In mathematics, a Beurling zeta function is an analogue of the
Riemann zeta function where the ordinary primes are replaced by a set of Beurling generalized primes: any sequence of real numbers greater than 1 that tend to infinity. These were introduced by .
A Beurling generalized integer is a number that can be written as a product of Beurling generalized primes. Beurling generalized the usual
prime number theorem to Beurling generalized primes. He showed that if the number ''N''(''x'') of Beurling generalized integers less than ''x'' is of the form ''N''(''x'') = ''Ax'' + O(''x'' log
−''γ''''x'') with ''γ'' > 3/2 then the number of Beurling generalized primes less than ''x'' is asymptotic to ''x''/log ''x'', just as for ordinary primes,
but if ''γ'' = 3/2 then this conclusion need not hold.
See also
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Abstract analytic number theory Abstract analytic number theory is a branch of mathematics which takes the ideas and techniques of classical analytic number theory and applies them to a variety of different mathematical fields. The classical prime number theorem serves as a pro ...
References
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Zeta and L-functions
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