Geodesic
Asymptotics for the sum of powers of distances between power residues modulo a~prime
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 83-95

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For fixed $q\in(0,4)$, prime $p\to\infty$, and $d\le\exp(c\sqrt{\ln p})$, where $c>0$ is a constant, we obtain the asymptotics for the sum of $q$th powers of distances between neighboring residues of degree $d$ modulo $p$. 
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     author = {M. Z. Garaev and S. V. Konyagin and Yu. V. Malykhin},
     title = {Asymptotics for the sum of powers of distances between power residues modulo a~prime},
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     year = {2012},
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M. Z. Garaev; S. V. Konyagin; Yu. V. Malykhin. Asymptotics for the sum of powers of distances between power residues modulo a~prime. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 83-95. http://geodesic.mathdoc.fr/item/TM_2012_276_a6/