On a Problem of Mordell with Primitive Roots

Abstract

We consider the sums of the form $$S=\sum _{x=1}^N\exp ((ax+b_1{g}_1^x+...+b_r{g}_r^x)/p)$$, where $p$ is prime and $g_1,...,g_r$ are primitive roots $(\mathrm{mod}p)$. An almost forty years old problem of L. J. Mordell asks to find a nontrivial estimate of $S$ when at least two of the coefficients $b_1,...,b_r$ are not divizible by $p$. Here we obtain a nontrivial bound of the average of these sums when $g_1$ runs over all primitive roots $(\mathrm{mod}p)$.

7 pages, added detailed proof of Lemma 2

Cited by in corpus (8)

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• Primitive Roots In Short Intervals
• Simple Proof of the Primitive Root Conjecture
• Asymptotic For Primitive Roots Producing Polynomials
• Topics In Primitive Roots
• Generalized Artin Primitive Root Conjecture
• Simultaneous Elements Of Prescribed Multiplicative Orders
• Primitive Root Conjecture in Arithmetic Progressions