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ID: INPUT{id.yaml}

Title: >
  Values of the Hurwitz zeta function at pairs of rational numbers

Definition: >
  The list contains values $\zeta(s,x)$ of the Riemann zeta function
  at rational numbers $s$ and $x$.

Parameters:
  x:
    type: Q
    constraints: $x > 0$
  s:
    type: Q
    constraints: $s \neq 1$
   
Comments:
  comment-Hurwitz-zeta-definition: >
    The Hurwitz zeta function $\zeta(s,x)$
    is the meromorphic continuation (with respect to $s$) of the series 
    $\sum_{n=0}^\infty (n+x)^{-s}$, $\Re(s) > 1$, $\Re(x) > 0$.
  comment-pole: >
    At $s=1$, $\zeta(s,x)$ has a simple pole (for any fixed $x$).
  comment-Riemann-zeta: >
    $\zeta(s,1)$ is the Riemann zeta function.
  
Formulas:

Programs: 
  program-sage:
    language: Sage
    code: |
      numbers = {(c/d, a/b): hurwitz_zeta(a/b, c/d) 
                      for b,a,d,c in cartesian_product(([1..5],[-5..5],[1..5],[1..5])) 
                      if gcd(a,b) == 1 and a/b != 1 and gcd(c,d) == 1}

References:

Links:
  Wiki:
    title: "Wikipedia: Hurwitz zeta function"
    url: https://en.wikipedia.org/wiki/Hurwitz_zeta_function

Similar tables:
  
Keywords:
  
Tags:
- special values
- L-function

Data properties:
  type: R
  complete: no
  
Display properties:
  number-header: $\zeta(s,x)$

Numbers: INPUT{numbers.yaml}

— preview —

Values of the Hurwitz zeta function at pairs of rational numbers

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Numbers

$x$

$s$

$\zeta(s,x)$

INPUT{numbers.yaml} (not shown in preview)

Definition

The list contains values $\zeta(s,x)$ of the Riemann zeta function at rational numbers $s$ and $x$.

Parameters

$x$

— rational number ($x > 0$)

$s$

— rational number ($s \neq 1$)

Comments

(1)

The Hurwitz zeta function $\zeta(s,x)$ is the meromorphic continuation (with respect to $s$) of the series $\sum_{n=0}^\infty (n+x)^{-s}$, $\Re(s) > 1$, $\Re(x) > 0$.

(2)

At $s=1$, $\zeta(s,x)$ has a simple pole (for any fixed $x$).

(3)

$\zeta(s,1)$ is the Riemann zeta function.

Programs

(P1)

Sage

numbers = {(c/d, a/b): hurwitz_zeta(a/b, c/d) 
                for b,a,d,c in cartesian_product(([1..5],[-5..5],[1..5],[1..5])) 
                if gcd(a,b) == 1 and a/b != 1 and gcd(c,d) == 1}

Links

[1]

Wikipedia: Hurwitz zeta function

Data properties

Entries are of type: real number

Table is complete: no