Analytic number theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers.^[1] 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.^[1]^[2] 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 and Waring's problem).

deals with the distribution of the prime numbers, 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.^[4]

Multiplicative 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.^[5]

Additive 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.^[3]

History[edit]

Precursors[edit]

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 of the quotient of the two functions π(x) and x / ln(x) as x approaches infinity is 1:

Automorphic L-function

Automorphic form

Langlands program

Maier's matrix method

(1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001

Apostol, Tom M.

; Choi, Stephen; Rooney, Brendan; Weirathmueller, Andrea, eds. (2008), The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, CMS Books in Mathematics, New York: Springer, doi:10.1007/978-0-387-72126-2, ISBN 978-0-387-72125-5

Borwein, Peter

(2000), Multiplicative number theory, Graduate Texts in Mathematics, vol. 74 (3rd revised ed.), New York: Springer-Verlag, ISBN 978-0-387-95097-6, MR 1790423

Davenport, Harold

(1974), Riemann's Zeta Function, New York: Dover Publications, ISBN 978-0-486-41740-0, MR 0466039

Edwards, H. M.

Tenenbaum, Gérald (1995), Introduction to Analytic and Probabilistic Number Theory, Cambridge studies in advanced mathematics, vol. 46, , ISBN 0-521-41261-7

Cambridge University Press

Ayoub, Introduction to the Analytic Theory of Numbers

H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I : Classical Theory

H. Iwaniec and E. Kowalski, Analytic Number Theory.

D. J. Newman, Analytic number theory, Springer, 1998

On specialized aspects the following books have become especially well-known:

Certain topics have not yet reached book form in any depth. Some examples are (i) Montgomery's pair correlation conjecture and the work that initiated from it, (ii) the new results of Goldston, Pintz and Yilidrim on small gaps between primes, and (iii) the Green–Tao theorem showing that arbitrarily long arithmetic progressions of primes exist.