content_image/416240.jpg content_image/416241.jpg content_image/416246.jpg content_image/416242.jpg content_image/416243.jpg content_image/416244.jpg content_image/416245.jpg content_image/416249.jpg content_image/416248.jpg content_image/416239.jpg content_image/416247.jpg content_image/416250.jpg
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 : 

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Here we have defined 

<img src='content_image/416241.jpg'>

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]] 

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 $^{2}$) 

<img src='content_image/416246.jpg'>

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) 

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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] 

in terms of 

<img src='content_image/416249.jpg'>

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 Fig. 2 we plot the three additional terms in the sec- ond line of Eq. (3.20) where we have defined the short- hand 

<img src='content_image/416248.jpg'>

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 − $^{1}$. 

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