= Gaussian sum truncation bound
{c}
{title2=$R_N(b)=2\sum_{n=N+1}^{\infty}e^{-\pi bn^2}$}
For $b>0$ and a nonnegative <integer> $N$, define the symmetric <partial sum> and its positive remainder by
$$
S_N(b)=1+2\sum_{n=1}^{N}e^{-\pi bn^2},\qquad R_N(b)=S(b)-S_N(b).
$$
Successive omitted terms have ratio $e^{-\pi b(2n+1)}\leq e^{-\pi b(2N+3)}$ for $n\geq N+1$. Bounding them by a <geometric series> gives
$$
0<R_N(b)\leq B_N(b):=\frac{2e^{-\pi b(N+1)^2}}{1-e^{-\pi b(2N+3)}}.
$$
Consequently the physical approximation to $S(a)$ has error at most $B_N(a)$, while the dual approximation $a^{-1/2}S_N(1/a)$ has error at most $a^{-1/2}B_N(1/a)$. The prefactor must be included when setting an absolute error tolerance.
For a truncation error below $10^{-12}$, at $a=0.04$ the physical side first passes this bound with $N=15$: $31$ lattice points give $B_{15}(0.04)<2.172\times10^{-14}$, whereas $N=14$ gives $B_{14}(0.04)>10^{-12}$. The dual side needs only $k=0$. At $a=25$ the physical side needs only $n=0$, whereas the dual side first passes the bound with $N=14$: $29$ lattice points give $25^{-1/2}B_{14}(0.04)<2.146\times10^{-13}$. These counts measure symmetric lattice terms; using the <even function> symmetry reduces the distinct <exponential function> evaluations to $N+1$. Truncation bounds do not include rounding error in a numerical implementation.
\Image[/poisson-gaussian-tail-error-work.png]
{title=Positive Gaussian tails and rigorous bounds versus retained lattice points, showing the efficient side at reciprocal scales}
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