So I'm trying to understand what appears to me to be a paradox in the Ashcroft & Mermin Solid State Physics book. In Eqn 5.8 it states $\exp(i\vec{k}\cdot\vec{R})=1$ where $\vec{R}$ is a direct lattice vector, $$\vec{R}=n_1\vec{a}_1+n_2\vec{a}_2+n_3\vec{a}_3 , $$ with $\vec{a}_i$ being a Bravais lattice vector and $n_i\in Z$ ($Z$ is the integers). Given how the reciprocal lattice is defined as $\vec{b}_i=2\pi\vec{a}_j\times\vec{a}_k$ (Eqn. 5.3) we can write any $$\vec{k}=k_1\vec{b}_1+k_2\vec{b}_2+k_3\vec{b}_3$$ where $k_i\in Z$. From this definition we can see $$\vec{k}\cdot\vec{R}=2\pi(k_1n_1+k_2n_2+k_3n_3)$$ and $$\exp(i\vec{k}\cdot\vec{R})=\left(\exp(i2\pi)\right)^{k_1n_1+k_2n_2+k_3n_3}=1^{k_1n_1+k_2n_2+k_3n_3}=1$$
Then later in Chapter 8, when he uses the Born-von Karman boundary conditions to define $\vec{k}=x_1\vec{b}_1+x_2\vec{b}_2+x_3\vec{b}_3$ (Eqn. 8.20); where $x_i=m_i/N_i;\ m_i\in Z$ (Eqn 8.26) and $N_i$ is the number of unit cells in that dimension. Clearly $x_i$ does not have to be $\in Z$, and now $\exp(i\vec{k}\cdot\vec{R})\neq 1$ for any arbitary $\vec{k}$.
The whole point I'm trying to reconcile is when working with Wannier Functions, how to treat terms such as $$\sum_k e^{i\vec{k}\cdot\vec{R}} \ \ {\rm and} \ \ \sum_R e^{i\vec{k}\cdot\vec{R}}$$ which I think can be evaluated to delta functions. When $\vec{k}$ is defined as rational multiples of the reciprocal Bravais lattice vectors it becomes obvious it's like summing the nth roots of unity, which is clear to me how they evaluate to Kronecker deltas. If $\exp(i\vec{k}\cdot\vec{R})=1$ always however, the sums would always just sum to the number of unit cells.
