Can anyone please tell me (give me a reference, preferably) if there is any explicit determination of sums of the form $g(n,\chi):=\sum_{r=1}^{q}\chi(r)e(\frac{rn+r^2}{q})$ where $\chi$ is a Dirichlet character modulo $q$, not necessarily primitive? Here $e(x)$ denotes $e^{2\pi ix}$.

Thanks in advance.


1 Answer 1


That is not a quadratic Gauss sum and it most likely does not have an explicit formula. Indeed in the simple case $q=p$ a prime, by the Lefschetz trace formula, it is the trace of Frobenius of a rank $2$ Galois representation, which means it is the sum of two complex numbers of absolute value $\sqrt{p}$. But there should be no easy way to compute the numbers.

To confirm this, we can look at how the sum varies as a function of $n$ as a sheaf on $\mathbb A^1$. It is rank $2$ and lisse, hence its geometric monodromy group contains $SL_2$, which means it doesn't reduce to some simpler sheaf.

However two observations might be helpful: If $q$ is a product of relatively prime numbers, you can factor your sum as a product. If $q$ is a prime power, you can split your sum into a sum over congruence classes mod a smaller power where $rn+r^2$ simplifies to a linear function and so simplify the expression.

  • $\begingroup$ Thank you @Will Sawin; I call it a quadratic Gauss sum because this is somewhat like the quadratic Gauss sum (except for the coefficient $\chi$). $\endgroup$
    – usr203050
    Aug 28, 2014 at 6:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.