Regularising Infinite Products by the Asymptotics of Finite Products

Abstract

We introduce a regularisation process for a sequence of real numbers computing the constant coefficient in an asymptotic expansion associated with the original sequence. We give explicit computations in several interesting particular cases. This approach originates in a geometric context.

Appendix: The Abel Summation Formula

For the material in this section, see Chapter I.0 of Tenenbaum [16]. We recall Abel’s Partial Summation Formula. We use it in the proof of Proposition 4.3 with $a_n = 1$ and $A(x) = [x]$.

Lemma 4.4

(Abel Formula) Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, and set

$$ A(x) := \sum _{n \le x} a_n. $$

Let $\phi \in \mathcal {C}^1 \bigl ( {\mathbb {R}}^+ \bigr )$ and $x \ge 1$. Then we have

$$ \sum _{n \le x} a_n \phi (n) = A(x) \phi (x) - \int _1^x A(t) \phi '(t) \, \mathrm {d}t. $$

We also need the following easy result in the course of the proofs of Lemmas 4.1, 4.2 and 4.3.

Lemma 4.5

Let $N \ge 1$ be an integer and $f \in \mathcal {C}^1([1, N])$. Let $g_0(t) = \{ t \} - \frac{1}{2}$, $g_1(t) = \frac{1}{2} \{ t \}^2 - \frac{1}{2} \{ t \} + \frac{1}{12}$ and $g_2(t) = \frac{1}{6} \{ t \}^3 - \frac{1}{4} \{ t \}^2 + \frac{1}{12} \{ t \}$, so that $g_0$, $g_1$, $g_2 \in \mathcal {C}^0({\mathbb {R}}) \cap \mathcal {C}^2({\mathbb {R}}\setminus {\mathbb {Z}})$ and $g_1'(t) = g_0(t)$ and $g_2'(t) = g_1(t)$ for $t \notin {\mathbb {Z}}$. We have

$$\begin{aligned} \int _1^N g_0(t) f(t) \,\mathrm {d}t = \frac{1}{12} \bigl ( f(N) - f(1) \bigr ) - \int _1^N g_1(t) f'(t) \, \mathrm {d}t. \end{aligned}$$

(4.4)

If $f \in \mathcal {C}^2([1, N])$, then

$$\begin{aligned} \int _1^N g_0(t) f(t) \,\mathrm {d}t = \frac{1}{12} \bigl ( f(N) - f(1) \bigr ) + \int _1^N g_2(t) f''(t) \, \mathrm {d}t. \end{aligned}$$

(4.5)

For the proof, we just remark that we can integrate by parts repeatedly over intervals of the form $[n, n + 1]$ for $n = 1$, 2, ..., $N - 1$, and then add the results. We exploit the fact that $g_0$, $g_1$ and $g_2$ are periodic functions with period 1, and that they have average 0 over a period.