In mathematical analysis and analytic number theory, Lambert summation is a summability method for summing infinite series related to Lambert series specially relevant in analytic number theory.
Definition Define the Lambert kernel by L ( x ) = log ( 1 / x ) x 1 − x {\displaystyle L(x)=\log(1/x){\frac {x}{1-x}}} with L ( 1 ) = 1 {\displaystyle L(1)=1} . Note that L ( x n ) > 0 {\displaystyle L(x^{n})>0} is decreasing as a function of n {\displaystyle n} when 0 < x < 1 {\displaystyle 0<x<1} . A sum ∑ n = 0 ∞ a n {\displaystyle \sum _{n=0}^{\infty }a_{n}} is Lambert summable to A {\displaystyle A} if lim x → 1 − ∑ n = 0 ∞ a n L ( x n ) = A {\displaystyle \lim _{x\to 1^{-}}\sum _{n=0}^{\infty }a_{n}L(x^{n})=A} , written ∑ n = 0 ∞ a n = A ( L ) {\displaystyle \sum _{n=0}^{\infty }a_{n}=A\,\,(\mathrm {L} )} .
Abelian and Tauberian theorem Abelian theorem: If a series is convergent to A {\displaystyle A} then it is Lambert summable to A {\displaystyle A} . Tauberian theorem: Suppose that ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} is Lambert summable to A {\displaystyle A} . Then it is Abel summable to A {\displaystyle A} . In particular, if ∑ n = 0 ∞ a n {\displaystyle \sum _{n=0}^{\infty }a_{n}} is Lambert summable to A {\displaystyle A} and n a n ≥ − C {\displaystyle na_{n}\geq -C} then ∑ n = 0 ∞ a n {\displaystyle \sum _{n=0}^{\infty }a_{n}} converges to A {\displaystyle A} . The Tauberian theorem was first proven by G. H. Hardy and John Edensor Littlewood but was not independent of number theory, in fact they used a number-theoretic estimate which is somewhat stronger than the prime number theorem itself. The unsatisfactory situation around the Lambert Tauberian theorem was resolved by Norbert Wiener.
Examples
∑ n = 1 ∞ μ ( n ) n = 0 ( L ) {\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}=0\,(\mathrm {L} )} , where μ is the Möbius function. Hence if this series converges at all, it converges to zero. Note that the sequence μ ( n ) n {\displaystyle {\frac {\mu (n)}{n}}} satisfies the Tauberian condition, therefore the Tauberian theorem implies ∑ n = 1 ∞ μ ( n ) n = 0 {\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}=0} in the ordinary sense. This is equivalent to the prime number theorem.
∑ n = 1 ∞ Λ ( n ) − 1 n = − 2 γ ( L ) {\displaystyle \sum _{n=1}^{\infty }{\frac {\Lambda (n)-1}{n}}=-2\gamma \,\,(\mathrm {L} )} where Λ {\displaystyle \Lambda } is von Mangoldt function and γ {\displaystyle \gamma } is Euler's constant. By the Tauberian theorem, the ordinary sum converges and in particular converges to − 2 γ {\displaystyle -2\gamma } . This is equivalent to ψ ( x ) ∼ x {\displaystyle \psi (x)\sim x} where ψ {\displaystyle \psi } is the second Chebyshev function.
See also Lambert series Abel–Plana formula Abelian and tauberian theorems
References Jacob Korevaar (2004). Tauberian theory. A century of developments. Grundlehren der Mathematischen Wissenschaften. Vol. 329. Springer-Verlag. p. 18. ISBN 3-540-21058-X. Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge: Cambridge Univ. Press. pp. 159–160. ISBN 978-0-521-84903-6. Norbert Wiener (1932). "Tauberian theorems". Ann. of Math. 33 (1). The Annals of Mathematics, Vol. 33, No. 1: 1–100. doi:10.2307/1968102. JSTOR 1968102.