Hardy-Littlewood pigeon substitute

The Tauber set of Hardy-Littlewood is a mathematical sentence that particularly in the areas of complex analysis and analytic number theory applies. It deals with a relatively simple criterion with which the asymptotic growth of a number theoretic function can be determined from the properties of the function it generates .

The theorem is named after Alfred Tauber and was proven in 1914 by the British mathematicians Godfrey Harold Hardy and John Edensor Littlewood . In 1930 Jovan Karamata's proof was greatly simplified.

formulation

The power series converges for all . It applies to a positive number ${\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}}$ ${\displaystyle |z|<1}$ ${\displaystyle \alpha \geq 0}$

${\displaystyle (1-x)^{\alpha }f(x)\longrightarrow A\quad {\text{wenn}}\quad x\nearrow 1}$.

Continue to apply

${\displaystyle na_{n}\geq -Cn^{\alpha }}$

with one constant for all . Then applies ${\displaystyle C>0}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle \sum _{i=1}^{n}a_{i}\sim {\frac {A}{\Gamma (1+\alpha )}}n^{\alpha },}$

where here denotes the gamma function . ${\displaystyle \Gamma }$

literature

• Jacob Korevaar: Tauberian Theory. A century of developments. Basic teaching of the mathematical sciences, Springer-Verlag, Berlin Heidelberg New York, ISBN 3-540-21058-X .

Individual evidence

1. ↑ Jacob Korevaar: Tauberian Theory. A century of developments , Theorem 7.4