Dirichlet Series Convergence Lemma/Lemma

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Lemma to Dirichlet Series Convergence Lemma

Let $\ds \map f s = \sum_{n \mathop = 1}^\infty \frac{a_n} {n^s}$ be a Dirichlet series.

Suppose that for some $s_0 = \sigma_0 + i t_0 \in \C$, $\map f {s_0}$ has bounded partial sums:

$(1): \quad \ds \size {\sum_{n \mathop = 1}^N a_n n^{-s_0} } \le M$

for some $M \in \R$ and all $N \ge 1$.


Then for every $s = \sigma + i t \in \C$ with $\sigma > \sigma_0$:

$\ds \size {\sum_{n \mathop = m}^N a_n n^{-s} } \le 2 M m^{\sigma_0 - \sigma} \paren {1 + \frac {\size {s - s_0} } {\sigma - \sigma_0} }$


Proof

From the Summation by Parts formula:

$\ds \sum_{n \mathop = m}^N f_n g_n = f_N G_N - f_m G_{m-1} - \sum_{n \mathop = m}^{N - 1} \map {G_n} {f_{n + 1} - f_n}$

Let:

$g_n = a_n n^{-s_0}$
$f_n = n^{s_0 - s}$

For $N \ge 1$, the quantities $G_N$ are the partial sums $(1)$.

Thus $G_N \le M$ for all $N \ge 1$.

We have:

\(\ds \size {\sum_{n \mathop = m}^N \frac {a_n} {n^s} }\) \(=\) \(\ds \size {\sum_{n \mathop = m}^N f_n g_n}\)
\(\ds \) \(\le\) \(\ds \size {f_N G_N} + \size {f_m G_{m - 1} } + \sum_{n \mathop = m}^{N - 1} \size {\map {G_n} {f_{n + 1} - f_n} }\) using partial summation and the Triangle Inequality
\(\ds \) \(\le\) \(\ds M \size {N^{s_0 - s} } + M \size {m^{s_0 - s} } + M \sum_{n \mathop = m}^{N - 1} \size {\paren {\paren {n + 1}^{s_0 - s} - n^{s_0 - s} } }\) using the given bound on the partial sums
\(\ds \) \(=\) \(\ds M N^{\sigma_0 - \sigma} + M m^{\sigma_0 - \sigma} + M \sum_{n \mathop = m}^{N - 1} \size {\paren {s - s_0} \int_n^{n + 1} t^{s_0 - s - 1} }\)
\(\ds \) \(\le\) \(\ds M N^{\sigma_0 - \sigma} + M m^{\sigma_0 - \sigma} + M \sum_{n \mathop = m}^{N - 1} \size {s - s_0} \int_n^{n + 1} \size { t^{s_0 - s - 1} }\) Modulus of Complex Integral
\(\ds \) \(=\) \(\ds M N^{\sigma_0 - \sigma} + M m^{\sigma_0 - \sigma} + M \sum_{n \mathop = m}^{N - 1} \size {s - s_0} \int_n^{n + 1} t^{\sigma_0 - \sigma - 1}\)
\(\ds \) \(=\) \(\ds M N^{\sigma_0 - \sigma} + M m^{\sigma_0 - \sigma} + M \size {s - s_0} \int_m^N t^{\sigma_0 - \sigma -1}\)
\(\ds \) \(=\) \(\ds M N^{\sigma_0 - \sigma} + M m^{\sigma_0 - \sigma} + M \frac {\size {s - s_0} } {\sigma - \sigma_0} \paren {m^{\sigma_0 - \sigma}- N^{\sigma_0 - \sigma} }\)
\(\ds \) \(\le\) \(\ds M N^{\sigma_0 - \sigma} + M m^{\sigma_0 - \sigma} + M \frac {\size {s - s_0} } {\sigma - \sigma_0} \paren {m^{\sigma_0 - \sigma} + N^{\sigma_0 - \sigma} }\)

Finally, because $N \ge m$ and $\sigma_0 - \sigma < 0$, we have:

$N^{\sigma_0 - \sigma} + m^{\sigma_0 - \sigma} \le 2 m^{\sigma_0 - \sigma}$

Therefore:

\(\ds \size {\sum_{n \mathop = m}^N a_n n^{-s} }\) \(\le\) \(\ds M N^{\sigma_0 - \sigma} + M m^{\sigma_0 - \sigma} + M \frac {\size {s - s_0} } {\sigma - \sigma_0} \paren {m^{\sigma_0 - \sigma} + N^{\sigma_0 - \sigma} }\)
\(\ds \) \(\le\) \(\ds 2 M m^{\sigma_0 - \sigma} + 2 M \frac {\size {s - s_0} } {\sigma - \sigma_0} m^{\sigma_0 - \sigma}\)
\(\ds \) \(=\) \(\ds 2 M m^{\sigma_0 - \sigma} \paren {1 + \frac {\size {s - s_0} } {\sigma - \sigma_0} }\)

Hence the result.

$\blacksquare$


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