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The Lipschitz formula for the reciprocal-power sums

Statement

For every integer $k \geq 2$ and every $τ \in H$ , with $q = e^{2 π i τ}$ , $\sum_{n \in Z} \frac{1}{(τ + n)^{k}} = \frac{(- 2 π i)^{k}}{(k - 1)!} \sum_{r = 1}^{\infty} r^{k - 1} q^{r},$ where the series on the left converges absolutely and the identity is independent of the order of summation.

Facts & Assumptions

Given: An integer $k \geq 2$ , a point $τ \in H$ , and $q = e^{2 π i τ}$ .

[F1]

$π \cot (π z) = \frac{1}{z} + \sum_{n \geq 1} (\frac{1}{z - n} + \frac{1}{z + n})$ , and the series converges locally uniformly on $C ∖ Z$ (The Mittag-Leffler expansion of pi cotangent).

[F2]

If the partial sums of $\sum_{j} g_{j}$ of holomorphic functions converge locally uniformly to $g$ , then $g$ is holomorphic and $g^{(m)} = \sum_{j} g_{j}^{(m)}$ for every $m$ , the derivative series again converging locally uniformly (A locally uniformly convergent series of holomorphic functions may be differentiated term by term, Complex analytic functions as locally representable by convergent power series).

[F3]

$\sin z = \frac{e^{i z} - e^{- i z}}{2 i}$ , $\cos z = \frac{e^{i z} + e^{- i z}}{2}$ , and $\cot z = \cos z / \sin z$ wherever $\sin z \neq 0$ (Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential, Tangent, cotangent, secant, and cosecant on their exact natural domains).

[F4]

$\exp (w + w^{'}) = \exp (w) \exp (w^{'})$ , $\exp (2 w) = \exp (w)^{2}$ , $\exp (- w) = 1 / \exp (w)$ , and $∣ \exp (x + i y) ∣ = e^{x}$ for real $x, y$ ( $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

[F5]

$\exp^{'} = \exp$ ; the chain rule and the algebra of derivatives give $\frac{d^{m}}{d w^{m}} e^{a w} = a^{m} e^{a w}$ and $\frac{d^{m}}{d w^{m}} \frac{1}{w + n} = \frac{(- 1)^{m} m!}{(w + n)^{m + 1}}$ on their domains (The complex exponential is entire and its complex derivative is itself, The chain rule for complex derivatives, Linearity, product, reciprocal, and quotient rules for complex derivatives).

[F6]

Convergence in $C$ is convergence in the metric $d (z, w) = ∣ z - w ∣$ (The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane); an absolutely convergent complex series converges, and every rearrangement of it has the same sum (Every absolutely convergent complex series converges, and rearrangements preserve its sum, Complex series, absolute convergence, complex power series, and radius of convergence).

[F7]

Let $k \geq 2$ . The real series $\sum_{n \geq 1} n^{- k}$ converges (For rational $p > 0$ , $\sum 1 / k^{p}$ converges iff $p > 1$ ), so $\sum_{n \geq 1} ∣ τ + n ∣^{- k}$ converges by comparison (If $0 \leq a_{k} \leq b_{k}$ eventually, convergence of $\sum b_{k}$ gives convergence of $\sum a_{k}$ , and divergence of $\sum a_{k}$ gives divergence of $\sum b_{k}$ , If $\sum ∣ a_{k} ∣$ converges then $\sum a_{k}$ converges); for $∣ r ∣ < 1$ the real series $\sum_{m \geq 0} r^{m}$ converges and $r^{m} \to 0$ (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges, For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ ).

