## The Series

Consider the series identity

$$\Phi(s) = \sum_{n=1}^\infty \frac{\mu(n) (\log n)^k}{n^s} \sum_{r \in R(n)} \zeta(s,r/n) = \sum_{n=1}^\infty \frac{\Lambda_k'(f(n))}{n^s}$$

$$R(n) = \left\{ 1 \leq r \leq n : f(r) \equiv 0 \bmod n \right\}$$ $$\Lambda_k'(n) = \sum_{d|n} \mu(d) (\log d)^k$$

Where $f(n) = f_1(n)...f_k(n)$ is an integer valued polynomial with $f(n) > 0$ when $n > 0$, and $f_i(n)$ are distinct and irreducible over $\mathbb{Q}$. The polynomial $f(n)$ must also satisfy the Bunyakovsky property which can be phrased as $|R(n)| < n$ for all $n > 1$. The function $\Lambda_k'(n)$ is a slight variation of the generalized von Mangoldt function and is nonzero only when $n$ has at most $k$ distinct prime factors.

To see why the equality of series holds, it suffices to observe that $d \mid f(n)$ if and only if $n \equiv r \bmod d$ for some $r \in R(n)$. Expanding the $d^{-s} \zeta(s, r/d)$ includes only the positive integers that lie on this arithmetic progression with coefficient $\mu(d) (\log d)^k$. Therefore, summing over all $r/d$ will contribute the term $\mu(d) (\log d)^k n^{-s}$ for all $d \mid f(n)$. Collecting terms by denominator gives the second series.

## Observations

The series for $\Phi(s)$ are absolutely convergent for $\Re(s) > 1$ and $\lim_{s \to 1^+} \Phi(s)$ will diverge only if $f(n)$ takes on values with at most $k$ distinct prime factors infinitely often with sufficient frequency. This is very similar to the statement of Schinzel's hypothesis H, which generalizes a number of other conjectures.

By considering the exponential series

$$F(t) = \sum_{n=1}^\infty \mu(n) (\log n)^k \sum_{r \in R(n)} \frac{e^{(n-r) t}}{e^{n t} - 1} = \sum_{n=1}^\infty \Lambda_k'(f(n))e^{-n t}$$

$$\Gamma(s) \Phi(s) = \int_0^\infty t^{s-1} F(t) dt$$

I think it should be possible, using properties of Mellin transforms, to show that $\Phi(s)$ has a pole at $s=1$ by showing that $F(t) \sim \frac{C}{t}$ as $t \to 0$ where

$$\lim_{t \to 0^+} t F(t) = \sum_{n=1}^\infty \frac{\mu(n) \omega(n) (\log n)^k}{n} = C $$ $$ \omega(n) = |R(n)|$$

Which can then be evaluated using

$$ G(s) = \sum_{n=1}^\infty \frac{\mu(n) \omega(n)}{n^s} = \prod_p 1 -\frac{\omega(p)}{p^s} $$

$$ C = \lim_{s \to 1} (-1)^k G^{(k)}(s) = (-1)^k k! \lim_{s \to 1} \zeta(s)^k G(s) = (-1)^k k! \prod_p \frac{1 - \omega(p)/p}{(1 - 1/p)^k} $$

Where the final product is the constant from the Bateman-Horn conjecture.

## Questions

- Has this series, or have other series of this form, been considered elsewhere? Specifically, series which sum over the local roots of a polynomial like this.
- Are these observations correct? Does this imply $\Phi(s)$ diverges as $s \to 1^+$?
- Are there any techniques that may be useful in finding an asymptotic series for $F(t)$ as $t \to 0$ in terms of $t^n$ or in otherwise meromorphically continuing $\Phi(s)$ beyond $\Re(s)>1$? The higher order terms seem to depend nontrivially on the values of $r \in R(n)$.
- How, if at all, are these series and questions about their asymptotic behavior related to sieve theory? It seems like they should be, but I can't see how.

provedbehavior at $s = 1$ matches what you expect, since that would imply infinitude of the prime values and you know this is still an open problem. (In my paper, the "gap" is Assumption 24.) $\endgroup$