• 검색 결과가 없습니다.

Riemann Zeta Function MinMax Re

N/A
N/A
Protected

Academic year: 2022

Share "Riemann Zeta Function MinMax Re"

Copied!
20
0
0

로드 중.... (전체 텍스트 보기)

전체 글

(1)

Search Site

Algebra

Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry

History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology

Alphabetical Index Interactive Entries Random Entry New in MathWorld MathWorld Classroom About MathWorld Send a Message to the Team Order book from Amazon

12,646 entries Sun Mar 4 2007

Calculus and Analysis > Special Functions > Riemann Zeta Function Foundations of Mathematics > Mathematical Problems > Unsolved Problems Recreational Mathematics > Interactive Entries > webMathematica Examples MathWorld Contributors > Cloitre

MathWorld Contributors > Drane MathWorld Contributors > Huvent MathWorld Contributors > Plouffe MathWorld Contributors > Sondow MathWorld Contributors > Terr MathWorld Contributors > Trott

Riemann Zeta Function

Min Max

Re Register for Unlimited Interactive Examples >>

Im

Other Wolfram Sites:

Wolfram Research Integrator Tones Functions Site Wolfram Science more…

Latest Mathematica Information >>

Download Mathematica Trial >>

Complete Mathematica Documentation >>

Show your math savvy with a MathWorld T-shirt.

Read the latest Technical Software News.

Compute the Riemann zeta function anywhere in the complex plane with Mathematica.

오후

-15 15

-15 15 Replot

(2)

The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day. The Riemann zeta function is defined over the complex plane for one complex variable, which is conventionally denoted (instead of the usual ) in deference to the notation used by Riemann in his 1859 paper that founded the study of this function (Riemann 1859). It is implemented in Mathematica as Zeta[s].

The plot above shows the "ridges" of for and . The fact that the ridges appear to decrease monotonically is not a coincidence, since monotonic decrease in fact implies the Riemann hypothesis (Zvengrowski and Saidak 2003; Borwein and Borwein 2003, pp. 95-96).

On the real line with , the Riemann zeta function can be defined by the integral

(1)

where is the gamma function. If is an integer , then we have the identity

(2)

so

(3)

To evaluate , let so that and plug in the above identity to obtain

(4)

(5)

(6)

오후

(3)

Integrating the final expression in (6) gives , which cancels the factor and gives the most common form of the Riemann zeta function,

(7)

which is sometimes known as a p-series.

The Riemann zeta function can also be defined in terms of multiple integrals by

(8)

and as a Mellin transform by

(9)

for , where is the fractional part (Balazard and Saias 2000).

It appears in the unit square integral

(10)

valid for (Guillera and Sondow 2005). For a nonnegative integer, this formula is due to Hadjicostas (2002), and the special cases and are due to Beukers (1979).

Note that the zeta function has a singularity at , where it reduces to the divergent harmonic series.

The Riemann zeta function satisfies the reflection functional equation

(11)

(Hardy 1999, p. 14; Krantz 1999, p. 160), a similar form of which was conjectured by Euler for real (Euler, read in 1749, published in 1768; Ayoub 1974; Havil 2003, p. 193). A symmetrical form of this functional equation is given by

(12)

(Ayoub 1974), which was proved by Riemann for all complex (Riemann 1859).

As defined above, the zeta function with a complex number is defined for . However, has a unique analytic continuation to the entire complex plane, excluding the point , which corresponds to a simple pole with complex residue 1 (Krantz 1999, p. 160). In particular, as , obeys

(13)

where is the Euler-Mascheroni constant (Whittaker and Watson 1990, p. 271).

오후

(4)

To perform the analytic continuation for , write

(14)

(15)

(16)

so rewriting in terms of immediately gives

(17)

Therefore,

(18)

Here, the sum on the right-hand side is exactly the Dirichlet eta function (sometimes also called the alternating zeta function). While this formula defines for only the right half-plane , equation (◇) can be used to analytically continue it to the rest of the complex plane. Analytic continuation can also be performed using Hankel functions. A globally convergent series for the Riemann zeta function (which provides the analytic continuation of to the entire complex plane except ) is given by

(19)

(Havil 2003, p. 206), where is a binomial coefficient, which was conjectured by Knopp around 1930, proved by Hasse (1930), and rediscovered by Sondow (1994). This equation is related to renormalization and random variates (Biane et al. 2001) and can be derived by applying Euler's series transformation with to equation (18).

