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Riemann zeta function
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zeta(s) =sum_{n=1}^inftyfrac{1}{n^s}
which converges when the real part of {{mvar|s}} is greater than 1. More general representations of {{math|Î¶(s)}} for all {{mvar|s}} are given below. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.As a function of a real variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis, which was not available at the time. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers.This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is considered by many mathematicians to be the most important unsolved problem in pure mathematics.WEB, Bombieri, Enrico,weblink The Riemann Hypothesis â€“ official problem description, Clay Mathematics Institute, 2014-08-08, The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, {{math|Î¶(2)}}, provides a solution to the Basel problem. In 1979 Roger ApÃ©ry proved the irrationality of {{math|Î¶(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 {{mvar|L}}-functions and {{mvar|L}}-functions, are known.Definition
(File:Ueber die Anzahl der Primzahlen unter einer gegebenen GrÃ¶sse.pdf|thumb|Bernhard Riemann's article On the number of primes below a given magnitude.)The Riemann zeta function {{math|Î¶(s)}} is a function of a complex variable {{math|s {{=}} Ïƒ + it}}. (The notation {{mvar|s}}, {{mvar|Ïƒ}}, and {{mvar|t}} is used traditionally in the study of the zeta function, following Riemann.)The zeta function can be expressed by the following integral:
zeta(s) = frac{1}{Gamma(s)} int_0^infty frac{x ^ {s-1}}{e ^ x - 1} , mathrm{d}x
where
Gamma(s) = int_0^infty x^{s-1},e^{-x} , mathrm{d}x
is the gamma function.For the special case where the real part of {{mvar|s}} is greater than 1, {{math|Î¶(s)}} always converges, and can be simplified to the following infinite series:
zeta(s) = sum_{n=1}^infty n^{-s} = frac{1}{1^s} + frac{1}{2^s} + frac{1}{3^s} + cdots quad sigma = operatorname{Re}(s) > 1.
The Riemann zeta function is defined as the analytic continuation of the function defined for {{math|Ïƒ > 1}} by the sum of the preceding series.Leonhard Euler considered the above series in 1740 for positive integer values of {{mvar|s}}, and later Chebyshev extended the definition to {{math|Re(s) > 1}}.BOOK, Devlin, Keith, Keith Devlin, The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, Barnes & Noble, 2002, New York, 43â€“47, 978-0-7607-8659-8, The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for {{mvar|s}} such that {{math|Ïƒ > 1}} and diverges for all other values of {{mvar|s}}. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values {{math|s â‰ 1}}. For {{math|s {{=}} 1}} the series is the harmonic series which diverges to {{math|+âˆž}}, and
lim_{sto 1}(s-1)zeta(s)=1.
Thus the Riemann zeta function is a meromorphic function on the whole complex {{mvar|s}}-plane, which is holomorphic everywhere except for a simple pole at {{math|s {{=}} 1}} with residue 1.Specific values
For any positive even integer {{math|2n}}:
zeta(2n) = frac{(-1)^{n+1}B_{2n}(2pi)^{2n}}{2(2n)!}
where {{math|B2n}} is the {{math|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 {{mvar|K}}-theory of the integers; see Special values of {{mvar|L}}-functions.For nonpositive integers, one has
zeta(-n)=(-1)^n frac{B_{n+1}}{n+1}
for {{math|n â‰¥ 0}} (using the NIST convention that {{math|B1 {{=}} âˆ’{{sfrac|1|2}}}})In particular, {{mvar|Î¶}} vanishes at the negative even integers because {{math|Bm {{=}} 0}} for all odd {{mvar|m}} other than 1. These are the so-called "trivial zeros" of the zeta function.Via analytic continuation, one can show that: - zeta(-1) = -tfrac{1}{12}
This gives a way to assign a finite result to the divergent series 1 + 2 + 3 + 4 + â‹¯, which has been used in certain contexts such as string theory.BOOK, Polchinski, Joseph, Joseph Polchinski, String Theory, Volume I: An Introduction to the Bosonic String, Cambridge University Press, 1998, 22, 978-0-521-63303-1,
- zeta(0) = -tfrac{1}{2};
Similarly to the above, this assigns a finite result to the series 1 + 1 + 1 + 1 + â‹¯.
- zetabigl(tfrac12bigr) approx -1.46035 45088 09586 81289 ({{OEIS2C|id=A059750}})
This is employed in calculating of kinetic boundary layer problems of linear kinetic equations.JOURNAL, A. J., Kainz, U. M., Titulaer, An accurate two-stream moment method for kinetic boundary layer problems of linear kinetic equations, 1855â€“1874, J. Phys. A: Math. Gen., 25, 7, 1992, 1992JPhA...25.1855K, 10.1088/0305-4470/25/7/026,
- zeta(1) = 1 + tfrac{1}{2} + tfrac{1}{3} + cdots = infty;
If we approach from numbers larger than 1, this is the harmonic series. But its Cauchy principal value
lim_{varepsilon to 0} frac{zeta(1+varepsilon)+zeta(1-varepsilon)}{2}
exists which is the Eulerâ€“Mascheroni constant {{math|Î³ {{=}} 0.5772â€¦}}.- zetabigl(tfrac32bigr) approx 2.61237 53486 85488 34335; ({{OEIS2C|id=A078434}})
This is employed in calculating the critical temperature for a Boseâ€“Einstein condensate in a box with periodic boundary conditions, and for spin wave physics in magnetic systems.
