L-series of functions on ZMod N
#
We show that if N
is a positive integer and Φ : ZMod N → ℂ
, then the L-series of Φ
has
analytic continuation (away from a pole at s = 1
if ∑ j, Φ j ≠ 0
).
The most familiar case is when Φ
is a Dirichlet character, but the results here are valid
for general functions; for the specific case of Dirichlet characters see
Mathlib.NumberTheory.LSeries.DirichletContinuation
.
Main definitions #
ZMod.LFunction Φ s
: the meromorphic continuation of the function∑ n : ℕ, Φ n * n ^ (-s)
.
Main theorems #
ZMod.LFunction_eq_LSeries
: if1 < re s
then theLFunction
coincides with the naiveLSeries
.ZMod.differentiableAt_LFunction
:ZMod.LFunction Φ
is differentiable ats ∈ ℂ
if eithers ≠ 1
or∑ j, Φ j = 0
.ZMod.LFunction_one_sub
: the functional equation relatingLFunction Φ (1 - s)
toLFunction (𝓕 Φ) s
, where𝓕
is the Fourier transform.
If Φ
is a periodic function, then the L-series of Φ
converges for 1 < re s
.
The unique meromorphic function ℂ → ℂ
which agrees with ∑' n : ℕ, Φ n / n ^ s
wherever the
latter is convergent. This is constructed as a linear combination of Hurwitz zeta functions.
Note that this is not the same as LSeries Φ
: they agree in the convergence range, but
LSeries Φ s
is defined to be 0
if re s ≤ 1
.
Equations
- ZMod.LFunction Φ s = ↑N ^ (-s) * ∑ j : ZMod N, Φ j * HurwitzZeta.hurwitzZeta (ZMod.toAddCircle j) s
Instances For
The L-function of a function on ZMod 1
is a scalar multiple of the Riemann zeta function.
The L-function of Φ
has a residue at s = 1
equal to the average value of Φ
.
The LFunction
of the function x ↦ e (j * x)
, where e : ZMod N → ℂ
is the standard additive
character, is expZeta (j / N)
.
Note this is not at all obvious from the definitions, and we prove it by analytic continuation from the convergence range.
Explicit formula for the L-function of 𝓕 Φ
, where 𝓕
is the discrete Fourier transform.