Hasse (1930) also proved the related globally (but more slowly) convergent series

(20)

that, unlike (19), can also be extended to a generalization of the Riemann zeta function known as the Hurwitz zeta function . is defined such that

(21)

(If the singular term is excluded from the sum definition of , then as well.) Expanding about gives

오후

(5)

(22)

where are the so-called Stieltjes constants.

The Riemann zeta function can also be defined in the complex plane by the contour integral

(23)

for all , where the contour is illustrated above (Havil 2003, pp. 193 and 249-252).

Zeros of come in (at least) two different types. So-called "trivial zeros" occur at all negative even integers , - 4, -6, ..., and "nontrivial zeros" at certain

(24)

for in the "critical strip" . The Riemann hypothesis asserts that the nontrivial Riemann zeta function zeros of all have real part , a line called the "critical line." This is now known to be true for the first

roots.

The plot above shows the real and imaginary parts of (i.e., values of along the critical strip) as is varied from 0 to 35 (Derbyshire 2004, p. 221).

The Riemann zeta function can be split up into

오후

(6)

(25)

where and are the Riemann-Siegel functions.

The Riemann zeta function is related to the Dirichlet lambda function and Dirichlet eta function by

(26)

and

(27)

(Spanier and Oldham 1987).

It is related to the Liouville function by

(28)

(Lehman 1960, Hardy and Wright 1979). Furthermore,

(29)

where is the number of distinct prime factors of (Hardy and Wright 1979, p. 254).

For a positive even integer -2, -4, ...,

(30)

giving the first few as

(31)

(32)

(33)

(34)

(Sloane's A117972 and A117973). For ,

(35)

where is the Glaisher-Kinkelin constant. Using equation (◇) gives the derivative

오후

(7)

(36)

which can be derived directly from the Wallis formula (Sondow 1994). can also be derived directly from the Euler-Maclaurin summation formula (Edwards 2001, pp. 134-135). In general, can be expressed analytically in terms of , , the Euler-Mascheroni constant , and the Stieltjes constants , with the first few examples being

(37) (38)

Derivatives can also be given in closed form, for example,

(39) (40)

(Sloane's A114875).

The derivative of the Riemann zeta function for is defined by

(41)

can be given in closed form as

(42) (43)

(Sloane's A073002), where is the Glaisher-Kinkelin constant (given in series form by Glaisher 1894).

The series for about is

(44)

where are Stieltjes constants.

In 1739, Euler found the rational coefficients in in terms of the Bernoulli numbers. Which, when combined with the 1882 proof by Lindemann that is transcendental, effectively proves that is transcendental.

The study of is significantly more difficult. Apéry (1979) finally proved to be irrational, but no similar results are known for other odd . As a result of Apéry's important discovery, is sometimes called Apéry's constant.

Rivoal (2000) and Ball and Rivoal (2001) proved that there are infinitely many integers such that is irrational, and subsequently that at least one of , , ..., is irrational (Rivoal 2001). This result was subsequently tightened by Zudilin (2001), who showed that at least one of , , , or is irrational.

A number of interesting sums for , with a positive integer, can be written in terms of binomial coefficients as the binomial sums

오후

(8)

(45)

(46)

(47)

(Guy 1994, p. 257; Bailey et al. 2006, p. 70). Apéry arrived at his result with the aid of the sum formula above. A relation of the form

(48)

has been searched for with a rational or algebraic number, but if is a root of a polynomial of degree 25 or less, then the Euclidean norm of the coefficients must be larger than , and if if algebraic of degree 25 or less, then the norm of coefficients must exceed (Bailey et al. 2006, pp. 70-71, updating Bailey and Plouffe).