- zeta(2) = 1 + frac{1}{2^2} + frac{1}{3^2} + cdots = frac{pi^2}{6} approx 1.64493 40668 48226 43647;! ({{OEIS2C|id=A013661}})
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?BOOK, C. Stanley Ogilvy, C. S., Ogilvy, J. T., Anderson, Excursions in Number Theory, 29â€“35, Dover Publications, 1988, 0-486-25778-9,
- zeta(3) = 1 + frac{1}{2^3} + frac{1}{3^3} + cdots approx 1.20205 69031 59594 28540; ({{OEIS2C|id=A002117}})
This number is called ApÃ©ry's constant.
- zeta(4) = 1 + frac{1}{2^4} + frac{1}{3^4} + cdots = frac{pi^4}{90} approx 1.08232 32337 11138 19152; ({{OEIS2C|id=A0013662}})
This appears when integrating Planck's law to derive the Stefanâ€“Boltzmann law in physics.
Euler product formula
The connection between the zeta function and prime numbers was discovered by Euler, who proved the identity
sum_{n=1}^inftyfrac{1}{n^s} = prod_{p text{ prime}} frac{1}{1-p^{-s}},
where, by definition, the left hand side is {{math|Î¶(s)}} and the infinite product on the right hand side extends over all prime numbers {{mvar|p}} (such expressions are called Euler products):
prod_{p text{ prime}} frac{1}{1-p^{-s}} = frac{1}{1-2^{-s}}cdotfrac{1}{1-3^{-s}}cdotfrac{1}{1-5^{-s}}cdotfrac{1}{1-7^{-s}}cdotfrac{1}{1-11^{-s}} cdots frac{1}{1-p^{-s}} cdots
Both sides of the Euler product formula converge for {{math|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 {{math|s {{=}} 1}}, diverges, Euler's formula (which becomes {{math|âˆp {{sfrac|p|p âˆ’ 1}}}}) implies that there are infinitely many primes.BOOK, Charles Edward, Sandifer, How Euler Did It, Mathematical Association of America, 2007, 193, 978-0-88385-563-8, The Euler product formula can be used to calculate the asymptotic probability that {{mvar|s}} randomly selected integers are set-wise coprime. Intuitively, the probability that any single number is divisible by a prime (or any integer) {{mvar|p}} is {{math|{{sfrac|1|p}}}}. Hence the probability that {{mvar|s}} numbers are all divisible by this prime is {{math|{{sfrac|1|p{{isup|s}}}}}}, and the probability that at least one of them is not is {{math|1 âˆ’ {{sfrac|1|p{{isup|s}}}}}}. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors {{mvar|n}} and {{mvar|m}} if and only if it is divisible by {{mvar|nm}}, an event which occurs with probability {{math|{{sfrac|1|nm}}}}). Thus the asymptotic probability that {{mvar|s}} numbers are coprime is given by a product over all primes,
prod_{p text{ prime}} left(1-frac{1}{p^s}right) = left( prod_{p text{ prime}} frac{1}{1-p^{-s}} right)^{-1} = frac{1}{zeta(s)}.
(More work is required to derive this result formally.)JOURNAL, J. E., Nymann, On the probability that {{mvar, k, positive integers are relatively prime|journal=Journal of Number Theory|volume=4|year=1972|pages=469â€“473|doi=10.1016/0022-314X(72)90038-8|issue=5|bibcode = 1972JNT.....4..469N }}Riemann's functional equation
The zeta function satisfies the functional equation:
zeta(s) = 2^spi^{s-1} sinleft(frac{pi s}{2}right) Gamma(1-s) zeta(1-s),
where {{math|Î“(s)}} is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points {{mvar|s}} and {{math|1 âˆ’ s}}, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that {{math|Î¶(s)}} has a simple zero at each even negative integer {{math|s {{=}} âˆ’2n}}, known as the trivial zeros of {{math|Î¶(s)}}. When {{mvar|s}} is an even positive integer, the product {{math|sin({{sfrac|{{pi}}s|2}})Î“(1 âˆ’ s)}} on the right is non-zero because {{math|Î“(1 âˆ’ s)}} has a simple pole, which cancels the simple zero of the sine factor.{{Collapse top|title=Proof of functional equation}}A proof of the functional equation proceeds as follows:We observe that if sigma > 0, then
int_0^infty x^{{1over2}{s} - 1}e^{-n^2pi x}, dx = {Gamma left({sover2}right)over{n^spi^{sover2}}}.
