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NumberTheory`Ramanujan`

The Riemann

function


is the simplest of all Dirichlet series. Ramanujan studied the Dirichlet series


where the are the coefficients of

in the series expansion


Just as there is the Riemann hypothesis that all of the nontrivial zeros of the lie on the critical line , there is a conjecture due to Ramanujan that all of the nontrivial zeros of lie on the critical line . The function

satisfies the functional equation


Just as

can be split into


where is RiemannSiegelZ and is RiemannSiegelTheta,

can split into


where


and



Functions related to the Ramanujan -Dirichlet series.

The Ramanujan -Dirichlet series is rather difficult to evaluate, especially far up the critical line

. It is only within the last few years that even a few of the zeros have been verified to lie on the critical line. This package does not provide any great new method to help with this effort, but it does use the fastest methods known.

  • This loads the package.
  • In[1]:= <<NumberTheory`Ramanujan`

  • This gives

    .
  • In[2]:= RamanujanTau[5]

    Out[2]=

  • This gives the first five terms in the generating function RamanujanTauGeneratingFunction[z].
  • In[3]:= Sum[RamanujanTau[n] z^n, {n, 5}]

    Out[3]=

  • The generating function can be evaluated numerically.
  • In[4]:= RamanujanTauGeneratingFunction[.1]

    Out[4]=

  • The generating function can be evaluated numerically even at some points outside the radius of convergence. This uses a functional equation to achieve analytic continuation.
  • In[5]:= RamanujanTauGeneratingFunction[.99]

    Out[5]=

  • Here is the value of the

    -Dirichlet series near the first zero on the critical line.
  • In[6]:= RamanujanTauDirichletSeries[6 + 9.22I]

    Out[6]=

  • This is the value of

    near the same zero.
  • In[7]:= z = RamanujanTauZ[9.22]

    Out[7]=

  • This is the value of

    .
  • In[8]:= theta = RamanujanTauTheta[9.22]

    Out[8]=

  • Here is the value of the

    -Dirichlet series again.
  • In[9]:= z Exp[-I theta]

    Out[9]=