In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, analytic number theory is a branch of
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
that uses methods from
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
to solve problems about the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. It is often said to have begun with
Peter Gustav Lejeune Dirichlet's 1837 introduction of
Dirichlet ''L''-functions to give the first proof of
Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is ...
. It is well known for its results on
prime numbers (involving the
Prime Number Theorem and
Riemann zeta function) and
additive number theory (such as the
Goldbach conjecture
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.
The conjecture has been shown to hold ...
and
Waring's problem).
Branches of analytic number theory
Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique.
*
Multiplicative number theory deals with the distribution of the
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s, such as estimating the number of primes in an interval, and includes the prime number theorem and
Dirichlet's theorem on primes in arithmetic progressions.
*
Additive number theory is concerned with the additive structure of the integers, such as
Goldbach's conjecture that every even number greater than 2 is the sum of two primes. One of the main results in additive number theory is the solution to
Waring's problem.
History
Precursors
Much of analytic number theory was inspired by the
prime number theorem. Let π(''x'') be the
prime-counting function that gives the number of primes less than or equal to ''x'', for any real number ''x''. For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that ''x'' / ln(''x'') is a good approximation to π(''x''), in the sense that the
limit
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of the ''quotient'' of the two functions π(''x'') and ''x'' / ln(''x'') as ''x'' approaches infinity is 1:
:
known as the asymptotic law of distribution of prime numbers.
Adrien-Marie Legendre conjectured in 1797 or 1798 that π(''a'') is approximated by the function ''a''/(''A'' ln(''a'') + ''B''), where ''A'' and ''B'' are unspecified constants. In the second edition of his book on number theory (1808) he then made a more precise conjecture, with ''A'' = 1 and ''B'' ≈ −1.08366.
Carl Friedrich Gauss considered the same question: "Im Jahr 1792 oder 1793", according to his own recollection nearly sixty years later in a letter to Encke (1849), he wrote in his logarithm table (he was then 15 or 16) the short note "Primzahlen unter
". But Gauss never published this conjecture. In 1838
Peter Gustav Lejeune Dirichlet came up with his own approximating function, the
logarithmic integral li(''x'') (under the slightly different form of a series, which he communicated to Gauss). Both Legendre's and Dirichlet's formulas imply the same conjectured asymptotic equivalence of π(''x'') and ''x'' / ln(''x'') stated above, although it turned out that Dirichlet's approximation is considerably better if one considers the differences instead of quotients.
Dirichlet
Johann Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series an ...
is credited with the creation of analytic number theory,
a field in which he found several deep results and in proving them introduced some fundamental tools, many of which were later named after him. In 1837 he published
Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is ...
, using
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. In proving the theorem, he introduced the
Dirichlet characters and
L-functions
In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give r ...
.
In 1841 he generalized his arithmetic progressions theorem from integers to the ring of Gaussian integers