Nbviewer.ipython.org. Retrieved 4 January 2017.
- ^ S2CID 216323223.
Theorem 2 implies that ζ has an essential singularity at infinity
- ^ Bombieri, Enrico. "The Riemann Hypothesis – official problem description" (PDF). Clay Mathematics Institute. Archived from the original (PDF) on 22 December 2015. Retrieved 8 August 2014.
- ISBN 978-0-7607-8659-8.
- ISBN 978-0-88385-563-8.
- ^ Blagouchine, I.V. (1 March 2018). The history of the functional equation of the zeta-function. Seminar on the History of Mathematics. St. Petersburg, RU: Steklov Institute of Mathematics; "online PDF". Archived from the original on 2 May 2018. Retrieved 2 May 2018.
- S2CID 125198685. Archived from the originalon 2 May 2018. Retrieved 2 May 2018.
- S2CID 115910600.
- Eric Weisstein. "Riemann Zeta Function Zeros". Retrieved 24 April 2021.
- ^ The L-functions and Modular Forms Database. "Zeros of ζ(s)".
- .
- .
- MR 0670132.
- S2CID 121144007.
- S2CID 117968965.
- ISBN 978-0-521-63303-1.
- .
- ^ Further digits and references for this constant are available at OEIS: A059750.
- doi:10.1080/0025570X.1998.11996638. Archived from the originalon 4 June 2011. Retrieved 29 May 2006.
- ISBN 0-486-25778-9.
- ^ Voronin, S. M. (1975). "Theorem on the Universality of the Riemann Zeta Function". Izv. Akad. Nauk SSSR, Ser. Matem. 39: 475–486. Reprinted in Math. USSR Izv. (1975) 9: 443–445.
- JSTOR 43736941.
- S2CID 120930513.
- ISBN 978-3-540-26526-9.
- ^ Karatsuba, A. A. (2001). "Lower bounds for the maximum modulus of ζ(s) in small domains of the critical strip". Mat. Zametki. 70 (5): 796–798.
- S2CID 250796539.
- ^ Karatsuba, A. A. (1996). "Density theorem and the behavior of the argument of the Riemann zeta function". Mat. Zametki (60): 448–449.
- ^ Karatsuba, A. A. (1996). "On the function S(t)". Izv. Ross. Akad. Nauk, Ser. Mat. 60 (5): 27–56.
- ^ a b Knopp, Konrad (1947). Theory of Functions, Part Two. New York, Dover publications. pp. 51–55.
- Zbl 0315.10035.
- ISBN 3-540-65399-6.
- MR 2115723.
- ^ "A series representation for the Riemann Zeta derived from the Gauss-Kuzmin-Wirsing Operator" (PDF). Linas.org. Retrieved 4 January 2017.
- ^ Bibcode:2016arXiv160602044B.
- ^ S2CID 120392534.
- .
- .
- Comptes rendus hebdomadaires des séances de l'Académie des Sciences(in French). 182: 1075–1077.
- ^ Maślanka, Krzysztof (1997). "The Beauty of Nothingness". Acta Cosmologica. XXIII–I: 13–17.
- .
- .
- S2CID 252780397.
- Bibcode:2003math......7215B.
- Bibcode:2006math......3713M.
- .
- ISBN 978-0-8218-2167-1. Archived from the original(PDF) on 26 July 2011. Retrieved 25 November 2017.
- ^ Mező, István (2013). "The primorial and the Riemann zeta function". The American Mathematical Monthly. 120 (4): 321.
- S2CID 55741906.
- ^ "A220335 - OEIS". oeis.org. Retrieved 17 April 2019.
- ^
arXiv:2211.13570.
- MR 0961614..
- ^ "Work on spin-chains by A. Knauf, et. al". Empslocal.ex.ac.uk. Retrieved 4 January 2017.
- ^ Gene Ward Smith. "Nearest integer to locations of increasingly large peaks of abs(zeta(0.5 + i*2*Pi/log(2)*t)) for increasing real t". The On-Line Encyclopedia of Integer Sequences. Retrieved 4 March 2022.