As a result, if sigma > 1 then
frac{Gammaleft(frac s 2right)zeta(s)}{pi^{s/2}} = sum_{n=1}^infty intlimits_0^infty x^{{sover 2}-1} e^{-n^2 pi x}, dx = int_0^infty x^{{sover 2}-1} sum_{n=1}^infty e^{-n^2 pi x}, dx.
With the inversion of the limiting processes justified by absolute convergence (hence the stricter requirement on sigma)For convenience, let
psi(x) := sum_{n=1}^infty e^{-n^2 pi x}
Then zeta(s) = {pi^{sover2}overGamma({s over 2})} intlimits_0^infty x^{{1over2}{s} - 1}psi(x), dxGiven that sum_{n=-infty}^infty {e^{-n^2 pi x}} = {1 over sqrt x}sum_{n=-infty}^infty {e^{-n^2 pi over x}}Then 2 psi(x)+1 = {1over sqrt x} left{2psi left ( {1 over x} right )+1right}Hence pi^{-{s over 2}} Gamma left ( {s over 2} right ) zeta (s) = int_0^1 x^{hide}sover 2}-1} psi(x) , dx + int_1^infty x^{{sover 2}-1} psi(x) , dxThis is equivalent to intlimits_0^1 x^{{sover 2}-1} left{ {1over sqrt x} psi left ( {1 over x} right ) + {1over 2 sqrt x} - {1 over 2} right} , dx + intlimits_1^infty x^{{sover 2}-1} psi(x) , dxOr
{} & {1 over {s({s-1})}} + intlimits_1^infty left ({x^{-{{s}over 2}-{1over 2}} + x^{{{s}over 2}-1}} right ) psi(x) , dx
end{align}which is convergent for all s, so holds by analytic continuation. Furthermore, the RHS is unchanged if s is changed to 1 âˆ’ s. Hence
pi^{-{s over 2}} Gamma left ( {s over 2} right ) zeta (s) = pi^{-{1 over 2} + {s over 2}} Gamma left ( {1 over 2} - {s over 2} right ) zeta (1-s)
which is the functional equation.BOOK, The Theory of the Riemann Zeta-function, Oxford Science Publications, {{date, 1986, |edition=2nd|pages=21â€“22|author=E. C. Titchmarsh|location=Oxford|isbn=0-19-853369-1}} Attributed to Bernhard Riemann.{{Collapse bottom}}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):
eta(s)= sum_{n=1}^infty frac{(-1)^{n+1}}{n^s} = left(1-{2^{1-s}}right)zeta(s).
Incidentally, this relation gives an equation for calculating {{math|Î¶(s)}} in the region 0 < {{math|Re(s)}} < 1, i.e.
zeta(s)=frac{1}{1-{2^{1-s}}}sum_{n=1}^infty frac{(-1)^{n+1}}{n^s} ,
where the η-series is convergent (albeit non-absolutely) in the larger half-plane {{math|s > 0}} (for a more detailed survey on the history of the functional equation, see e.g. BlagouchineI. V. Blagouchine The history of the functional equation of the zeta-function. Seminar on the History of Mathematics, Steklov Institute of Mathematics at St. Petersburg, 1 March 2018. PDFI. V. Blagouchine Rediscovery of Malmstenâ€™s integrals, their evaluation by contour integration methods and some related results. The Ramanujan Journal, vol. 35, no. 1, pp. 21-110, 2014. Addendum: vol. 42, pp. 777â€“781, 2017. PDF).Riemann also found a symmetric version of the functional equation applying to the xi-function:
xi(s) = frac{1}{2}pi^{-frac{s}{2}}s(s-1)Gammaleft(frac{s}{2}right)zeta(s),!
which satisfies:
xi(s) = xi(1 - s).!
