\right]= \frac{1}{16\pi^2\ell^4_A}\bigg( -\frac{2\ell_A^4}{t^2} - \frac{8s^2 \ell_A^2}{t}+ 5s^4 - \frac{5}{2}s^2 - \frac{13}{240}\bigg)~,$$ for a massless fermion with spin $s$. We inserted a sign for the fermion by hand in order to take statistics into account.

The $t=0$ poles in the massless heat kernels are the same for bosons and fermions (up to the sign that was inserted for fermions) as we expected since that is the case for each of the underlying massive representations. On the other hand, some of the terms that are finite as $t\to 0$ differ, also as expected. This feature is the origin of the apparent lack of pattern in the heat kernel coefficients that is evident when we consider the finite parts of $K_{\textrm{massless}}$ for the first few spins in table \[table:Kmassless\].

     **Spin**      $ 16\pi^2 \ell^4_A K^{\rm finite}_{\textrm{massless}}$          $a$
  --------------- -------------------------------------------------------- --------------------
        $0$                           $-\frac{2}{15}$                        $\frac{1}{180}$
   $\frac{1}{2}$                      $-\frac{11}{30}$                       $\frac{11}{720}$
        $1$                           $-\frac{62}{15}$                       $\frac{31}{180}$
   $\frac{3}{2}$                      $\frac{589}{30}$                      $-\frac{589}{720}$
        $2$                          $-\frac{1142}{15}$                     $\frac{571}{180}$

  : The values of $K_{\textrm{massless}}$ computed in AdS$_4$ and the corresponding $a$ anomalies. All entries including the scalar $s=0$ refers to two degrees of freedom.[]{data-label="table:Kmassless"}

Our results for the finite parts of the heat kernel $K(t)$ in AdS$_4$ are identical to the $a_4$ coefficients introduced in (\[eq:Kdef\]) up to 