- ^ William A. Sethares (2005). Tuning, Timbre, Spectrum, Scale (2nd ed.). Springer-Verlag London. p. 74.
...there are many different ways to evaluate the goodness, reasonableness, fitness, or quality of a scale...Under some measures, 12-tet is the winner, under others 19-tet appears best, 53-tet often appears among the victors...
- ^ Most of the formulas in this section are from § 4 of J. M. Borwein et al. (2000)
References
- Apostol, T. M. (2010). "Zeta and Related Functions". In MR 2723248..
- doi:10.1016/S0377-0427(00)00336-8. Archived from the original(PDF) on 13 December 2013.
- Cvijović, Djurdje; Klinowski, Jacek (2002). "Integral Representations of the Riemann Zeta Function for Odd-Integer Arguments". J. Comput. Appl. Math. 142 (2): 435–439. MR 1906742.
- Cvijović, Djurdje; Klinowski, Jacek (1997). "Continued-fraction expansions for the Riemann zeta function and polylogarithms". Proc. Amer. Math. Soc. 125 (9): 2543–2550. .
- ISBN 0-486-41740-9. Has an English translation of Riemann's paper.
- .
- Hardy, G. H. (1949). Divergent Series. Clarendon Press, Oxford.
- S2CID 120392534. (Globally convergent series expression.)
- Ivic, A. (1985). The Riemann Zeta Function. John Wiley & Sons. ISBN 0-471-80634-X.
- Motohashi, Y. (1997). Spectral Theory of the Riemann Zeta-Function. Cambridge University Press. ISBN 0521445205.
- Karatsuba, A. A.; Voronin, S. M. (1992). The Riemann Zeta-Function. Berlin: W. de Gruyter.
- Mező, István; Dil, Ayhan (2010). "Hyperharmonic series involving Hurwitz zeta function". Journal of Number Theory. 130 (2): 360–369. S2CID 122707401.
- ISBN 978-0-521-84903-6.
- ISBN 0-387-98308-2.
- Raoh, Guo (1996). "The Distribution of the Logarithmic Derivative of the Riemann Zeta Function". Proceedings of the London Mathematical Society. s3–72: 1–27. .
- Riemann, Bernhard (1859). "Über die Anzahl der Primzahlen unter einer gegebenen Grösse". Monatsberichte der Berliner Akademie.. In Gesammelte Werke, Teubner, Leipzig (1892), Reprinted by Dover, New York (1953).
- Sondow, Jonathan (1994). "Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series" (PDF). Proc. Amer. Math. Soc. 120 (2): 421–424. .
- Titchmarsh, E. C. (1986). Heath-Brown (ed.). The Theory of the Riemann Zeta Function (2nd rev. ed.). Oxford University Press.
- Whittaker, E. T.; Watson, G. N. (1927). A Course in Modern Analysis (4th ed.). Cambridge University Press. Ch. 13.
- Zhao, Jianqiang (1999). "Analytic continuation of multiple zeta functions". Proc. Amer. Math. Soc. 128 (5): 1275–1283. MR 1670846.
External links
Media related to Riemann zeta function at Wikimedia Commons
- "Zeta-function". Encyclopedia of Mathematics. EMS Press. 2001 [1994].
- Riemann Zeta Function, in Wolfram Mathworld — an explanation with a more mathematical approach
- Tables of selected zeros Archived 17 May 2009 at the Wayback Machine
- Prime Numbers Get Hitched A general, non-technical description of the significance of the zeta function in relation to prime numbers.
- X-Ray of the Zeta Function Visually oriented investigation of where zeta is real or purely imaginary.
- Formulas and identities for the Riemann Zeta function functions.wolfram.com
- Riemann Zeta Function and Other Sums of Reciprocal Powers, section 23.2 of Abramowitz and Stegun
- Frenkel, Edward. "Million Dollar Math Problem" (video). Brady Haran. Archived from the original on 11 December 2021. Retrieved 11 March 2014.
- Mellin transform and the functional equation of the Riemann Zeta function—Computational examples of Mellin transform methods involving the Riemann Zeta Function
- Visualizing the Riemann zeta function and analytic continuation a video from 3Blue1Brown