(Riemann's original {{math|Î¾(t)}} was slightly different.)Zeros, the critical line, and the Riemann hypothesis
File:Zero-free region for the Riemann zeta-function.svg|right|thumb|300px|Apart from the trivial zeros, the Riemann zeta function has no zeros to the right of {{math|Ïƒ {{=}} 1}} and to the left of {{math|Ïƒ {{=}} 0}} (neither can the zeros lie too close to those lines). Furthermore, the non-trivial zeros are symmetric about the real axis and the line {{math|Ïƒ {{=}} {{sfrac|1|2}}}} and, according to the Ïƒ {{=}} {{sfrac|1|2}}}}.right|thumb|300px|This image shows a plot of the Riemann zeta function along the critical line for real values of {{mvar|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.(File:RiemannCriticalLine.svg|thumb|300px|The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivial zeros can be seen at Im(s) = Â±14.135, Â±21.022 and Â±25.011.)The functional equation shows that the Riemann zeta function has zeros at {{nowrap|âˆ’2, âˆ’4,â€¦}}. These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from {{math|sin {{sfrac|Ï€s|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 {{math|{s âˆˆ â„‚ : 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 {{mvar|s}} has {{math|Re(s) {{=}} {{sfrac|1|2}}}}. In the theory of the Riemann zeta function, the set {{math|{s âˆˆ â„‚ : Re(s) {{=}} {{sfrac|1|2}}}}} is called the critical line. For the Riemann zeta function on the critical line, see {{mvar|Z}}-function.The Hardyâ€“Littlewood conjectures
In 1914, Godfrey Harold Hardy proved that {{math|Î¶ ({{sfrac|1|2}} + it)}} has infinitely many real zeros.Hardy and John Edensor Littlewood formulated two conjectures on the density and distance between the zeros of {{math|Î¶ ({{sfrac|1|2}} + it)}} on intervals of large positive real numbers. In the following, {{math|N(T)}} is the total number of real zeros and {{math|N0(T)}} the total number of zeros of odd order of the function {{math|Î¶ ({{sfrac|1|2}} + it)}} lying in the interval {{math|(0, T]}}.{{numbered listÎµ > 0}}, there exists a {{math|T0(Îµ) > 0}} such that when
T geq T_0(varepsilon) quadtext{ and }quad H=T^{frac14+varepsilon},
the interval {{math|(T, T + H]}} contains a zero of odd order.Îµ > 0}}, there exists a {{mathT0(Îµ) > 0}} and {{math>cÎµ > 0}} such that the inequality
N_0(T+H)-N_0(T) geq c_varepsilon H
holds when
T geq T_0(varepsilon) quadtext{ and }quad H=T^{frac12+varepsilon}.
}}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 the theory of numbers. The prime number theorem is equivalent to the fact that there are no zeros of the zeta function on the {{math|Re(s) {{=}} 1}} line.JOURNAL, Harold G., Diamond, Elementary methods in the study of the distribution of prime numbers, Bulletin of the American Mathematical Society, 7, 3, 1982, 553â€“89, 670132, 10.1090/S0273-0979-1982-15057-1, A better resultJOURNAL, Ford, K., 2002, Vinogradov's integral and bounds for the Riemann zeta function, Proc. London Math. Soc., 85, 3, 565â€“633, 10.1112/S0024611502013655, that follows from an effective form of Vinogradov's mean-value theorem is that {{math|Î¶ (Ïƒ + it) â‰ 0}} whenever {{math|{{abs|t}} â‰¥ 3}} and
sigmage 1-frac{1}{57.54(log{|t|})^frac23(log{log{|t|}})^frac13}.
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 ({{math|Î³n}}) contains the imaginary parts of all zeros in the upper half-plane in ascending order, then
lim_{nrightarrowinfty}left(gamma_{n+1}-gamma_nright)=0.
The critical line theorem asserts that a positive proportion of the nontrivial zeros lies on the critical line. (The Riemann hypothesis would imply that this proportion is 1.)In the critical strip, the zero with smallest non-negative imaginary part is {{math|{{sfrac|1|2}} + 14.13472514â€¦i}} ({{OEIS2C|A058303}}). The fact that
zeta(s)=overline{zeta(overline{s})}
for all complex {{math|s â‰ 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 {{math|Re(s) {{=}} {{sfrac|1|2}}}}.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 MÃ¶bius function {{math|Î¼(n)}}:
frac{1}{zeta(s)} = sum_{n=1}^infty frac{mu(n)}{n^s}
for every complex number {{mvar|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 {{mvar|s}} is greater than {{sfrac|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.JOURNAL, Voronin, S. M., 1975, Theorem on the Universality of the Riemann Zeta Function, Izv. Akad. Nauk SSSR, Ser. Matem., 39, 475â€“486, Reprinted in Math. USSR Izv. (1975) 9: 443â€“445. More recent work has included effective versions of Voronin's theorem and extending it to Dirichlet L-functions.BOOK, Steuding, JÃ¶rn, 2007, Value-Distribution of L-Functions, 1877, Berlin, Springer, 19, 978-3-540-26526-9, Lecture Notes in Mathematics
, 10.1007/978-3-540-44822-8, 1711.06671,
, 10.1007/978-3-540-44822-8, 1711.06671,
Estimates of the maximum of the modulus of the zeta function
Let the functions {{math|F(T;H)}} and {{math|G(s0;Î”)}} be defined by the equalities
F(T;H) = max_{|t-T|le H}left|zetaleft(tfrac{1}{2}+itright)right|,qquad G(s_{0};Delta) = max_{|s-s_{0}|leDelta}|zeta(s)|.