Therefore, no such sums for are known for .

The identity

(49)

(50)

(51)

(52)

for is complex number not equal to a nonzero integer gives an Apéry-like formula for even positive (Bailey et al.

2006, pp. 72-77).

The Riemann zeta function may be computed analytically for even using either contour integration or Parseval's theorem with the appropriate Fourier series. An unexpected and important formula involving a product over the primes was first discovered by Euler in 1737,

(53)

(54)

(55)

오후

(9)

(56) (57)

Here, each subsequent multiplication by the th prime leaves only terms that are powers of . Therefore,

(58)

which is known as the Euler product formula (Hardy 1999, p. 18; Krantz 1999, p. 159), and called "the golden key" by Derbyshire (2004, pp. 104-106). The formula can also be written

(59)

where and are the primes congruent to 1 and 3 modulo 4, respectively.

For even ,

(60)

where is a Bernoulli number (Mathews and Walker 1964, pp. 50-53; Havil 2003, p. 194). Another intimate connection with the Bernoulli numbers is provided by

(61)

for , which can be written

(62)

for . (In both cases, only the even cases are of interest since trivially for odd .) Rewriting (62),

(63)

for , 3, ... (Havil 2003, p. 194), where is a Bernoulli number, the first few values of which are , 1/120, , 1/240, ... (Sloane's A001067 and A006953).

Although no analytic form for is known for odd ,

(64)

where is a harmonic number (Stark 1974). In addition, can be expressed as the sum limit

(65)

오후

(10)

for , 5, ... (Apostol 1973, given incorrectly in Stark 1974).

For the Möbius function,

(66)

(Havil 2003, p. 209).

The values of for small positive integer values of are

(67) (68) (69) (70) (71) (72) (73) (74) (75) (76)

Euler gave to for even (Wells 1986, p. 54), and Stieltjes (1993) determined the values of , ..., to 30 digits of accuracy in 1887. The denominators of for , 2, ... are 6, 90, 945, 9450, 93555, 638512875, ...

(Sloane's A002432). The numbers of decimal digits in the denominators of for , 1, ... are 1, 5, 133, 2277, 32660, 426486, 5264705, ... (Sloane's A114474).

An integral for positive even integers is given by

(77)

and integrals for positive odd integers are given by

(78)

(79)

(80)

(81)

where is an Euler polynomial and is a Bernoulli polynomial (Cvijovic and Klinowski 2002; J. Crepps, pers.

comm., Apr. 2002).

오후

(11)

The value of can be computed by performing the inner sum in equation (◇) with ,

(82)

to obtain

(83)

where is the Kronecker delta.

Similarly, the value of can be computed by performing the inner sum in equation (◇) with ,

(84)

which gives

(85)

(86)

This value is related to a deep result in renormalization theory (Elizalde et al. 1994, Elizalde 1995, Bloch 1996, Lepowski 1999).

It is apparently not known if the value

(87)

(Sloane's A059750) can be expressed in terms of known mathematical constants. This constant appears, for example, in Knuth's series.

Rapidly converging series for for odd were first discovered by Ramanujan (Zucker 1979, Zucker 1984, Berndt 1988, Bailey et al. 1997, Cohen 2000). For and ,

(88)

where is again a Bernoulli number and is a binomial coefficient. The values of the left-hand sums (divided by ) in (88) for , 7, 11, ... are 7/180, 19/56700, 1453/425675250, 13687/390769879500,

7708537/21438612514068750, ... (Sloane's A057866 and A057867). For and , the corresponding formula is slightly messier,

오후

(12)

(89)

(Cohen 2000).

Defining

(90)

the first few values can then be written

(91) (92) (93) (94) (95) (96) (97) (98) (99) (100)

(Plouffe 1998).

Another set of related formulas are

(101)

(102)

(103)

(104)

오후

(13)

(105)

(Plouffe 2006).

Multiterm sums for odd include

(106)

(107)

(108)

(109)

(Borwein and Bradley 1997, 1997; Bailey et al. 2006, p. 71), where is a generalized harmonic number.

