The Riemann zeta function can be defined by the integral

(1) |

(2) |

(3) |

(4) |

where is the Gamma Function. Integrating the final expression in (4) gives , which cancels the factor and gives the most common form of the Riemann zeta function,

(5) |

(6) |

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 few
nontrivial 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) |

(8) |

(9) |

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

(10) |

(11) |

(12) |

(13) |

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 the product of Primes was first discovered by Euler in 1737,

(15) |

(16) |

(17) |

(18) |

(19) |

Two sum identities involving are

(20) |

(21) |

(22) |

(23) |

(24) |

(25) |

(26) |

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) |

(33) |

(34) |

Euler gave to for Even , 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).

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

(35) |

(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-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 |

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 de Moivre's Identity or related identities.

Consider the Fourier Series of

(46) |

(47) | |||

(48) | |||

(49) |

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

(50) |

(51) |

But , and , so

(52) |

Now, if ,

(53) |

so the Fourier Series is

(54) |

(55) |

(56) |

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) |

(59) |

(60) |

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

(61) |

To evaluate the integral, rotate the coordinate system by so

(62) | |||

(63) |

and

(64) | |||

(65) |

Then

(66) |

(67) |

Make the substitution

(68) | |||

(69) | |||

(70) |

so

(71) |

(72) |

(73) |

But

(74) |

(75) |

Combining and gives

(76) |

**References**

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. Sequence
A002432/M4283
in ``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.

© 1996-9

1999-05-25