Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable. Where it is asserted that some list of integers is finite, the question is whether in principle the list could be printed out after a machine computation.An early example of an ineffective result was J. E. Littlewood's theorem of 1914, that in the prime number theorem the differences of both (x) and (x) with their asymptotic estimates change sign infinitely often.

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