“Riemann Zeta Function”的概念、定义、习题解题方法及翻译-数学辞典

单词

Riemann Zeta Function

释义

Riemann Zeta Function

The Riemann zeta function can be defined by the integral

(1)

where

. If

is an Integer

, then

(2)

(3)

Let

, then

and



			(4)

where

is the Gamma Function. Integrating the final expression in (4) gives

, which cancelsthe factor

and gives the most common form of the Riemann zeta function,

(5)

, the zeta function reduces to the Harmonic Series (which diverges), and therefore has a singularity. In theComplex Plane, trivial zeros occur at

, ..., and nontrivial zeros at

(6)

for

. The figures below show the structure of

by plotting

and

The Riemann Hypothesis asserts that the nontrivial Roots of all have Real Part, a line called the ``Critical Strip.'' This is known to be true for the first roots (Brent et al. 1982). The above plot shows for between 0 and 60. As can be seen, the first fewnontrivial zeros occur at , 21.022040, 25.010858, 30.424876, 32.935062, 37.586178, ... (Wagon 1991,pp. 361-362 and 367-368).

The Riemann zeta function can also be defined in terms of Multiple Integrals by

(7)

The Riemann zeta function can be split up into

(8)

where

and

are the Riemann-Siegel Functions. An additional identity is

(9)

where

is the Euler-Mascheroni Constant.

The Riemann zeta function is related to the Dirichlet Lambda Function and Dirichlet Eta Function by

(10)

and

(11)

(Spanier and Oldham 1987). It is related to the Liouville Function

(12)

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

(13)

where

is the number of different prime factors of

(Hardy and Wright 1979).

A generalized Riemann zeta function known as the Hurwitz Zeta Function can also be defined such that

(14)

The Riemann zeta function may be computed analytically for Even using either Contour Integration or Parseval's Theorem with the appropriate Fourier Series. An interesting formula involving theproduct of Euler in 1737,


			(15)

(16)

(17)

Here, each subsequent multiplication by the next Prime

leaves only terms which are Powers of

.Therefore,

(18)

where

runs over all Euler's

product formula can also bewritten

(19)

Two sum identities involving are

(20)

(21)

The Riemann zeta function is related to the Gamma Function

(22)

was proved to be transcendental for all even

by Euler.

Apéry (1979) proved

to be Irrational with the aid of the

sum formula below. As a result,

issometimes called Apéry's Constant.

(23)

(24)

(25)

(Guy 1994, p. 257). A relation of the form

(26)

has been searched for with

a Rational or Algebraic Number, but if

is a Rootof a Polynomial of degree 25 or less, then the Euclidean norm of the coefficients must be larger than

(Bailey, Bailey and Plouffe). Therefore, no such sums are known for

are known for

The zeta function is defined for , but can be analytically continued to as follows:


	(27)

(28)

(29)

The Derivative of the Riemann zeta function is defined by

(30)

(31)

For Even ,

(32)

where

is a Bernoulli Number. Another intimate connection with the Bernoulli Numbers is provided by

(33)

No analytic form for

is known for Odd

, but

can be expressed as the sum limit

(34)

(Stark 1974). The values for the first few integral arguments are

Euler

gave

for Even

, and Stieltjes (1993) determined the values of

,...,

to 30 digits of accuracy in 1887. The denominators of

for

, 2, ... are6, 90, 945, 9450, 93555, 638512875, ... (Sloane's A002432).

Using the LLL Algorithm, Plouffe (inspired by Zucker 1979, Zucker 1984, and Berndt 1988) has found some beautifulinfinite sums for with Odd . Let

(35)

then

			(36)
			(37)
			(38)
			(39)
			(40)
			(41)
			(42)
			(43)
			(44)
			(45)

The inverse of the Riemann Zeta Function is the asymptotic density of th-powerfree numbers (i.e.,Squarefree numbers, Cubefree numbers, etc.). The following table gives the number of th-powerfreenumbers 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

The value for can be found using a number of different techniques (Apostol 1983, Choe 1987, Giesy 1972, Holme 1970,Kimble 1987, Knopp and Schur 1918, Kortram 1996, Matsuoka 1961, Papadimitriou 1973, Simmons 1992, Stark 1969, Stark 1970,Yaglom and Yaglom 1987). The problem of finding this value analytically is sometimes known as the Basler Problem(Castellanos 1988). Yaglom and Yaglom (1987), Holme (1970), and Papadimitrou (1973) all derive the result from deMoivre's Identity or related identities.