G. Huvent (2002) found the beautiful formula

(110)

A number of sum identities involving include

(111)

(112)

(113)

오후

(14)

(114)

Two surprising sums involving are given by

(115)

(116)

where is the Euler-Mascheroni constant (Havil 2003, pp. 109 and 111-112). Equation (115) can be generalized to

(117)

(T. Drane, pers. comm., Jul. 7, 2006) for .

Other unexpected sums are

(118)

(Tyler and Chernhoff 1985; Boros and Moll 2004, p. 248) and

(119)

(118) is a special case of

(120)

where is a Hurwitz zeta function (Danese 1967; Boros and Moll 2004, p. 248).

Considering the sum

(121)

then

(122)

where is the natural logarithm of 2, which is a particular case of

오후

(15)

(123)

where is the digamma function and is the Euler-Mascheroni constant, which can be derived from

(124)

(B. Cloitre, pers. comm., Dec. 11, 2005; cf. Borwein et al. 2000, eqn. 27).

An additional set of sums over is given by

(125)

(126)

(127) (128) (129)

(130)

(131) (132) (133)

(134) (135)

(Sloane's A093720, A076813, and A093721), where is a modified Bessel function of the first kind, is a regularized hypergeometric function. These sums have no known closed-form expression.

오후

(16)

The inverse of the Riemann zeta function , plotted above, is the asymptotic density of th-powerfree numbers (i.

e., squarefree numbers, cubefree numbers, etc.). The following table gives the number of th-powerfree numbers for several values of .

2 0.607927 7 61 608 6083 60794 607926

3 0.831907 9 85 833 8319 83190 831910

4 0.923938 10 93 925 9240 92395 923939

5 0.964387 10 97 965 9645 96440 964388

6 0.982953 10 99 984 9831 98297 982954

SEE ALSO: Abel's Functional Equation, Berry Conjecture, Critical Line, Critical Strip, Debye Functions, Dirichlet Beta Function, Dirichlet Eta Function, Dirichlet Lambda Function, Euler Product, Harmonic Series, Hurwitz Zeta Function, Khinchin's Constant, Lehmer's Phenomenon, Montgomery's Pair Correlation Conjecture, p-Series, Periodic Zeta Function, Prime Number Theorem, Psi Function, Riemann Hypothesis, Riemann P-Series, Riemann-Siegel Functions, Riemann-von Mangoldt Formula, Riemann Zeta Function zeta(2), Riemann Zeta Function Zeros, Stieltjes Constants, Voronin Universality Theorem, Xi-Function. [Pages Linking Here]

RELATED WOLFRAM SITES: http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/

Portions of this entry contributed by Jonathan Sondow (author's link)

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Riemann Zeta Function and Other Sums of Reciprocal Powers." §23.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807-808, 1972.

Adamchik, V. S. and Srivastava, H. M. "Some Series of the Zeta and Related Functions." Analysis 18, 131-144, 1998.

Aizenberg, L.; Adamchik, V.; and Levit, V. E. "Approaching the Riemann Hypothesis with Mathematica." http://library.wolfram.com/

infocenter/Articles/3268/.

Apéry, R. "Irrationalité de et ." Astérisque 61, 11-13, 1979.

오후

(17)

Apostol, T. M. "Another Elementary Proof of Euler's Formula for ." Amer. Math. Monthly 80, 425-431, 1973.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 332-335, 1985.

Ayoub, R. "Euler and the Zeta Function." Amer. Math. Monthly 81, 1067-1086, 1974.

Bailey, D. H. "Multiprecision Translation and Execution of Fortran Programs." ACM Trans. Math. Software. To appear.

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Natick, MA: A. K.

Peters, 2006. http://crd.lbl.gov/~dhbailey/expmath/maa-course/hyper-ema.pdf.

Bailey, D. and Plouffe, S. "Recognizing Numerical Constants." http://www.cecm.sfu.ca/organics/papers/bailey/.

Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "On the Khinchin Constant." Math. Comput. 66, 417-431, 1997.

