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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

  1. 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.
  2. Are these observations correct? Does this imply $\Phi(s)$ diverges as $s \to 1^+$?
  3. 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 $r \in R(n)$.
  4. 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.
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    $\begingroup$ I am not sure what you mean that the question is posted twice... on my user page it only lists one question. Was the other removed? $\endgroup$ – Liam Eagen Dec 21 '18 at 21:33
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    $\begingroup$ See math.uconn.edu/~kconrad/articles/hlconst.pdf for an analogue of your question using Dirichlet series instead of a power series in $e^{-t}$ and using products of von Mangoldt functions at polynomial values. I think it is unrealistic to expect anyone has unconditionally proved behavior 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.) $\endgroup$ – KConrad Dec 21 '18 at 21:37
  • $\begingroup$ The product you call $G(s)$ is essentially the same as the product I wrote as $G(s)$ in my paper linked to in my previous comment. See Definition 30. $\endgroup$ – KConrad Dec 21 '18 at 21:42
  • $\begingroup$ Thanks for the reference KConrad. Your paper seems very relevant, I will take a look. $\endgroup$ – Liam Eagen Dec 21 '18 at 22:09
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In the 1960's Turán wrote several papers on a function-theoretic sieve. He managed to express the number of prime twins in terms of roots of $L$-series. He began like you did by expressing $\Lambda$ as a sum over divisors. Then he did not consider the behaviour for $s\searrow 1$, but used complex integration to get an exact formula. Next he shifted the path of integration. The surprising thing in his approach is that the contribution of most roots can be shown to be negligible. Unfortunately, as far as I know up to now no results on prime twins have been established by following this road. The most obvious obstacle is the fact that despite his surprising reductions, even under GRH a trivial bound for the contribution of the roots is far too large. So one would have to prove massive cancellation between different roots, which is extremely difficult.

One reference would be P. Turán, On some conjectures in the theory of numbers, Proc. London Math. Soc. (3) 14a (1965), 288–299.

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