Axioms for the Selberg class
  1. Analyticity: tex2html_wrap_inline19 is an entire function of finite order for some non-negative integer m
  2. Ramanujan Hypothesis: tex2html_wrap_inline23 for any fixed tex2html_wrap_inline25
  3. Functional equation: there must be a function tex2html_wrap_inline27 of the form

    displaymath29

    where tex2html_wrap_inline31 , Q>0, tex2html_wrap_inline35 , and tex2html_wrap_inline37 such that

    displaymath39

    satisfies

    displaymath41

    where tex2html_wrap_inline43 .

  4. Euler product: tex2html_wrap_inline45 , and

    displaymath47

    where tex2html_wrap_inline49 unless n is a positive power of a prime and tex2html_wrap_inline53 for some tex2html_wrap_inline55 .


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Thu Jun 19 15:06:15 CDT 1997