Balazard, M. and Saias, E. "The Nyman-Beurling Equivalent Form for the Riemann Hypothesis." Expos. Math. 18, 131-138, 2000.

Balazard, M.; Saias, E.; and Yor, M. "Notes sur la fonction de Riemann, 2." Adv. Math. 143, 284-287, 1999.

Ball, K. and Rivoal, T. "Irrationalité d'une infinité valeurs de la fonction zêta aux entiers impairs." Invent. Math. 146, 193-207, 2001.

Berndt, B. C. Ch. 14 in Ramanujan's Notebooks, Part II. New York: Springer-Verlag, 1988.

Beukers, F. "A Note on the Irrationality of and ." Bull. London Math. Soc. 11, 268-272, 1979.

Biane, P.; Pitman, J.; and Yor, M. "Probability Laws Related to the Jacobi Theta and Riemann Zeta Functions, and Brownian Excursions." Bull.

Amer. Math. Soc. 38, 435-465, 2001.

Bloch, S. "Zeta Values and Differential Operators on the Circle." J. Algebra 182, 476-500, 1996.

Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England:

Cambridge University Press, 2004.

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Natick, MA: A. K. Peters, pp. 95-96 and 99- 100, 2003.

Borwein, D. and Borwein, J. "On an Intriguing Integral and Some Series Related to ." Proc. Amer. Math. Soc. 123, 1191-1198, 1995.

Borwein, J. M. and Bradley, D. M. "Searching Symbolically for Apéry-Like Formulae for Values of teh Riemann Zeta Function." ACM SIGSAM Bull.

Algebraic Sym. Manip. 30, 2-7, 1996.

Borwein, J. M. and Bradley, D. M. "Empirically Determined Apéry-Like Formulae for ." Exp. Math. 6, 181-194, 1997.

Borwein, J. M.; Bradley, D. M.; and Crandall, R. E. "Computational Strategies for the Riemann Zeta Function." J. Comput. Appl. Math. 121, 247- 296, 2000.

Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988.

Choudhury, B. K. "The Riemann Zeta-Function and Its Derivatives." Proc. Roy. Soc. London Ser. A 450, 477-499, 1995.

Cohen, H. "High Precision Computation of Hardy-Littlewood Constants." Preprint. http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi.

Conrey, J. B. "The Riemann Hypothesis." Not. Amer. Math. Soc. 50, 341-353, 2003. http://www.ams.org/notices/200303/fea-conrey-web.

pdf.

Cvijovic, D. and Klinowski, J. "Integral Representations of the Riemann Zeta Function for Odd-Integer Arguments." J. Comput. Appl. Math. 142, 435-439, 2002.

Danese, A. E. "Solution to Problem 1801. A Zeta-Function Identity." Amer. Math. Monthly 74, 80-81, 1967.

오후

(18)

Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, 1980.

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.

Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.

Elizalde, E. Ten Physical Applications of Spectral Zeta Functions. Berlin: Springer-Verlag, 1995.

Elizalde, E.; Odintsov, S. D.; Romeo, A.; Bytsenko, A. A.; and Zerbini, S. Zeta Regularization Techniques With Applications. River Edge, NJ:

World Scientific, 1994.

Euler, L. "Remarques sur un beau rapport entre les series des puissances tant directes que réciproques." Mémoires de l'academie des sciences de Berlin 17, 83-106, 1768. Reprinted in Opera Omnia, Series 1, Vol. 15 pp. 70-90.

Glaisher, J. W. L. "On the Constant which Occurs in the Formula for ." Messenger Math. 24, 1-16, 1894.

Guillera, J. and Sondow, J. "Double Integrals and Infinite Products for Some Classical Constants Via Analytic Continuations of Lerch's Transcendent." 16 June 2005 http://arxiv.org/abs/math.NT/0506319.

Guy, R. K. "Series Associated with the -Function." §F17 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 257- 258, 1994.

Hadjicostas, P. "Some Generalizations of Beukers' Integrals." Kyungpook Math. J. 42, 399-416, 2002.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.

Hardy, G. H. and Wright, E. M. "The Zeta Function." §17.2 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 245-247 and 255, 1979.

