Asymptotics of ζ s
as s → 1
#
The goal of this file is to evaluate the limit of ζ s - 1 / (s - 1)
as s → 1
.
Main results #
tendsto_riemannZeta_sub_one_div
: the limit ofζ s - 1 / (s - 1)
, at the filter of punctured neighbourhoods of 1 inℂ
, exists and is equal to the Euler-Mascheroni constantγ
.
Outline of arguments #
We consider the sum F s = ∑' n : ℕ, f (n + 1) s
, where s
is a real variable and
f n s = ∫ x in n..(n + 1), (x - n) / x ^ (s + 1)
. We show that F s
is continuous on [1, ∞)
,
that F 1 = 1 - γ
, and that F s = 1 / (s - 1) - ζ s / s
for 1 < s
.
By combining these formulae, one deduces that the limit of ζ s - 1 / (s - 1)
at 𝓝[>] (1 : ℝ)
exists and is equal to γ
. Finally, using this and the Riemann removable singularity criterion
we obtain the limit along punctured neighbourhoods of 1 in ℂ
.
## Definitions
Sum of finitely many term
s.
Equations
- ZetaAsymptotics.term_sum s N = Finset.sum (Finset.range N) fun (n : ℕ) => ZetaAsymptotics.term (n + 1) s
Instances For
Topological sum of term
s.
Equations
- ZetaAsymptotics.term_tsum s = ∑' (n : ℕ), ZetaAsymptotics.term (n + 1) s
Instances For
Evaluation of the sum for s = 1
#
The topological sum of ZetaAsymptotics.term (n + 1) 1
over all n : ℕ
is 1 - γ
. This is
proved by directly evaluating the sum of the first N
terms and using the limit definition of γ
.
Evaluation of the sum for 1 < s
#
Continuity of the sum #
First version of the limit formula, with a limit over real numbers tending to 1 from above.
The function ζ s - 1 / (s - 1)
tends to γ
as s → 1
.