%0 Journal Article
%T On Perron’s Formula and the Prime Numbers
%A Michael M. Anthony
%J Advances in Pure Mathematics
%P 487-494
%@ 2160-0384
%D 2024
%I Scientific Research Publishing
%R 10.4236/apm.2024.146027
%X The Riemann hypothesis is intimately connected to the counting functions for the primes. In particular, Perron&#8217;s explicit formula relates the prime counting function to fixed points of iterations of the explicit formula with particular relations involving the trivial and non-trivial roots of the Riemann Zeta function and the Primes. The aim of the paper is to demonstrate this relation at the fixed points of iterations of explicit formula, defined by functions of the form <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mrow>
    <mi>lim</mi></mrow>
   <mrow>
    <mi>T</mi><mo>&#x2208;</mo><mi>&#x039D;</mi><mo>&#x2192;</mo><mi>&#x221E;</mi></mrow>
  </msub>
  <msup>
   <mi>f</mi>
   <mi>T</mi>
  </msup>
  <mrow><mo>(</mo>
   <mrow>
    <msub>
     <mi>z</mi>
     <mi>w</mi>
    </msub>
    </mrow>
  <mo>)</mo></mrow><mo>=</mo><msub>
   <mi>z</mi>
   <mi>w</mi>
  </msub>
  </mrow>
</math>

,where, <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>z</mi>
   <mi>w</mi>
  </msub>
  </mrow>
</math>

 is a real number.
%K Perron
%K Fixed Points
%K Iterations
%K Number Theory
%K Riemann Hypothesis
%K Iterations
%K Invariance
%K Primes
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=133965