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The binomial series gives a meromorphic continuation of any series of the form $\sum_{n=N}^\infty g(n) f(n)^{-s}$ for some polynomials $g(x) = \sum_{j=0}^r c_j x^j$ and $f(x) = x^r \prod_{k=1}^d (x-a_k)$

$$f(n)^{-s} = n^{-(d+r)s} \prod_{k=1}^d (1-a_kn^{-1})^{-s} = n^{-(d+r)s} \prod_{k=1}^d \left(\sum_{m=0}^\infty {-s \choose m} (-a_k)^m n^{-m}\right) \\= \sum_{l=0}^\infty h_l(s) n^{-s(d+r)-l}$$ $$\sum_{n=N}^\infty g(n) f(n)^{-s} =\sum_{j=0}^r c_j \sum_{l=0}^\infty h_l(s) (\zeta(s(d+r)+l-j)-\sum_{n=1}^{N-1} n^{-s(d+r)-l+j})$$ Where for $N > \max_k \frac{1}{|a_k|}$ the last series converges locally uniformly (away from the poles) for every $s$

The binomial series gives a meromorphic continuation of any series of the form $\sum_{n=N}^\infty g(n) f(n)^{-s}$ for some polynomials $g(x) = \sum_{j=0}^t c_j x^j$ and $f(x) = x^r \prod_{k=1}^d (x-a_k)$

$$f(n)^{-s} = n^{-(d+r)s} \prod_{k=1}^d (1-a_kn^{-1})^{-s} = n^{-(d+r)s} \prod_{k=1}^d \left(\sum_{m=0}^\infty {-s \choose m} (-a_k)^m n^{-m}\right) \\= \sum_{l=0}^\infty h_l(s) n^{-s(d+r)-l}$$ $$\sum_{n=N}^\infty g(n) f(n)^{-s} =\sum_{j=0}^t c_j \sum_{l=0}^\infty h_l(s) (\zeta(s(d+r)+l-j)-\sum_{n=1}^{N-1} n^{-s(d+r)-l+j})$$ Where for $N > \max_k \frac{1}{|a_k|}$ the last series converges locally uniformly (away from the poles) for every $s$

The binomial series gives a meromorphic continuation of any series of the form $\sum_{n=N}^\infty f(n)^{-s}$ for some polynomial $f(x) = x^r \prod_{k=1}^d (x-a_k)$

$$f(n)^{-s} = n^{-(d+r)s} \prod_{k=1}^d (1-a_kn^{-1})^{-s} = n^{-(d+r)s} \prod_{k=1}^d \left(\sum_{m=0}^\infty {-s \choose m} (-a_k)^m n^{-m}\right) \\= \sum_{l=0}^\infty h_l(s) n^{-s(d+r)-l}$$ $$\sum_{n=N}^\infty f(n)^{-s} = \sum_{l=0}^\infty h_l(s) (\zeta(s(d+r)+l)-\sum_{n=1}^{N-1} n^{-s(d+r)-l})$$ Where for $N > \max_k \frac{1}{|a_k|}$ the last series converges locally uniformly (away from the poles) for every $s$

The binomial series gives a meromorphic continuation of any series of the form $\sum_{n=N}^\infty g(n) f(n)^{-s}$ for some polynomials $g(x) = \sum_{j=0}^r c_j x^j$ and $f(x) = x^r \prod_{k=1}^d (x-a_k)$

$$f(n)^{-s} = n^{-(d+r)s} \prod_{k=1}^d (1-a_kn^{-1})^{-s} = n^{-(d+r)s} \prod_{k=1}^d \left(\sum_{m=0}^\infty {-s \choose m} (-a_k)^m n^{-m}\right) \\= \sum_{l=0}^\infty h_l(s) n^{-s(d+r)-l}$$ $$\sum_{n=N}^\infty g(n) f(n)^{-s} =\sum_{j=0}^r c_j \sum_{l=0}^\infty h_l(s) (\zeta(s(d+r)+l-j)-\sum_{n=1}^{N-1} n^{-s(d+r)-l+j})$$ Where for $N > \max_k \frac{1}{|a_k|}$ the last series converges locally uniformly (away from the poles) for every $s$

The binomial series gives a meromorphic continuation of any series of the form $\sum_{n=N}^\infty f(n)^{-s}$ for some polynomial $f(x) = x^r \prod_{k=1}^d (x-a_k)$

$$f(n)^{-s} = n^{-(d+r)s} \prod_{k=1}^d (1-a_kn^{-1})^{-s} = n^{-(d+r)s} \prod_{k=1}^d \left(\sum_{m=0}^\infty {-s \choose m} (-a_k n)^{-sm}\right) \\= \sum_{l=0}^\infty h_l(s) n^{-s(d+r+l)}$$ $$\sum_{n=N}^\infty f(n)^{-s} = \sum_{l=0}^\infty h_l(s) (\zeta(s(d+r+l))-\sum_{n=1}^{N-1} n^{-s(d+r+l)})$$ Where for $N > \max_k \frac{1}{|a_k|}$ the last series converges locally uniformly (away from the poles) for every $s$

The binomial series gives a meromorphic continuation of any series of the form $\sum_{n=N}^\infty f(n)^{-s}$ for some polynomial $f(x) = x^r \prod_{k=1}^d (x-a_k)$

$$f(n)^{-s} = n^{-(d+r)s} \prod_{k=1}^d (1-a_kn^{-1})^{-s} = n^{-(d+r)s} \prod_{k=1}^d \left(\sum_{m=0}^\infty {-s \choose m} (-a_k)^m n^{-m}\right) \\= \sum_{l=0}^\infty h_l(s) n^{-s(d+r)-l}$$ $$\sum_{n=N}^\infty f(n)^{-s} = \sum_{l=0}^\infty h_l(s) (\zeta(s(d+r)+l)-\sum_{n=1}^{N-1} n^{-s(d+r)-l})$$ Where for $N > \max_k \frac{1}{|a_k|}$ the last series converges locally uniformly (away from the poles) for every $s$