Elliott nd problem erdos concerning power residue sums

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Elliott, P. D. T. A., A problem of Erdös concerning power residue sums. Acta Arith. 13 (1967), 131--149.

For a positive integer k let n_k(p) be the least positive kth power non-residue modulo p when p = 1 (mod k), and zero otherwise. Erdos had proved that the sum of n_2(p) over p < x is asymptotic to c x / log x and conjectured the same shape of result for every k; Theorem 1 of this note proves it, showing that for each k > 0 and each constant a < 4 e^(1 - 1/k) the sum of n_k(p)^a over p < x is asymptotic to C_{k,a} x / log x, and that when k is an odd prime the constant is the explicit series C_{k,a} = sum over r >= 1 of k^(-r) q_r^a, where q_r is the rth rational prime. The proof has three parts and rests on algebraic-number-theory lemmas about linear disjointness of field extensions (Lemma 1: two extensions of a field G, one finite and normal, are linearly disjoint over G exactly when their common subfield is G) and on when a radical lies in a cyclotomic field (Lemma 2, via Galois theory), combined with prime-ideal-theorem densities for the splitting conditions under which given small primes are all kth power residues modulo p; the larger values of n_k(p) are handled with a generalization of Selberg's sieve, Linnik's large sieve and the bound n_k(p) < c(eps) p^(zeta_k + eps), zeta_k = (1/4) e^(1/k - 1), which Lemma 16 obtains by Vinogradov's method with Burgess's character-sum estimate. Elliott notes the result had been stated without proof by Barban. The paper is the reference for the asymptotic behavior of least power non-residues averaged over primes in Erdos problem 980.

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Bears on. #980

Results to transcribe.

• Theorem 1 (p. 131): For each integer k > 0 and a < 4 e^(1-1/k), the sum of n_k(p)^a over primes p < x is asymptotic to C_{k,a} x / log x for a constant C_{k,a}, with C_{k,a} = sum_r k^(-r) q_r^a, q_r the rth prime, when k is an odd prime.
• Lemma 1 (p. 132): Two extensions E, F of a field G, one of them finite and normal, are linearly disjoint over G if and only if the intersection of E and F is G.
• Lemma 2 (p. 133): For positive integers l, k and a rational t that is not a perfect power of a rational number and whose negative -t is not a rational square, an l-th root of t can lie in the cyclotomic field Q(k-th roots of unity) only if l = 1 or 2; when l = 2, t must in addition be composed of squares and of primes dividing k.