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Results tagged with cv.complex-variables
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user 5101
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
4
votes
1
answer
235
views
Conditional convergence of Artin $L$-functions
Let $k$ be a number field and $V$ a non-trivial irreducible Artin representation over $k$. Consider the associated Artin $L$-function with corresponding Euler product decomposition $L(V,s)= \prod_v L_ …
- 105k
1
vote
Accepted
Existence of analytic continuation of Dirichlet series corresponding to the indicator sequen...
Serre deals with problems of this type in the paper:
Serre -Divisibilité de certaines fonctions arithmétiques
The fact you want should follow from the results in Sections 1 and 2.
Alternatively, th …
- 21.4k
1
vote
Accepted
Logarithms of $L$-functions of irreducible characters of Galois group
Yes this is the fact that for a non-trivial irreducible Artin character, the associated Artin L-function is holomorphic and non-zero on $\rm{re}\, s \geq 1$.
For the trivial character, one just obtai …
- 21.4k
2
votes
Accepted
Dirichlet series without order term
These are called Dirichlet polynomials.
They arise in many places in analytic number theory. For example, in approximate functional equations of $L$-functions.
- 21.4k
18
votes
Accepted
Anything special (historical?) about surface $x\cdot y\cdot z\ +\ x+y+z=0$?
The corresponding projective surface
$$S: xyz + (x + y + z)w^2=0 \subset \mathbb{P}^3,$$
is a singular cubic surface - singular cubic surfaces are special. It has three singularities, each of which h …
- 21.4k
4
votes
Accepted
Converse to a theorem of Landau on Dirichlet series
I'm not sure you can hope for much. For example consider the case $c_n=1$ if $n$ is not a square, and $c_n=-1$ otherwise. The associated Dirichlet series has a pole at $s=1$, but of course the terms a …
- 21.4k