History of Kloosterman sums modulo a prime

     more than the second moment $p^2-p-1$ in CITE{formula-moments}. No simple closed     form for $K(a;p)$ at a prime is known.'-  comment-twisted: The issue this table answers CITE{issue13} asks for Kloosterman-    sums of Dirichlet characters, $K(a,b,\chi;p)=\sum_{x=1}^{p-1}\chi(x)\,e^{2\pi-    i(ax+b\bar x)/p}$, which for the trivial character $\chi$ is $K(a,b;p)$ and for-    a nontrivial $\chi$ is in general not real; for the quadratic character it is-    a Salié sum. The entries here are the sums for the trivial character. The LMFDB's-    page for a character CITE{LMFDB} computes its twisted sums to ten decimals, and-    the values it gives for the trivial character agree with the entries.+  comment-twisted: The twisted Kloosterman sum $K(a,b,\chi;p)=\sum_{x=1}^{p-1}\chi(x)\,e^{2\pi+    i(ax+b\bar x)/p}$ is $K(a,b;p)$ for the trivial character $\chi$ and in general+    not real for a nontrivial one; for the quadratic character it is a Salié sum.+    The entries here are the sums for the trivial character. The LMFDB's page for+    a character CITE{LMFDB} computes its twisted sums to ten decimals, and the values+    it gives for the trivial character agree with the entries. Formulas:   formula-two-parameters: $K(a,b;m)=K(b,a;m)$, and $K(ac,b;m)=K(a,bc;m)$ for $\gcd(c,m)=1$;

     bib: H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical       Society Colloquium Publications 53, 2004.-  issue13:-    bib: 'numberdb-data issue #13, "Kloosterman sums of Dirichlet characters", https://github.com/numberdb/numberdb-data/issues/13' Tags: - algebraic