Some Notes on the Dirichlet Lambda Function Johar M. Ashfaque
Let λ(s) = be the Dirichlet lambda function, β(s) =
∞ X
1 (2n + 1)s n=0 ∞ X
(−1)n (2n + 1)s n=0
be its alternating form.
1
Introduction
According to Varadarajan, a long time ago Pietro Mengoli posed the problem of finding the sum of the series ∞ X 1 1 1 = 1 + + + · · ·. 2 n 4 9 n=1 This problem was first solved by Euler in a letter to Bernoulli as 1+ In general, let
1 1 π2 + +···= . 4 9 6 ∞ X 1 ζ(s) = s n n=1
be Riemann’s zeta function, Euler also proved the following formula ζ(2k) =
(−1)k−1 B2k 22k 2k π 2(2k)!
where B2k are the Bernoulli numbers.
2
Relation to Riemann Zeta
In general λ(s) =
λ(2) = ζ(2) −
1 3 π2 ζ(2) = ζ(2) = 22 4 8
λ(4) = ζ(4) −
1 15 π4 ζ(4) = ζ(4) = 24 16 96
∞ X
1 = (1 − 2−s )ζ(s), s (2n + 1) n=0
1
Re(s) > 1.