Here {{mvar|T}} is a sufficiently large positive number, {{math|0 < H â‰ª ln ln T}}, {{math|s0 {{=}} Ïƒ0 + iT}}, {{math|{{sfrac|1|2}} â‰¤ Ïƒ0 â‰¤ 1}}, {{math|0 < Î” < {{sfrac|1|3}}}}. Estimating the values {{mvar|F}} and {{mvar|G}} from below shows, how large (in modulus) values {{math|Î¶(s)}} can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip {{math|0 â‰¤ Re(s) â‰¤ 1}}.The case {{math|H â‰« ln ln T}} was studied by Kanakanahalli Ramachandra; the case {{math|Î” > c}}, where {{math|c}} is a sufficiently large constant, is trivial.Anatolii Karatsuba proved,JOURNAL, A. A., Karatsuba, Lower bounds for the maximum modulus of {{math, Î¶(s), in small domains of the critical strip | pages=796â€“798| journal= Mat. Zametki| volume=70|issue=5| year=2001}}JOURNAL, A. A., Karatsuba, Lower bounds for the maximum modulus of the Riemann zeta function on short segments of the critical line, 99â€“104, Izv. Ross. Akad. Nauk, Ser. Mat., 68, 8, 2004, in particular, that if the values {{mvar|H}} and {{math|Î”}} exceed certain sufficiently small constants, then the estimates
F(T;H) ge T^{- c_1},qquad G(s_0; Delta) ge T^{-c_2},
hold, where {{math|c1}} and {{math|c2}} are certain absolute constants.The argument of the Riemann zeta function
The function
S(t) = frac{1}{pi}arg{zetaleft(tfrac12+itright)}
is called the argument of the Riemann zeta function. Here {{math|arg Î¶({{sfrac|1|2}} + it)}} is the increment of an arbitrary continuous branch of {{math|arg Î¶(s)}} along the broken line joining the points {{math|2}}, {{math|2 + it}} and {{math|{{sfrac|1|2}} + it}}.There are some theorems on properties of the function {{math|S(t)}}. Among those resultsJOURNAL, A. A., Karatsuba, Density theorem and the behavior of the argument of the Riemann zeta function, 448â€“449, Mat. Zametki, 60, 1996, JOURNAL, A. A., Karatsuba, On the function {{math, S(t), | pages=27â€“56| journal= Izv. Ross. Akad. Nauk, Ser. Mat. |volume=60 |issue=5 |year=1996}} are the mean value theorems for {{math|S(t)}} and its first integral
S_1(t) = int_0^t S(u) , mathrm{d}u
on intervals of the real line, and also the theorem claiming that every interval {{math|(T, T + H]}} for
H ge T^{frac{27}{82}+varepsilon}
contains at least
Hsqrt[3]{ln T}e^{-csqrt{lnln T}}
points where the function {{math|S(t)}} changes sign. Earlier similar results were obtained by Atle Selberg for the case
Hge T^{frac12+varepsilon}.
Representations
Dirichlet series
An extension of the area of convergence can be obtained by rearranging the original series.BOOK, Konrad, Knopp, Theory of Functions, 1945, 51â€“55, The series
zeta(s)=frac{1}{s-1}sum_{n=1}^infty left(frac{n}{(n+1)^s}-frac{n-s}{n^s}right)
converges for {{math|Re(s) > 0}}, while
zeta(s) =frac{1}{s-1}sum_{n=1}^inftyfrac{n(n+1)}{2}left(frac{2n+3+s}{(n+1)^{s+2}}-frac{2n-1-s}{n^{s+2}}right)
converges even for {{math|Re(s) > âˆ’1}}. In this way, the area of convergence can be extended to {{math|Re(s) > âˆ’k}} for any negative integer {{math|âˆ’k}}.Mellin-type integrals
{{unreferenced section|date=December 2014}}The Mellin transform of a function {{math|f(x)}} is defined as
int_0^infty f(x)x^s, frac{mathrm{d}x}{x}
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 {{mvar|s}} is greater than one, we have
Gamma(s)zeta(s) =int_0^inftyfrac{x^{s-1}}{e^x-1} ,mathrm{d}x,
where {{math|Î“}} denotes the gamma function. By modifying the contour, Riemann showed that
2sin(pi s)Gamma(s)zeta(s) =ioint_H frac{(-x)^{s-1}}{e^x-1},mathrm{d}x
for all {{mvar|s}} (where {{mvar|H}} denotes the Hankel contour).Starting with the integral formula zeta(n) {Gamma(n)} =int_{0}^{infty} frac{x ^ {n-1}}{e ^ x - 1} mathrm{d}x, one can showWEB,weblink Evaluating the definite integral..., math.stackexchange.com, by substitution and iterated differentation for natural kge 2
int_{0}^{infty} frac{x ^ {n}e^x}{(e ^ x - 1)^k} mathrm{d}x = frac{n!}{ (k-1)!}zeta^nprod_{j=0}^{k-2}left(1-frac jzetaright)
using the notation of umbral calculus where each power zeta^r is to be replaced by zeta(r), so e.g. for k=2 we have int_{0}^{infty} frac{x ^ {n}e^x}{(e ^ x - 1)^2} mathrm{d}x = {n!}zeta(n) , while for k=4 this becomes
int_{0}^{infty} frac{x ^ {n}e^x}{(e ^ x - 1)^4} mathrm{d}x = frac{n!}{ 6} bigl( zeta^{n-2} -3zeta^{n-1} +2zeta^n bigr)= n!frac{ zeta(n-2) -3zeta(n-1) +2zeta(n) }{ 6}.
