Analytic number theory

In mathematics, analytic number theory is a part of number theory that uses ideas from mathematical analysis to solve questions about whole numbers.^[1] It is believed to have started in 1837 when Peter Gustav Lejeune Dirichlet used Dirichlet L-functions to give the first proof of Dirichlet's theorem about arithmetic progressions.^[1]^[2] This branch of math is well-known for its work on prime numbers (like the Prime Number Theorem and the Riemann zeta function) and on additive number theory (such as the Goldbach conjecture and Waring's problem).

Branches of analytic number theory

Analytic number theory has two main parts, based on the type of questions they try to answer.^[3]

Multiplicative number theory looks at how prime numbers are spread out, such as counting how many primes are between two numbers. It includes the Prime Number Theorem and Dirichlet's theorem on primes in arithmetic progressions.^[4]
Additive number theory studies how whole numbers can be added together, like Goldbach's conjecture that says every even number greater than 2 is the sum of two prime numbers. One important result in this part is solving Waring's problem.^[5]

↑ ^1.0 ^1.1 Apostol, Tom M. (2000). Introduction to analytic number theory. Undergraduate texts in mathematics (11th print ed.). New York: Springer. p. 7. ISBN 978-0-387-90163-3. MR 0434929. Zbl 0335.10001.
↑ Davenport, Harold; Montgomery, Hugh L. (2000). Multiplicative number theory. Graduate texts in mathematics (3 ed.). New York: Springer. p. 1. ISBN 978-0-387-95097-6. MR 1790423.
↑ "Wayback Machine" (PDF). faculty.math.illinois.edu. Retrieved 2025-01-08.
↑ Davenport, Harold (1980). "Multiplicative Number Theory". Graduate Texts in Mathematics. 74. doi:10.1007/978-1-4757-5927-3. ISBN 978-1-4757-5929-7. ISSN 0072-5285.
↑ Nathanson, Melvyn B. (1996). Additive Number Theory the Classical Bases. Graduate Texts in Mathematics Ser. New York, NY: Springer New York. pp. vii–viii. ISBN 978-0-387-94656-6.