10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 −10 2 −10 1 −10 0 −10 −1 −10 −2 Dimensionless temperature  τ Di m ensionless energy e( ,p=3) τ FIG. 3: Dimensionless energy e ( τ, p ) for the case p = 3, i.e. opening angle α = π/ 3 approximated by a “brute force” cal- culation truncating the sums. The zero-temperature limit and high- τ asymptote are shown as dashed lines. IV. REGAINING THE LIMIT OF ZERO TEMPERATURE Comparing Eq. (3.20) with the zero temperature result derived in Ref. [34], and previously known for the cylin- drical shell [8, 10], it is not obvious that our expression simplifies to the zero temperature result as τ → 0. In this section we show that upon careful examination the correct limit is in fact obtained. Let us write down the zero temperature result ˜ E 0 for general p in its regularized form suitable for comparison 1 (c.f. Ref. [34], Eq. (4.14)): ˜ E 0 = 1 2 ¯ E 0 + ∞ ∑ m =1 ¯ E m + 1 32 πna 2 ln(2 π/p ); (4.1a) ¯ E 0 = 1 4 πna 2 ∫ ∞ 0 dxx [ ln(1 − x $^{2}$λ2$_{0}$) + x 4 4(1 + x $^{2}$)3 ] ; (4.1b) ¯ E m 0 = 1 4 πna 2 ∫ ∞ 0 dxx [ ln(1 − x $^{2}$λ2 $_{mp}$) + x 4 4( m $^{2}$p2 + x $^{2}$)3 ] . (4.1c) The finite temperature quantity ¯ E of Eq. (3.5) is ana- lytic as τ → 0 and inverse application of the transition (2.5) simply gives us ¯ E τ → 0 −→ 1 2 ¯ E 0 + ∞ ∑ m =1 ¯ E $_{m}$. (4.2) What remains is essentially to determine the low τ be- havior of K ( τ ) and S ( τ, a ) to check that the last term of Eq. (4.1a) may be regained. 1 These definitions of ¯ E$_{0}$ and ¯ E$_{m}$$_{0}$ differ from those of Ref. [34] by a prefactor n . 5 To study the behavior of S it is convenient to employ the symmetry relation S ( τ, p ) = S ( p, τ ) which gives S ( τ, p ) = 2 τ ( γ − ln 4 πτ p ) + 8 τ ∞ ∑ l =1 σ$_{0}$ ( l ) K$_{0}$ (2 πlp/τ ) . (4.3) For large arguments K$_{0}$ ( x ) ∝ $^{exp(}$− x ), so the sum over l is exponentially small as τ → 0. This immediately gives the asymptotic behavior: S ( τ, p ) ∼ 2 τ [ γ − ln(2 π/p $^{)}$− $^{ln 2}$− ln τ ] , τ → 0 . (4.4) Next we turn to K ( τ ). Using the Euler-Maclaurin for- mula (e.g. Ref. [39], p.806) we have K ( τ ) = ∞ ∑ l =0 κ ( τ + lτ ) = 1 τ ∫ ∞ τ dt κ ( t ) + 1 2 κ ( τ ) − ∞ ∑ l =1 τ 2 l − $^{1}$B$_{2}$$_{l}$ (2 l )! κ (2 l − $^{1)}$( τ ) (4.5) with κ ( t ) = t − √$_{1 +}$ t 2 $_{t}$√$_{1 +}$ t 2 . (4.6) The integral has the solution ∫ ∞ τ dt κ ( t ) = ln 2 τ τ $_{+}$√$_{1 +}$ τ 2 = ln τ $^{+ ln 2}$− τ + 1 $_{6}$τ 3 + ... (4.7) To leading order in τ , thus, the sum in Eq. (4.5) reads 1 τ ∑ ∞ l =1 B$_{2}$$_{l}$/ 2 l . As is typically the case for series expan- sions close to non-analytical points, the series is formally divergent. It can, however, be regularized by means of Borel summation [41]. For a highly similar problem and details on how to approach it, see [42]. We show in Ap- pendix A that the Borel regularized sum evaluates to ∞ ∑ l =1 B$_{2}$$_{l}$ 2 l = γ − 1 $_{2}$. (4.9) Thus we have found the low- τ expansion of K ( τ ): K ( τ ) ∼ 1 τ (ln τ $^{+ ln 2}$− γ $^{)}$− 1 2 + . . . , τ → 0 . (4.10) Further terms cancel at least to order τ $^{2}$, the leading order correction being at least of order τ $^{4}$. as τ → 0. Moreover, κ ( τ ) = − 1 /τ + 1 − 1 $_{2}$τ 2 + .. , and upon inspection we recognize that τ 2 l − 1 κ (2 l − $^{1)}$( τ ) = (2 l − 1)! τ $^{+ (}$−1) l [ (2 l )! 2 $^{l}$l ! ] 2 τ 2 l + ... (4.8)