We can also find expressions which relate to prime numbers and the prime number theorem. If {{math|Ï€(x)}} is the prime-counting function, then
ln zeta(s) = s int_0^infty frac{pi(x)}{x(x^s-1)},mathrm{d}x,
for values with {{math|Re(s) > 1}}.A similar Mellin transform involves the Riemann prime-counting function {{math|J(x)}}, which counts prime powers {{math|pn}} with a weight of {{math|{{sfrac|1|n}}}}, so that
J(x) = sum frac{pileft(x^frac{1}{n}right)}{n}.
Now we have
ln zeta(s) = sint_0^infty J(x)x^{-s-1},mathrm{d}x.
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 {{math|Ï€(x)}} can be recovered from it by MÃ¶bius inversion.Theta functions
The Riemann zeta function can be given by a Mellin transformBOOK, JÃ¼rgen, Neukirch, Algebraic number theory, Springer, 1999, 422, 3-540-65399-6,
2pi^{-frac{s}{2}}Gammaleft(frac{s}{2}right)zeta(s) = int_0^infty bigl(theta(it)-1bigr)t^{frac{s}{2}-1},mathrm{d}t,
in terms of Jacobi's theta function
theta(tau)= sum_{n=-infty}^infty e^{pi i n^2tau}.
However, this integral only converges if the real part of {{mvar|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 {{mvar|s}} except 0 and 1:
pi^{-frac{s}{2}}Gammaleft(frac{s}{2}right)zeta(s) = frac{1}{s-1}-frac{1}{s} +frac{1}{2} int_0^1 left(theta(it)-t^{-frac12}right)t^{frac{s}{2}-1},mathrm{d}t + frac{1}{2}int_1^infty bigl(theta(it)-1bigr)t^{frac{s}{2}-1},mathrm{d}t.
Laurent series
{{unreferenced section|date=December 2014}}The Riemann zeta function is meromorphic with a single pole of order one at {{math|s {{=}} 1}}. It can therefore be expanded as a Laurent series about {{math|s {{=}} 1}}; the series development is then
zeta(s)=frac{1}{s-1}+sum_{n=0}^infty frac{(-1)^ngamma_n}{n!}(s-1)^n.
The constants {{math|Î³n}} here are called the Stieltjes constants and can be defined by the limit
gamma_n = lim_{m rightarrow infty}{left(left(sum_{k = 1}^m frac{(ln k)^n}{k}right) - frac{(ln m)^{n+1}}{n+1}right)}.
The constant term {{math|Î³0}} is the Eulerâ€“Mascheroni constant.Integral
For all {{math|s âˆˆ â„‚}}, {{math|s â‰ 1}} the integral relation (cf. Abelâ€“Plana formula)
zeta(s) = frac{1}{s-1}+frac{1}{2}+2!int_0^{infty}!!!frac{sin(sarctan t)}{left(1+t^2right)^frac{s}{2}left(e^{2pi t}-1right)},mathrm{d}t
holds true, which may be used for a numerical evaluation of the zeta-function.Rising factorial
Another series development using the rising factorial valid for the entire complex plane is{{Citation needed|date=October 2015}}
zeta(s) = frac{s}{s-1} - sum_{n=1}^infty bigl(zeta(s+n)-1bigr)frac{s(s+1)cdots(s+n-1)}{(n+1)!}.
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 Gaussâ€“Kuzminâ€“Wirsing operator acting on {{math|xs âˆ’ 1}}; that context gives rise to a series expansion in terms of the falling factorial.WEB,weblink A series representation for the Riemann Zeta derived from the Gauss-Kuzmin-Wirsing Operator, Linas.org, 2017-01-04, Hadamard product
On the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansion
zeta(s) = frac{e^{left(log(2pi)-1-frac{gamma}{2}right)s}}{2(s-1)Gammaleft(1+frac{s}{2}right)} prod_rho left(1 - frac{s}{rho} right) e^frac{s}{rho},
where the product is over the non-trivial zeros {{mvar|Ï}} of {{math|Î¶}} and the letter {{mvar|Î³}} again denotes the Eulerâ€“Mascheroni constant. A simpler infinite product expansion is
zeta(s) = pi^frac{s}{2} frac{prod_rho left(1 - frac{s}{rho} right)}{2(s-1)Gammaleft(1+frac{s}{2}right)}.
This form clearly displays the simple pole at {{math|s {{=}} 1}}, the trivial zeros at âˆ’2, âˆ’4, â€¦ due to the gamma function term in the denominator, and the non-trivial zeros at {{math|s {{=}} Ï}}. (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 {{mvar|Ï}} and {{math|1 âˆ’ Ï}} should be combined.)Globally convergent series
A globally convergent series for the zeta function, valid for all complex numbers {{mvar|s}} except {{math|s {{=}} 1 + {{sfrac|2Ï€i|ln 2}}n}} for some integer {{mvar|n}}, was conjectured by Konrad Knopp and proven by Helmut Hasse in 1930 (cf. Euler summation):
zeta(s)=frac{1}{1-2^{1-s}} sum_{n=0}^infty frac {1}{2^{n+1}} sum_{k=0}^n binom{n}{k} frac{(-1)^k}{(k+1)^{s}}.
