Kloosterman paths of prime powers moduli

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What is drawn

For any power $q$ of a prime number (called the modulus), let us note $$ 1\leq j_1<\dotsc<j_{\varphi(q)}<q $$ the integers prime to $q$. For all $a$ and $b$ prime to $a$, and $j_\ell$ one of the integers prime to $q$ defined above, we define the Kloosterman partial sum $K_j(a,b)$ by $$ K_{j_{\ell}}(a,b;q)=\frac{1}{\sqrt{q}}\sum_{n=1}^{\ell}\exp\left(2i\pi\frac{aj_n+b\overline{j_n}}{q}\right) $$ where $\overline{j_n}$ is an inverse of $j_n$ modulo $q$. Kloosterman’s partial sums are complex numbers. We can therefore associate them with points that we plot, one after the other, starting with $K_{j_1}(a,b)$ then $K_{j_2}(a,b)$ up to $K_{j_{\varphi(q)}}(a,b;q)$.

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References

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