Proof

1.1F1F3F4F6F7givenalgebra

The function $z \mapsto π \cot (π z)$ is holomorphic on the open set $H \subseteq C ∖ Z$ , and by [F1] the partial sums of $z^{- 1} + \sum_{n \geq 1} ((z - n)^{- 1} + (z + n)^{- 1})$ converge locally uniformly on $H$ to it. Fix $w$ with $∣ w ∣ < 1$ : the telescoping identity $(1 - w) \sum_{m = 0}^{N} w^{m} = 1 - w^{N + 1}$ , the convergence $∣ w ∣^{N + 1} \to 0$ of [F7] and the metric description [F6] give $∣ (1 - w) \sum_{m \leq N} w^{m} - 1 ∣ \to 0$ ; dividing by the nonzero $∣ 1 - w ∣$ proves $\sum_{m \geq 0} w^{m} = 1 / (1 - w)$ . Now put $u = e^{i π z}$ and $Q = u^{2} = e^{2 π i z}$ by [F4]; then [F3] and [F4] give $\sin (π z) = (u - u^{- 1}) / (2 i) = (Q - 1) / (2 i u)$ and $\cos (π z) = (u + u^{- 1}) / 2 = (Q + 1) / (2 u)$ , so $π \cot (π z) = π i \frac{Q + 1}{Q - 1} = - π i (1 + 2 \sum_{r \geq 1} Q^{r})$ , because $∣ Q ∣ = e^{- 2 π Im z} < 1$ for $z \in H$ by [F4] and $\frac{1 + Q}{1 - Q} = 1 + 2 \sum_{r \geq 1} Q^{r}$ .

2.1F2F5step 1.1algebra

The series in 1.1 has holomorphic terms and locally uniformly convergent partial sums on $H$ , so [F2] applies and, for $m = k - 1$ , differentiating termwise gives $\frac{d^{k - 1}}{d z^{k - 1}} π \cot (π z) = (- 1)^{k - 1} (k - 1)! \sum_{n \in Z} (z + n)^{- k}$ on $H$ , the displayed sum being understood through the absolutely convergent paired series of [F1] together with the term $z^{- k}$ differentiated from $z^{- 1}$ ; here $\frac{d^{m}}{d z^{m}} (z + n)^{- 1} = (- 1)^{m} m! (z + n)^{- m - 1}$ by [F5]. On the other side the $Q$ -series of 1.1 consists of entire terms with locally uniformly convergent partial sums, so differentiating it $k - 1$ times termwise by [F2] and [F5] gives $- 2 π i (2 π i)^{k - 1} \sum_{r \geq 1} r^{k - 1} Q^{r}$ as functions of $z$ .

3.1F6F7step 2.1givenalgebra∎

Evaluating the two expressions of 2.1 at $z = τ$ , where $Q = e^{2 π i τ} = q$ , gives $(- 1)^{k - 1} (k - 1)! \sum_{n \in Z} (τ + n)^{- k} = - 2 π i (2 π i)^{k - 1} \sum_{r \geq 1} r^{k - 1} q^{r}$ . Dividing by the nonzero real number $(- 1)^{k - 1} (k - 1)!$ yields the stated identity, since $- 2 π i (2 π i)^{k - 1} / (- 1)^{k - 1} = (- 2 π i)^{k}$ . Finally, for $n \geq 2 ∣ τ ∣$ one has $∣ τ \pm n ∣ \geq n - ∣ τ ∣ \geq n / 2$ , so $\sum_{n \geq 1} ∣ τ \pm n ∣^{- k} \leq C_{τ} + 2^{k + 1} \sum_{n \geq 1} n^{- k} < \infty$ by [F7]; hence the family $((τ + n)^{- k})_{n \in Z}$ is absolutely summable and, by [F6], every enumeration of it converges to the same sum, which is the order-independence asserted in the Statement.

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Sources

D. Zagier, Elliptic Modular Forms and Their Applications, in The 1-2-3 of Modular Forms (Universitext, Springer, 2008) (standard reference, not scraped)
J. S. Milne, Modular Functions and Modular Forms (v1.31, 2017) (standard reference, not scraped)
C. T. McMullen, Advanced Complex Analysis, Math 213a course notes (Harvard, 2010) (standard reference, not scraped)