Hasse, H. "Ein Summierungsverfahren für die Riemannsche Zeta-Reihe." Math. Z. 32, 458-464, 1930.

Hauss, M. Verallgemeinerte Stirling, Bernoulli und Euler Zahlen, deren Anwendungen und schnell konvergente Reihen für Zeta Funktionen.

Aachen, Germany: Verlag Shaker, 1995.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.

Howson, A. G. "Addendum to: 'Euler and the Zeta Function' (Amer. Math. Monthly 81 (1974), 1067-1086) by Raymond Ayoub." Amer. Math.

Monthly 82, 737, 1975.

Huvent, G. "Autour de la primitive de ." Feb. 3, 2002. http://perso.orange.fr/gery.huvent/articlespdf/

Autour_primitive.pdf.

Huylebrouck, D. "Similarities in Irrationality Proofs for , , , and ." Amer. Math. Monthly 108, 222-231, 2001.

Ivic, A. A. The Riemann Zeta-Function. New York: Wiley, 1985.

Ivic, A. A. Lectures on Mean Values of the Riemann Zeta Function. Berlin: Springer-Verlag, 1991.

Jones, G. A. and Jones, J. M. "The Riemann Zeta Function." Ch. 9 in Elementary Number Theory. Berlin: Springer-Verlag, pp. 163-189, 1998.

Karatsuba, A. A. and Voronin, S. M. The Riemann Zeta-Function. Hawthorne, NY: De Gruyter, 1992.

Katayama, K. "On Ramanujan's Formula for Values of Riemann Zeta-Function at Positive Odd Integers." Acta Math. 22, 149-155, 1973.

Keiper, J. "The Zeta Function of Riemann." Mathematica Educ. Res. 4, 5-7, 1995.

오후

(19)

Knopp, K. "4th Example: The Riemann -Function." Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 51-57, 1996.

Krantz, S. G. "Riemann's Zeta Function." §13.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 158-159, 1999.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 35, 1983.

Lehman, R. S. "On Liouville's Function." Math. Comput. 14, 311-320, 1960.

Lepowski, J. "Vertex Operator Algebras and the Zeta Function." 29 Sep 1999. http://arxiv.org/abs/math.QA/9909178/.

Mathews, J. and Walker, R. L. Mathematical Methods of Physics, 2nd ed. Reading, MA: W. A. Benjamin/Addison-Wesley, pp. 50-53, 1970.

Patterson, S. J. An Introduction to the Theory of the Riemann Zeta-Function. New York: Cambridge University Press, 1988.

Pegg, E. Jr. and Weisstein, E. W. "Seven Mathematical Tidbits." MathWorld Headline News. Nov. 8, 2004. http://mathworld.wolfram.com/

news/2004-11-08/seventidbits/#3.

Plouffe, S. "Identities Inspired from Ramanujan Notebooks II." Jul. 21, 1998. http://www.lacim.uqam.ca/~plouffe/identities.html.

Plouffe, S. "Identities Inspired from Ramanujan Notebooks (Part 2)." Apr. 2006. http://www.lacim.uqam.ca/~plouffe/inspired2.pdf.

Riemann, G. F. B. "Über die Anzahl der Primzahlen unter einer gegebenen Grösse." Monatsber. Königl. Preuss. Akad. Wiss. Berlin, 671-680, Nov. 1859.

Reprinted in Das Kontinuum und Andere Monographen (Ed. H. Weyl). New York: Chelsea, 1972.

Rivoal, T. "La fonction Zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs." Comptes Rendus Acad. Sci. Paris 331, 267-270, 2000.

Rivoal, T. "Irrationalité d'au moins un des neuf nombres , , ..., ." 25 Apr 2001. http://arxiv.org/abs/math.NT/0104221/.

Sloane, N. J. A. Sequences A001067, A002432/M4283, A006953/M2039, A057866, A057867, A059750, A073002, A076813, A093720, A093721, A114474, A114875, A117972, and A117973 in "The On-Line Encyclopedia of Integer Sequences."

