Let $Okay/ mathbb Q $ be a finite Galois extension and let $X$ be a correct non-empty subset of the Galois group $G=Gal(Okay/ mathbb Q)$ that’s closed below conjugation. Think about a set of integer primes $P$ such that for all sufficiently massive primes $p$, the next equivalence holds

$$p in P iff textual content{ the conjugacy class of the Frobenius component }sigma_p textual content{ is contained in }X$$

Now let $E$ be a multiplicative set of pure numbers (that’s, for all coprime $m, n in mathbb N$, we’ve the equivalence $mn in E iff m in E$ or $n in E$) such that the set of prime numbers in $E$ is strictly the set $P$ above and let $E’ := mathbb N setminus E$ denote the complement of $E$. Think about the indicator sequence $(a_n)_{n geq 1}$ of $E’$ (in order that $a_n := 1 iff n in E’$ and $a_n=0$ in any other case) and let $F(s) := sum_{n geq 1} a_n n^{-s}$ be the Dirichlet Collection comparable to the sequence $(a_n)_{n geq 1}$.

I need to present that the operate $F$ analytically continues to a area of the shape given within the picture the place $delta>0$ is fastened, the circle across the level $1$ is of radius $epsilon < delta$ and the infinite branches $C$ and $D$ are outlined by

$$Re(s) = 1 – frac{a}Im(s)$$

(the place $a$ and $A$ are fastened constructive numbers, notice that the inside of the circle has been excluded from the aforementioned area) such that on this area we’ve

$$F(s) = O((log |Im(s)|)^A) textual content{ as } |Im(s)| rightarrow infty$$

The one outcomes of this type I’m considerably accustomed to are these on the analytic continuation of the same old Riemann Zeta Perform (which I learn in Apostol’s “Introduction to Analytic Quantity Idea”). Though I’ve obtained another instant observations (as an example: the pure and Dirichlet density of $P$ should each be $|X|/|G| in (0,1)$ by the Chebotarev Density Theorem and that the sequence $(a_n)$ must be multiplicative therefore we are able to get one thing akin to an `Euler-Product’ illustration of the Dirichlet Collection $F(s)$), I’ve no normal thought on get began on this drawback and I’d actually a proof or a reference containing an entire (and ideally not too inaccessaible) proof of the identical. Thanks.

**P.S.:** It says right here (Continuation as much as zero of a Dirichlet sequence with bounded coefficients) {that a} Dirichlet sequence with bounded coefficients needn’t be meromorphically continuable to the fitting of zero, however I have not discovered any constructive outcomes on M.O. on this course.