The series only appeared in an appendix to Hasse's paper, and did not become generally known until it was discussed by Jonathan Sondow in 1994.JOURNAL, Jonathan, Sondow, Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series, Proceedings of the American Mathematical Society, 1994, 120, 2, 421â€“424,weblink 10.1090/S0002-9939-1994-1172954-7, free, Hasse also proved the globally converging series
zeta(s)=frac 1{s-1}sum_{n=0}^infty frac 1{n+1}sum_{k=0}^nbinom {n}{k}frac{(-1)^k}{(k+1)^{s-1}}
in the same publication, but research by Iaroslav BlagouchineJOURNAL
, Blagouchine, Iaroslav V.
, 1501.00740
, 10.1016/j.jnt.2015.06.012
, Journal of Number Theory
, 365â€“396
, Expansions of generalized Euler's constants into the series of polynomials in {{pi, −2 and into the formal enveloping series with rational coefficients only
| volume = 158
| year = 2016}}JOURNAL
, Blagouchine, Iaroslav V.
, 1606.02044
,weblink
, Integers (Electronic Journal of Combinatorial Number Theory)
, 1â€“45
, Three Notes on Ser's and Hasse's Representations for the Zeta-functions
, 18A
, 2018, 2016arXiv160602044B,
has found that this latter series was actually first published by Joseph Ser in 1926.JOURNAL, Joseph, Ser, Joseph Ser, Sur une expression de la fonction Î¶(s) de Riemann, Upon an expression for Riemann's Î¶ function, 1926, Comptes rendus hebdomadaires des sÃ©ances de l'AcadÃ©mie des Sciences, 182, 1075â€“1077, French, New proofs for both of these results were offered by Demetrios Kanoussis in 2017.JOURNAL, A New Proof of H. Hasse's Global Expression for the Riemann's Zeta Function, 2017, Demetrios P., Kanoussis,weblink Other similar globally convergent series include
, 1501.00740
, 10.1016/j.jnt.2015.06.012
, Journal of Number Theory
, 365â€“396
, Expansions of generalized Euler's constants into the series of polynomials in {{pi, −2 and into the formal enveloping series with rational coefficients only
| volume = 158
| year = 2016}}JOURNAL
, Blagouchine, Iaroslav V.
, 1606.02044
,weblink
, Integers (Electronic Journal of Combinatorial Number Theory)
, 1â€“45
, Three Notes on Ser's and Hasse's Representations for the Zeta-functions
, 18A
, 2018, 2016arXiv160602044B,
sum_{n=0}^infty (-1)^n psi_{n+2}(a) sum_{k=0}^{n} (-1)^k binom{n}{k} (k+1)^{-s}right} ,quad Re(a)>-1
end{align}where {{math|H'n}} are the harmonic numbers, left[{cdot atop cdot}right] are the Stirling numbers of the first kind, (s-k)_k is the Pochhammer symbol, {{math|G'n}} are the Gregory coefficients, {{math|G{{su|b=n|p=(k)}}}} are the Gregory coefficients of higher order, {{math|Cn}} are the Cauchy numbers of the second kind ({{math|C1 {{=}} 1/2}}, {{math|C2 {{=}} 5/12}}, {{math|C3 {{=}} 3/8}},...), and {{math|ψn(a)}}are the Bernoulli polynomials of the second kind, see Blagouchine's paper.Peter Borwein has developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series suitable for high precision numerical calculations.BOOK, Peter, Borwein, Peter Borwein,weblink An Efficient Algorithm for the Riemann Zeta Function, Conference Proceedings, Canadian Mathematical Society, 2000, Constructive, Experimental, and Nonlinear Analysis, 27, 29â€“34, 978-0-8218-2167-1, Michel A., ThÃ©ra, American Mathematical Society, on behalf of the Canadian Mathematical Society, Providence, RI, Series representation at positive integers via the primorial
zeta(k)=frac{2^k}{2^k-1}+sum_{r=2}^inftyfrac{(p_{r-1}#)^k}{J_k(p_r#)}qquad k=2,3,ldots.