Sondow, J. "Analytic Continuation of Riemann's Zeta Function and Values at Negative Integers via Euler's Transformation of Series." Proc. Amer.

Math. Soc. 120, 421-424, 1994.

Spanier, J. and Oldham, K. B. "The Zeta Numbers and Related Functions." Ch. 3 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 25- 33, 1987.

Srivastava, H. M. "Some Simple Algorithms for the Evaluations and Representations of the Riemann Zeta Function at Positive Integer Arguments." J. Math. Anal. Appl. 246, 331-351, 2000.

Stark, E. L. "The Series , 3, 4, ..., Once More." Math. Mag. 47, 197-202, 1974.

Stieltjes, T. J. Oeuvres Complètes, Vol. 2 (Ed. G. van Dijk.) New York: Springer-Verlag, p. 100, 1993.

Titchmarsh, E. C. The Zeta-Function of Riemann. London: Cambridge University Press, 1930.

Titchmarsh, E. C. The Theory of the Riemann Zeta Function, 2nd ed. New York: Clarendon Press, 1987.

Tyler, D. and Chernhoff, P. "Problem 3103. An Odd Sum Reappears." Amer. Math. Monthly 92, 507, 1985.

Vardi, I. "The Riemann Zeta Function." Ch. 8 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 141-174, 1991.

Wagon, S. "The Riemann Zeta Function." §10.6 in Mathematica in Action. New York: W. H. Freeman, pp. 353-362, 1991.

오후

(20)

Watkins,M. R. "Inexplicable Secrets of Creation." http://www.maths.ex.ac.uk/~mwatkins/zeta/.

Weisstein, E. W. "Books about Riemann Zeta Function." http://www.ericweisstein.com/encyclopedias/books/RiemannZetaFunction.html.

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

Woon, S. C. "Generalization of a Relation Between the Riemann Zeta Function and Bernoulli Numbers." 24 Dec 1998. http://arxiv.org/abs/

math.NT/9812143/.

Zucker, I. J. "The Summation of Series of Hyperbolic Functions." SIAM J. Math. Anal. 10, 192-206, 1979.

Zucker, I. J. "Some Infinite Series of Exponential and Hyperbolic Functions." SIAM J. Math. Anal. 15, 406-413, 1984.

Zudilin, W. "One of the Numbers , , , Is Irrational." Uspekhi Mat. Nauk 56, 149-150, 2001.

Zvengrowski, P. and Saidak, F. "On the Modulus of the Riemann Zeta Function in the Critical Strip." Math. Slovaca 53, 145-172, 2003.

LAST MODIFIED: September 5, 2006 CITE THIS AS:

Sondow, Jonathan and Weisstein, Eric W. "Riemann Zeta Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/

RiemannZetaFunction.html

© 1999 CRC Press LLC, © 1999-2007 Wolfram Research, Inc. | Terms of Use

오후

참조

관련 문서

El reino Goryeo tenía relaciones comerciales con los comerciantes llamados ‘tajir’, palabra que indica a gente de la zona de Asia Central, la cual

그리 고 필리핀은 인터넷 연결이 약하기 때문에 수강신청을 하는 날은 한국에 있는 지인에게 부탁하는 것이 좋을 것

"Institutions, Political Poker, and Regime Evolution in France" in Kurt von Mettenheim, ed., Presidential Institutions and Democratic Politics:

In practice, "Plant systematics" involves relationships between plants and their evolution, especially at the higher levels, whereas "plant taxonomy" deals with

2. Pulse AUX/ iPod en la unidad o FUNCTION en el control remoto para seleccionar la función iPod. En el modo [iPod], puede utilizar el iPod con la pantalla del iPod

Les politiques d’intégration sociale des immigrés étrangers sont différentes dans chaque pays selon les valeurs sociales, les idéologies, les systèmes de

"The Law of Liability and Compensation for Oil Pollution Damage by the ocean ship"(Gesetz über die Haftung und Entschädigung für Ölverschmutzungsschäden

3 An Analytic Function of Constant Absolute Value Is Constant The Cauchy-Riemann equations also help in deriving general. properties