Definition:Analytic Number Theory

Definition

Analytic number theory is a branch of number theory which uses tools from analysis to solve problems about the integers.

Also see

• Results about analytic number theory can be found here.

Historical Note

Some consider that the field of analytic number theory was effectively created by Leonhard Paul Euler in $1737$, on the occasion of his discovery of two results:

Sum of Reciprocals of Powers as Euler Product

and:

Sum of Reciprocals of Primes is Divergent.

Others place its origins with Johann Peter Gustav Lejeune Dirichlet, with Dirichlet's Theorem on Arithmetic Sequences.

Other important results in the field include:

• the Prime Number Theorem
• the Bertrand-Chebyshev Theorem
• the Hardy-Littlewood Circle Method.

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.16$: The Sequence of Primes
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): analytic number theory