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	   We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial  			 		 $f \in \mathbb {Z}[x]$ 		. We use an explicit version of Mertens’ theorem for number fields to estimate a related sum over rational primes. For a given  			 		 $f \in \mathbb {Z}[x]$ 		, our result yields a finite list of primes that certifies the number of distinct irreducible factors of  f .

AN EFFECTIVE ANALYTIC FORMULA FOR THE NUMBER OF DISTINCT IRREDUCIBLE FACTORS OF A POLYNOMIAL