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Let $K/ \mathbb Q $ be a finite Galois extension and let $X$ be a proper non-empty subset of the Galois group $G=Gal(K/ \mathbb Q)$ that is closed under conjugation. Consider a set of integer primes $P$ such that for all sufficiently large primes $p$, the following equivalence holds $$p \in P \iff \text{ the conjugacy class of the Frobenius element }\sigma_p \text{ is contained in }X$$

Now let $E$ be a multiplicative set of natural numbers (that is, for all coprime $m, n \in \mathbb N$, we have the equivalence $mn \in E \iff m \in E$ or $n \in E$) such that the set of prime numbers in $E$ is exactly the set $P$ above and let $E' := \mathbb N \setminus E$ denote the complement of $E$. Consider the indicator sequence $(a_n)_{n \geq 1}$ of $E'$ (so that $a_n := 1 \iff n \in E'$ and $a_n=0$ otherwise) and let $F(s) := \sum_{n \geq 1} a_n n^{-s}$ be the Dirichlet Series corresponding to the sequence $(a_n)_{n \geq 1}$.

I want to show that the function $F$ analytically continues to a region of the form given in the image The Mentioned Region has been Coloured in Yellow. where $\delta>0$ is fixed, the circle around the point $1$ is of radius $\epsilon < \delta$ and the infinite branches $C$ and $D$ are defined by $$\Re(s) = 1 - \frac{a}{(\log (2+|\Im(s)|))^A}$$
(where $a$ and $A$ are fixed positive numbers, note that the interior of the circle has been excluded from the aforementioned region) such that in this region we have $$F(s) = O((\log |\Im(s)|)^A) \text{ as } |\Im(s)| \rightarrow \infty$$

The only results of this kind I am somewhat familiar with are those on the analytic continuation of the usual Riemann Zeta Function (which I read in Apostol's ``Introduction to Analytic Number Theory"). Although I have obtained some other immediate observations (for instance: the natural and Dirichlet density of $P$ must both be $|X|/|G| \in (0,1)$ by the Chebotarev Density Theorem and that the sequence $(a_n)$ should be multiplicative hence we can get something akin to an `Euler-Product' representation of the Dirichlet Series $F(s)$), I have no general idea on how to get started on this problem and I would really a proof or a reference containing a complete (and preferably not too inaccessible) proof of the same. Thank you.

P.S.: It says here (Continuation up to zero of a Dirichlet series with bounded coefficients) that a Dirichlet series with bounded coefficients need not be meromorphically continuable to the right of zero, but I haven't found any positive results on M.O. in this direction.

Edit (19-05-2020): I found the following result (although I don't know how to show this one either), which I think might be relevant:

If $f_P(s) = \sum_{p \not\in P} p^{-s}$, then $f_P$ extends into a holomorphic function to the right of the curve $C$ and $D$ (in the image) except for the real axis from $1-\delta$ to $1$, that is into a region of the form $$R:= \left\{ s \in \mathbb{C} \Bigg| \Re(s) \geq 1-\frac{a}{(\log T)^A}, \Im(s) \neq 0 \right\} \cup \Big((1, \infty) \times 0 \Big)$$
where it also satisfies the bound $f_P(s) = O(\log \log (2+|\Im(s)|))\text{ as }|\Im(s)| \rightarrow \infty$.

I could also show that the function $h(s):= \log F(s) - f_P(s)$ is holomorphic for $\Re(s) \geq 1$. But I am still not sure how I can complete the proof from here. I would appreciate a proof even if it assumes these the two results.

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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, this should also follow from the more general Proposition 2.2 in:

https://arxiv.org/abs/1810.06024

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  • $\begingroup$ I see Serre has stated the two results I mentioned above and has even proven the second one. However, I am stuck at the same place where he just says that "we can show by means of (1.7) and (2.5) ..." to get the analytic continuation into the region in the image above where the bound $F(s) = \log^A(2+|\Im(s)|)$ holds (he mentions this in the beginning of his proof of Proposition 2.8). Can you please tell me how that follows from (2.5) and (1.7)? Thanks. $\endgroup$ – asrxiiviii May 20 '20 at 2:50
  • $\begingroup$ Also, I have been unable to find either of the references (one due to Landau and the other due to Watson) he has mentioned in his proof of Proposition 2.8. $\endgroup$ – asrxiiviii May 20 '20 at 3:23
  • $\begingroup$ Furthermore, the two regions mentioned in 1.7 and 2.8 look similar but the region in 1.7 (which is the same as the region $R$ I mentioned in my 19-05 Edit) excludes the real axis till the point $1$, while the region in the picture above (same as the region considered in 2.8) excludes a small ball around $1$. I don't see a rigorous way of getting holomorphicity of $F(s)$ in the second region (region in the image above) using the holomorphicity of $\sum_{p \not\in P} p^{-s}$ in the first. $\endgroup$ – asrxiiviii May 20 '20 at 3:38

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