# How to calculate the residue of $1/f(z)$ at a numerical approximation to a root of $f(z)$?

The input

Residue[1/DirichletL[19,10,s],{s,s0}]


gives 0 even when s0 is a root. For example, from LMFDB, I found s0 = 0.5 + 1.51608375316006 I is an approximate root of DirichletL(19,10,s). (In LMFDB this character is actually indexed 18, though.)

For the Riemann zeta function, we can get around this by using ZetaZero to represent s0. What can be done for other $L$-functions?

• what is LMFDB?....from help it says about Residue Laurent expansion of expr. What is the Laurent expansion of 1/DirichletL[19,10,s]? does it have a Laurent expansion? – Nasser Feb 25 '14 at 3:20
• LMFDB is a database of information about $L$-functions and related structures: lmfdb.org. DirichletL[19,10,s] is a specific $L$-function, $L(\chi,s)$, where the modulus of the Dirichlet character $\chi$ is $19$. It has a simple zero at s0, so it's reciprocal should have a pole there (and therefore a Laurent expansion). – A l'Maeaux Feb 25 '14 at 3:30
• This...and this.. might be of help. You can calculate the steps manually to gain better understanding. – Sejwal Feb 25 '14 at 3:47

You can use Cauchy's theorem.

Define the approximate zero of your function :

zero = FindRoot[DirichletL[19, 10, s], {s, 0.5 + I}][[1, 2]]
(* 0.5 + 1.51608 I *)


Series will not consider this a pole of 1/DirichletL[19, 10, s] and I think this is why you get a zero residue.

However, integrating on a small square around that pole one finds :

Table[{eps,
NIntegrate[1/DirichletL[19, 10, s],
{s, zero + eps (1+I), zero + eps (-1+I), zero + eps (-1-I),
zero + eps (1-I), zero + eps (1+1 I)}]/(2 Pi I)},
{eps, 10^Range[0., -5, -1] }] Same for the Zeta function as a check :

Residue[1/Zeta[s], {s, ZetaZero}] // N
(* 1.2451 - 0.198218 I *)

Table[{eps,
NIntegrate[1/Zeta[s],
{s, ZetaZero + eps (1+I), ZetaZero + eps (-1+I),
ZetaZero + eps (-1-I), ZetaZero + eps (1-I),
ZetaZero + eps (1+I)}]/(2 Pi I)},
{eps, 10^Range[0, -5, -1] }] • Good method for a numerical solution. But it is possible to get the residue as a symbolic expression (as a function of s0) ? – Vaclav Kotesovec Feb 17 '17 at 21:46
• For example for ZetaZero we have the residue 1/Derivative[Zeta][ZetaZero]. But in general ? – Vaclav Kotesovec Feb 17 '17 at 21:53
• @VaclavKotesovec I don't know; unlike for the Zeta function, there is no symbolic expression of the zeros for DirichletL. – b.gates.you.know.what Feb 18 '17 at 9:07