Convergence of L-series #
We define LSeries.abscissaOfAbsConv f
(as an EReal
) to be the infimum
of all real numbers x
such that the L-series of f
converges for complex arguments with
real part x
and provide some results about it.
Tags #
L-series, abscissa of convergence
The abscissa x : EReal
of absolute convergence of the L-series associated to f
:
the series converges absolutely at s
when re s > x
and does not converge absolutely
when re s < x
.
Equations
- LSeries.abscissaOfAbsConv f = sInf (Real.toEReal '' {x : ℝ | LSeriesSummable f ↑x})
Instances For
If f
and g
agree on large n : ℕ
, then their LSeries
have the same
abscissa of absolute convergence.
If ‖f n‖
is O(n^x)
, then the abscissa of absolute convergence
of f
is bounded by x + 1
.
If f
is O(1)
, then the abscissa of absolute convergence of f
is bounded above by 1
.
If f
is real-valued and x
is strictly greater than the abscissa of absolute convergence
of f
, then the real series ∑' n, f n / n ^ x
converges.
If F
is a binary operation on ℕ → ℂ
with the property that the LSeries
of F f g
converges whenever the LSeries
of f
and g
do, then the abscissa of absolute convergence
of F f g
is at most the maximum of the abscissa of absolute convergence of f
and that of g
.