Analytic Number Theory
Analytic number theory
is the branch of
number theory
that uses methods from
mathematical analysis
. Its first major success was
Dirichlet
's application of analysis to prove
the existence of infinitely many primes in any arithmetic progression
. The proofs of the
prime number theorem
based on the
Riemann zeta function
is another milestone. The outline of the subject remains similar to the heyday of the subject in the 1930s.
Multiplicative number theory
deals with the distribution of the
prime numbers
, applying
Dirichlet series
as generating functions. It is assumed that the methods will eventually apply to the general
L-function
, though that theory is still largely conjectural.
Additive number theory
has as typical problems
Goldbach's conjecture
and
Waring's problem
. Methods have changed somewhat. The
circle method
of
Hardy
and
Littlewood
was conceived as applying to
power series
near the
unit circle
in the
complex plane
; it is now thought of in terms of finite exponential sums (that is, on the unit circle, but with the power series truncated). The needs of
diophantine approximation
are for auxiliary functions that aren't
generating functions
- their coefficients are constructed by use of a
pigeonhole principle
- and involve
several complex variables
. The fields of diophantine approximation and
transcendence theory
have expanded, to the point that the techniques have been applied to the
Mordell conjecture
. The biggest single technical change after 1950 has been the development of
sieve methods
as an auxiliary tool, particularly in multiplicative problems. These are
combinatorial
in nature, and quite varied. Also much cited are uses of
probabilistic
number theory - forms of random distribution assertions on the primes, for example: these have not received any definitive shape. The extremal branch of combinatorial theory has in return been much influenced by the value placed in analytic number theory on (often separate) quantitative upper and lower bounds.
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