This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.1)

ZetaZero

ZetaZero[k]
represents the k^(th) zero of the Riemann zeta function on the critical line.
ZetaZero
represents the k^(th) zero with imaginary part greater than .
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For positive k, ZetaZero[k] represents the zero of on the critical line that has the k^(th) smallest positive imaginary part.
  • For negative k, ZetaZero[k] represents zeros with progressively larger negative imaginary parts.
  • N gives a numerical approximation to the specified zero.
  • ZetaZero can be evaluated to arbitrary numerical precision.
  • ZetaZero automatically threads over lists.
Find numerically the position of the first zero:
Symbolic property:
Find numerically the position of the first zero:
In[1]:=
Click for copyable input
Out[1]=
 
Symbolic property:
In[1]:=
Click for copyable input
Out[1]=
Evaluate to high precision:
Find the first zero with the imaginary part greater than 1000:
ZetaZero threads element-wise over lists:
Negative order is interpreted as a reflected root of the Zeta function:
Plot distances between successive zeros:
Visualize the first 10 zeros:
Compute :
Show good Gram points, where RiemannSiegelZ changes sign for consecutive points:
Show a bad Gram point:
First occurrence of :
ZetaZero is not defined:
New in 6