Consider the Fourier Series of

(46)

which has coefficients given by


			(47)

			(48)
			(49)

where the latter is true since the integrand is Odd. Therefore, the Fourier Series is given explicitly by

(50)

Now,

is given by the Cosine Integral


			(51)

But

, and

, so


			(52)

Now, if


			(53)

so the Fourier Series is

(54)

Letting

gives

, so

(55)

and we have

(56)

Higher values of

can be obtained by finding

and proceeding as above.

The value can also be found simply using the Root Linear Coefficient Theorem. Consider the equation and expand sin in a Maclaurin Series

(57)

(58)

where

. But the zeros of

occur at

, ..., so the zeros of

occur at

, .... Therefore, the sum of the rootsequals the Coefficient of the leading term

(59)

which can be rearranged to yield

(60)

Yet another derivation (Simmons 1992) evaluates the integral using the integral




			(61)

To evaluate the integral, rotate the coordinate system by

			(62)
			(63)

and

			(64)
			(65)

Then

(66)

Now compute the integrals

and



			(67)

Make the substitution

			(68)
			(69)
			(70)

(71)

and

(72)

can also be computed analytically,



			(73)

But





	(74)



			(75)

Combining

and

gives

(76)

See also Abel's Functional Equation, Debye Functions, Dirichlet Beta Function, Dirichlet EtaFunction, Dirichlet Lambda Function, Harmonic Series, Hurwitz Zeta Function, Khintchine'sConstant, Lehmer's Phenomenon, Psi Function, Riemann Hypothesis, Riemann P-Series,Riemann-Siegel Functions, Stieltjes Constants, Xi Function

References

Riemann Zeta Function

Abramowitz, M. and Stegun, C. 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.

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

Apostol, T. M. ``A Proof that Euler Missed: Evaluating the Easy Way.'' Math. Intel. 5, 59-60, 1983.

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. and Plouffe, S. ``Recognizing Numerical Constants.'' http://www.cecm.sfu.ca/organics/papers/bailey/.

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

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

Brent, R. P.; van der Lune, J.; te Riele, H. J. J.; and Winter, D. T. ``On the Zeros of the Riemann Zeta Function in the Critical Strip. I.'' Math. Comput. 33, 1361-1372, 1979.

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

Choe, B. R. ``An Elementary Proof of .'' Amer. Math. Monthly 94, 662-663, 1987.

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

Edwards, H. M. Riemann's Zeta Function. New York: Academic Press, 1974.

Farmer, D. W. ``Counting Distinct Zeros of the Riemann Zeta-Function.'' Electronic J. Combinatorics 2, R1 1-5, 1995.http://www.combinatorics.org/Volume_2/volume2.html#R1.

Giesy, D. P. ``Still Another Proof that .'' Math. Mag. 45, 148-149, 1972.

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.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 255, 1979.

Holme, F. ``Ein enkel beregning av .'' Nordisk Mat. Tidskr. 18, 91-92 and 120, 1970.

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.

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.

Kimble, G. ``Euler's Other Proof.'' Math. Mag. 60, 282, 1987.

Knopp, K. and Schur, I. ``Über die Herleitug der Gleichung .'' Archiv der Mathematik u. Physik 27, 174-176, 1918.

Kortram, R. A. ``Simple Proofs for and .'' Math. Mag. 69, 122-125, 1996.

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

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

Matsuoka, Y. ``An Elementary Proof of the Formula .'' Amer. Math. Monthly 68, 486-487, 1961.

Papadimitriou, I. ``A Simple Proof of the Formula .'' Amer. Math. Monthly 80, 424-425, 1973.

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

Plouffe, S. ``Identities Inspired from Ramanujan Notebooks.'' http://www.lacim.uqam.ca/plouffe/identities.html.

Simmons, G. F. ``Euler's Formula by Double Integration.'' Ch. B. 24 in Calculus Gems: Brief Lives and Memorable Mathematics. New York: McGraw-Hill, 1992.

Sloane, N. J. A. SequenceA002432/M4283in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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.

Stark, E. L. ``Another Proof of the Formula .'' Amer. Math. Monthly 76, 552-553, 1969.

Stark, E. L. ``.'' Praxis Math. 12, 1-3, 1970.

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, 2nd ed. Oxford, England: Oxford University Press, 1987.

Titchmarsh, E. C. and Heath-Brown, D. R. The Theory of the Riemann Zeta-Function, 2nd ed. Oxford, England: Oxford University Press, 1986.

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

Wagon, S. ``The Evidence: Where Are the Zeros of Zeta of ?'' Math. Intel. 8, 57-62, 1986.

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

Yaglom, A. M. and Yaglom, I. M. Problem 145 in Challenging Mathematical Problems with Elementary Solutions, Vol. 2. New York: Dover, 1987.

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.

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