Suvarna Garge (Editor)

Rational zeta series

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

In mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or the Hurwitz zeta function. Specifically, given a real number x, the rational zeta series for x is given by

Contents

x = ∑ n = 2 ∞ q n ζ ( n , m )

where qn is a rational number, the value m is held fixed, and ζ(s, m) is the Hurwitz zeta function. It is not hard to show that any real number x can be expanded in this way.

Elementary series

For integer m>1, one has

x = ∑ n = 2 ∞ q n [ ζ ( n ) − ∑ k = 1 m − 1 k − n ]

For m=2, a number of interesting numbers have a simple expression as rational zeta series:

1 = ∑ n = 2 ∞ [ ζ ( n ) − 1 ]

and

1 − γ = ∑ n = 2 ∞ 1 n [ ζ ( n ) − 1 ]

where γ is the Euler–Mascheroni constant. The series

log ⁡ 2 = ∑ n = 1 ∞ 1 n [ ζ ( 2 n ) − 1 ]

follows by summing the Gauss–Kuzmin distribution. There are also series for π:

log ⁡ π = ∑ n = 2 ∞ 2 ( 3 / 2 ) n − 3 n [ ζ ( n ) − 1 ]

and

13 30 − π 8 = ∑ n = 1 ∞ 1 4 2 n [ ζ ( 2 n ) − 1 ]

being notable because of its fast convergence. This last series follows from the general identity

∑ n = 1 ∞ ( − 1 ) n t 2 n [ ζ ( 2 n ) − 1 ] = t 2 1 + t 2 + 1 − π t 2 − π t e 2 π t − 1

which in turn follows from the generating function for the Bernoulli numbers

x e x − 1 = ∑ n = 0 ∞ B n t n n !

Adamchik and Srivastava give a similar series

∑ n = 1 ∞ t 2 n n ζ ( 2 n ) = log ⁡ ( π t sin ⁡ ( π t ) )

A number of additional relationships can be derived from the Taylor series for the polygamma function at z = 1, which is

ψ ( m ) ( z + 1 ) = ∑ k = 0 ∞ ( − 1 ) m + k + 1 ( m + k ) ! ζ ( m + k + 1 ) z k k ! .

The above converges for |z| < 1. A special case is

∑ n = 2 ∞ t n [ ζ ( n ) − 1 ] = − t [ γ + ψ ( 1 − t ) − t 1 − t ]

which holds for |t| < 2. Here, ψ is the digamma function and ψ(m) is the polygamma function. Many series involving the binomial coefficient may be derived:

∑ k = 0 ∞ ( k + ν + 1 k ) [ ζ ( k + ν + 2 ) − 1 ] = ζ ( ν + 2 )

where ν is a complex number. The above follows from the series expansion for the Hurwitz zeta

ζ ( s , x + y ) = ∑ k = 0 ∞ ( s + k − 1 s − 1 ) ( − y ) k ζ ( s + k , x )

taken at y = −1. Similar series may be obtained by simple algebra:

∑ k = 0 ∞ ( k + ν + 1 k + 1 ) [ ζ ( k + ν + 2 ) − 1 ] = 1

and

∑ k = 0 ∞ ( − 1 ) k ( k + ν + 1 k + 1 ) [ ζ ( k + ν + 2 ) − 1 ] = 2 − ( ν + 1 )

and

∑ k = 0 ∞ ( − 1 ) k ( k + ν + 1 k + 2 ) [ ζ ( k + ν + 2 ) − 1 ] = ν [ ζ ( ν + 1 ) − 1 ] − 2 − ν

and

∑ k = 0 ∞ ( − 1 ) k ( k + ν + 1 k ) [ ζ ( k + ν + 2 ) − 1 ] = ζ ( ν + 2 ) − 1 − 2 − ( ν + 2 )

For integer n ≥ 0, the series

S n = ∑ k = 0 ∞ ( k + n k ) [ ζ ( k + n + 2 ) − 1 ]

can be written as the finite sum

S n = ( − 1 ) n [ 1 + ∑ k = 1 n ζ ( k + 1 ) ]

The above follows from the simple recursion relation Sn + Sn + 1 = ζ(n + 2). Next, the series

T n = ∑ k = 0 ∞ ( k + n − 1 k ) [ ζ ( k + n + 2 ) − 1 ]

may be written as

T n = ( − 1 ) n + 1 [ n + 1 − ζ ( 2 ) + ∑ k = 1 n − 1 ( − 1 ) k ( n − k ) ζ ( k + 1 ) ]

for integer n ≥ 1. The above follows from the identity Tn + Tn + 1 = Sn. This process may be applied recursively to obtain finite series for general expressions of the form

∑ k = 0 ∞ ( k + n − m k ) [ ζ ( k + n + 2 ) − 1 ]

for positive integers m.

Half-integer power series

Similar series may be obtained by exploring the Hurwitz zeta function at half-integer values. Thus, for example, one has

∑ k = 0 ∞ ζ ( k + n + 2 ) − 1 2 k ( n + k + 1 n + 1 ) = ( 2 n + 2 − 1 ) ζ ( n + 2 ) − 1

Expressions in the form of p-series

Adamchik and Srivastava give

∑ n = 2 ∞ n m [ ζ ( n ) − 1 ] = 1 + ∑ k = 1 m k ! S ( m + 1 , k + 1 ) ζ ( k + 1 )

and

∑ n = 2 ∞ ( − 1 ) n n m [ ζ ( n ) − 1 ] = − 1 + 1 − 2 m + 1 m + 1 B m + 1 − ∑ k = 1 m ( − 1 ) k k ! S ( m + 1 , k + 1 ) ζ ( k + 1 )

where B k are the Bernoulli numbers and S ( m , k ) are the Stirling numbers of the second kind.

Other series

Other constants that have notable rational zeta series are:

  • Khinchin's constant
  • Apéry's constant

    • References

    Rational zeta series Wikipedia
    (Text) CC BY-SA