Q:

Additive character sum of primitive elements over finite fields

Let $\mathbb{F}_q$ be a finite field with $q$ elements and multiplicative generator $\alpha$, let $\chi$ be a non-trivial additive character of $\mathbb{F}_q$, and let $N$ be a divisor of $q-1$. Consider the sum
$$
\sum_{(k, q-1) = 1} \chi \left( \alpha^{kN} \right). 
$$
Essentially the sum runs over all the elements of order $(q-1)/N$ (with some repetitions). 
Question: Is there anything special known about sums of this type?  I'm aware of the identity for multiplicative character sums whereby the sum of all primitive roots of unity yields the Moebius function.  Thanks in advance. 

A:

Undoubtedly you know how to write these sums in terms of monomial sums $\sum_{x\in\Bbb{F}_q^*}\chi(x^{N'})$ where $N\mid N'\mid q-1$. Just a Möbius inversion business.  Such monomial sums can be rewritten using Gauss sums 
$$S(\chi,\psi)=\sum_{x\in\Bbb{F}_q^*}\chi(x)\psi(x),$$
where $\psi$ is a multiplicative character such that $\psi^N=1$ is the trivial character.  This, for example, leads to an upper bound for the monomial sums, as it is known that in the non-trivial cases $|S(\chi,\psi)|=\sqrt q$.
I don't know about your general case.  Some information that may be of use to you has been uncovered by people working in the grey area between number theory/finite fields and coding theory.  In particular when $q$ is even the monomial sums $S(a):=\sum_{x\in\Bbb{F}_q^*}\chi(ax^{N'})$, $a$ a constant from $\Bbb{F}_q$, give the weight distribution of a certain cyclic code.  The papers that I am aware of are

L.D. Baumert and J. Mykkeltveit, Weight distributions of some irreducible cyclic codes, JPL Technical Report 32-1526,
Gerard van der Geer, Marcel van der Vlugt: Weight distributions for a certain class of codes and maximal curves.  Discrete Mathematics 106-107: 209-218 (1992) [I know that vdG & vdV have done some nice work in this area, but I'm not 100% that this paper is the one],
Two recursive algorithms for computing the weight distribution of certain irreducible cyclic codes,
Moisio, M.J. ; Väänänen, K.O.
Information Theory, IEEE Transactions on
Volume: 45 , Issue: 4 

The key idea is that in the binary case when values of $\chi$ are rational integers, the Galois group of the finite field gives relations between the sums $S(\chi,\psi)$ aand $S(\chi,\psi^{p^\ell})$ (they are equal, if $\chi$ is the trace character) for various choices of $\ell$. But the Galois theory of $\Bbb{Q}(\zeta_N)$ relates the sums $S(\chi,\psi)$ and $S(\chi,\psi^a)$, $\gcd(a,N)=1.$ The above authors are exploiting this in the cases when $\psi^{p^\ell}$
ranges either over all the generators of $\langle\psi\rangle$ or over a subgroup of index two.  IOW, when $p$ generates either the group of units in $\Bbb{Z}_N$ (primitive case) or an index two subgroup (semiprimitive case).  In those cases the Gauss sums are rational integers (resp.  algebraic integers in a known quadratic imaginary number field), and their exact values, or at least distribution of values, can be determined exactly. 
Admittedly those are very special cases, but may be you can use their techniques on a problem that interests you.