## Riemann Zeta Function

The Riemann zeta function can be defined by the integral

 (1)

where . If is an Integer , then

 (2)

so
 (3)

Let , then and
 (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)

At , the zeta function reduces to the Harmonic Series (which diverges), and therefore has a singularity. In the Complex 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 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)

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 by
 (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 the product of Primes was first discovered by 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 Primes. Euler's product formula can also be written
 (19)

Two sum identities involving are

 (20)

 (21)

The Riemann zeta function is related to the Gamma Function by
 (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, is sometimes 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 Root of 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)

As ,
 (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 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)

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-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)

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 roots equals 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 so
 (62) (63)

and
 (64) (65)

Then

 (66)

Now compute the integrals and .
 (67)

Make the substitution
 (68) (69) (70)

so
 (71)

and
 (72)

can also be computed analytically,
 (73)

But
 (74)
so
 (75)

Combining and gives
 (76)

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

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