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A classical conjecture predicts how often a polynomial in    takes prime values. The natural analogous conjecture for prime values of a polynomial  f ( T ) ∈  k [ u ][ T ], where  k  is a finite field, is false. The conjecture over  k [ u ] was modified in earlier work by introducing a correction factor that encodes unexpected periodicity of the Möbius function at the values of  f  on  k [ u ] when  f  ∈  k [ u ][ T p  ], where  p  is the characteristic of  k .

Prime specialization in higher genus I