4 The sum in (3.11) can be regularized by analytical con- tinuation. We will write it in the following form, us- ing symmetry properties with respect to m ↔ − m and j ↔ − j : Here we have defined wherein the double prime on the summation mark means that the term m = j = 0 is explicitly excluded, and Clearly, K ( τ ) is finite for all τ > 0. The function S ( τ, p ) may be regularized by use of the Chowla-Selberg formula [see e.g. Eq. (4.33) of Ref. [40]] Clearly, ( τ ) is finite for all τ > 0. where where ν are summed over the divisors of l and it is as- sumed that Δ > 0. K is again the modified Bessel func- tion of the second kind. The apparent pole as q → 1 2 now vanishes due to a cancellation between the first two terms of (3.15), and we find that letting q = 1 2 + s 2 and taking the limit s → 0 + (here a = p $^{2}$, b = 0 , c = τ $^{2}$) where γ = 0 . 577216 ... is Euler’s constant. Now σ$_{0}$ ( l ) is simply the number of positive divisors of l , σ$_{0}$ (1) = 1 , σ$_{0}$ (2) = σ$_{0}$ (3) = 2 , σ$_{0}$ (4) = 3 etc. Note that Eq. (3.18) Figure 3 shows a numerical calculation of e ( τ, p = 3) as a function of τ along with its high and low τ asymptotes (see derivations in the following sections). The calcula- tion was performed by “brute force” by truncating the sums after a number of terms, and has somewhat limited accuracy due to the large number of terms in the j sum in Eq. (3.20) required for small τ , scaling as τ − $^{1}$. In Fig. 2 we plot the three additional terms in the sec- ond line of Eq. (3.20) where we have defined the short- hand with ˜ e$_{m,j}$$^{,}$K , and S given in Eqs. (3.5), (3.14) and (3.18), respectively. The differentiations with respect to τ are now straightforward, should the full expanded expression be desirable. in terms of is valid for all τ ; although it appears most convenient for large τ , it is, by the symmetry property seen in Eq. (3.13), equally useful for small τ . We finally write down the final, regularized energy of the wedge (and, simultaneously, cylinder) at finite T , us- ing the convention used in Ref. [34] FIG. 2: The additional terms of the regularized energy in Eq. (3.20) which are subtracted from the double sum there in the case p = 3. Shown also is the sum of the three addi- tional terms and their low-temperature asymptotic value from Eq. (4.11). 10 −3 10 −2 10 −1 10 0 10 1 10 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 E$_{S}$ E$_{K}$ 3τ/64 E$_{S}$ + E$_{K}$ +3τ/64 -ln(2 ϖ/p)/4 Dimensionless temperature  τ lim s → 0 + ∞ ∑ m =0$^{′}$ ∞ ∑ j =0$^{′}$ 1 ˜ ϕ 1+ s = 1 4 lim s → 0 + ∑ m,j ∈ Z 1 ˜ ϕ 1+ s = 1 4 + 1 4 S ( τ, p ) + 1 2 K ( τ ) . (3.12) S ( τ, p ) = lim s → 0 + ∑ m,j ∈ Z$^{′′}$ 1 ϕ 1+ s = S ( p, τ ) , (3.13) K ( τ ) = lim s → 0 + ∞ ∑ j $_{=1}$[ 1 ˜ ϕ 1+ s − 1 ϕ 1+ s ]$_{m}$$_{=0}$ = ∞ ∑ j =1 jτ − √ 1 + j $^{2}$τ2 jτ 1 + j $^{2}$τ2 . (3.14) ∑ m,j ∈ Z$^{′′}$ ( am 2 + bmj + cj $^{2}$) − q = 2 ζ (2 q ) a − q + 2 2 q √$_{πa}$ q − $^{1}$Γ( q − 1 $_{2}$) ζ (2 q − 1) Γ( q )Δ q 1 2 + 2 q + 5 2 π q Γ( q )Δ 1 $_{2}$( q − 1 $_{2}$) √$_{a}$ ∞ ∑ l =1 l q − 1 2 σ$_{1}$$_{−}$$_{2}$$_{q}$ ( l ) × cos( lπb/a ) K$_{q}$$_{−}$ 1 $_{2}$( $_{πl}$√$_{Δ}$$_{/a}$$_{)}$$_{,}$ (3.15) Δ =4 ac − b $^{2}$, (3.16) σ$_{w}$ ( l ) = ∑ ν | l ν $^{w}$, (3.17) S ( τ, p ) = 2 p ( γ − ln 4 πp τ ) + 8 p ∞ ∑ l =1 σ$_{0}$ ( l ) K$_{0}$ (2 πlτ /p ) (3.18) ˆ T $^{=(3}$− 3 τ ∂$_{τ}$ + τ $^{2}$∂2 $_{τ}$); (3.21a) E$_{S}$ ( τ, p ) = τ 64 ˆ T S ( τ, p ); E$_{K}$ ( τ, p ) = τ 32 ˆ T K ( τ ) . (3.21b) e ( τ, p ) = 4 τ π ∞ ∑ m =0$^{′}$ ∞ ∑ j =0$^{′}$ ˜ e$_{m,j}$ ( τ, p ) − τ 64 $^{(3}$− 3 τ ∂$_{τ}$ + τ $^{2}$∂2 $^{$_{τ}$)[1 + 2}$K ( τ $^{) +}$S( τ, p )] . (3.20) ˜ E ( τ, p, a ) = 1 8 πna $_{2}$e ( τ, p ) , (3.19)