Here {{math|pn#}} is the primorial sequence and {{math|Jk}} is Jordan's totient function.JOURNAL, IstvÃ¡n, MezÅ‘, The primorial and the Riemann zeta function, The American Mathematical Monthly, 2013, 120, 4, 321, Series representation by the incomplete poly-Bernoulli numbers
The function {{mvar|Î¶}} can be represented, for {{math|Re(s) > 1}}, by the infinite series
zeta(s)=sum_{n=0}^infty B_{n,ge2}^{(s)}frac{(W_k(-1))^n}{n!},
where {{math|k âˆˆ {âˆ’1, 0}|}}, {{math|Wk}} is the {{mvar|k}}th branch of the Lambert {{mvar|W}}-function, and {{math|B{{su|b=n, â‰¥2|p=(Î¼)}}}} is an incomplete poly-Bernoulli number.JOURNAL, Takao, Komatsu, IstvÃ¡n, MezÅ‘, Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers, Publicationes Mathematicae Debrecen, 2016, 88, 3â€“4, 357â€“368, 10.5486/pmd.2016.7361, 1510.05799, The Mellin transform of the Engel map
The function :g(x) = x left( 1+leftlfloor x^{-1}rightrfloor right) -1 is iterated to find the coefficients appearing in Engel expansions.WEB,weblink A220335 - OEIS, oeis.org, 2019-04-17, The Mellin transform of the map g(x) is related to the Riemann zeta function by the formula
begin{align}
int_0^1 g (x) x^{s - 1} , dx & = sum_{n = 1}^infty
int_{frac{1}{n + 1}}^{frac{1}{n}} (x (n + 1) - 1) x^{s - 1} , d x[6pt]
& = sum_{n = 1}^infty frac{n^{- s} (s - 1) + (n + 1)^{- s - 1} (n^2 + 2 n + 1) + n^{- s - 1} s - n^{1 - s}}{(s + 1) s (n + 1)}[6pt]
& = frac{zeta (s)}{s + 1} - frac{1}{s (s + 1)}
end{align}
int_{frac{1}{n + 1}}^{frac{1}{n}} (x (n + 1) - 1) x^{s - 1} , d x[6pt]
& = sum_{n = 1}^infty frac{n^{- s} (s - 1) + (n + 1)^{- s - 1} (n^2 + 2 n + 1) + n^{- s - 1} s - n^{1 - s}}{(s + 1) s (n + 1)}[6pt]
& = frac{zeta (s)}{s + 1} - frac{1}{s (s + 1)}
end{align}
Numerical algorithms
For v=1,2,3,dots , the Riemann zeta function has for fixed sigma_{0}0, with explicit error bounds. For zeta(s), these are as follows:For a given argument s
with 0leqsigmaleq2
and 0 < t
one can approximate zeta(s) to any accuracy deltaleq0.05
by summing the first series to n = leftlceil 3.151 cdot vNrightrceil
, E_{1}left(sright)
and 0 < t
one can approximate zeta(s) to any accuracy deltaleq0.05
by summing the first series to n = leftlceil 3.151 cdot vNrightrceil
to m=leftlceil Nrightrceil
and neglecting E_{-1}left(sright)
, if one chooses v
and neglecting E_{-1}left(sright)
as the next higher integer of the unique solution of x-maxleft(frac{1-sigma}{2},0right)lnleft(frac{1}{2}+x+tauright)=lnfrac{8}{delta}
in the unknown x
, and from this N=1.11left(1+frac{frac{1}{2}+tau}{v}right)^{frac{1}{2}}. For t=0
in the unknown x
one can neglect E_{1}left(sright)
altogether. Under the mild condition tau>frac{5}{3}left(frac{3}{2}+lnfrac{8}{delta}right)
one needs at most 2+8sqrt{1+lnfrac{8}{delta}+maxleft(frac{1-sigma}{2},0right)lnleft(2tauright)}~sqrt{tau}
summands. Hence this algorithm is essentially as fast as the Riemann-Siegel formula. Similar algorithms are possible for Dirichlet L-functions.
altogether. Under the mild condition tau>frac{5}{3}left(frac{3}{2}+lnfrac{8}{delta}right)
one needs at most 2+8sqrt{1+lnfrac{8}{delta}+maxleft(frac{1-sigma}{2},0right)lnleft(2tauright)}~sqrt{tau}
summands. Hence this algorithm is essentially as fast as the Riemann-Siegel formula. Similar algorithms are possible for Dirichlet L-functions.
Applications
The zeta function occurs in applied statistics (see Zipf's law and Zipfâ€“Mandelbrot law).Zeta function regularization is used as one possible means of regularization of divergent series and divergent integrals in quantum field theory. In one notable example, the Riemannzeta-function shows up explicitly in one method of calculating the Casimir effect. The zeta function is also useful for the analysis of dynamical systems.WEB,weblink Work on spin-chains by A. Knauf, et. al, Empslocal.ex.ac.uk, 2017-01-04,Infinite series
The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.Most of the formulas in this section are from Â§ 4 of J. M. Borwein et al. (2000)- sum_{n=2}^inftybigl(zeta(n)-1bigr) = 1
- sum_{n=1}^inftybigl(zeta(2n)-1bigr)=frac{3}{4}
- sum_{n=1}^inftybigl(zeta(2n+1)-1bigr)=frac{1}{4}
- sum_{n=1}^infty(zeta(2n)-1),t^{2n} = frac{t^2}{t^2-1} + frac{1}{2} left(1- pi tcot(tpi)right)
- sum_{n=1}^infty(zeta(2n+1)-1),t^{2n} = frac{t^2}{t^2-1} + frac{1}{2}left(psi^0(t)+psi^0(-t) right) - gamma
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