Distribution of power residues and non-residues

2020 Mathematics Subject Classification: Primary: 11A15 Secondary: 11N69 [MSN][ZBL]

The distribution among the numbers $1,\ldots,m-1$ of those values of $x$ for which the congruence

$$y^n\equiv x\pmod{m},$$

$n>1$, is solvable (or unsolvable) in integers. Questions on the distribution of power residues and non-residues have been studied most fully in the case modulo a prime number $p$. Let $q=\mathrm{gcd}(n,p-1)$. Then the congruence $y^n\equiv x\pmod{p}$ is solvable for $(p-1)/q$ values of $x$ in the set $1,\ldots,p-1$ and unsolvable for the remaining $(q-1)(p-1)/q$ values (see Two-term congruence). However, comparatively little is known about how these values are distributed among the numbers $1,\ldots,p-1$.

The first results about the distribution of power residues were obtained by C.F. Gauss (see [Ga]) in 1796. From that time until the work of I.M. Vinogradov only isolated special results were obtained on questions concerning the distribution of power residues and non-residues. In 1915, Vinogradov (see [Vi]) proved a series of general results about the distribution of power residues and non-residues, and about primitive roots (cf. Primitive root) modulo $p$ among the numbers $1,\ldots,p$. In particular, he obtained the bound

$$N_{\mathrm{min}} < p^{\frac{1}{2\sqrt{e}}}(\ln p)^2$$

for the least quadratic non-residue $N_{\mathrm{min}}$, and the bound

$$N^*_{\mathrm{min}}\leq 2^{2k}\sqrt{p}\ln p,$$

where $k$ is the number of distinct prime divisors of $p-1$, for the least primitive root $N_{\mathrm{min}}^*$ modulo $p$.

In addition, he made a number of conjectures on the distribution of quadratic residues and non-residues (see Vinogradov hypotheses) which stimulated a series of investigations in this area. Thus, Yu.V. Linnik [Li] proved that for sufficiently large $N$, the number of prime numbers $p$ in the interval $[N^{\epsilon},N]$ for which $N_{\mathrm{min}}>p^{\epsilon}$ does not exceed a certain constant $C(\epsilon)$, depending only on $\epsilon>0$. Thus, the prime numbers $p$ for which $N_{\mathrm{min}}>p^{\epsilon}$, if they exist at all, are met only very rarely. Another significant step in the work on Vinogradov's conjectures was the theorem of D.A. Burgess [Bu]: For any fixed sufficiently small $\delta>0$, the maximal distance $d(p)$ between neighbouring quadratic non-residues satisfies the inequality

$$d(p)\leq A(\delta)p^{\frac{1}{4}+\delta}.$$

In particular, one has

$$N_{\mathrm{min}}\leq B(\delta)p^{\frac{1}{4\sqrt{e}}+\delta}.$$

In these inequalities, the constants $A(\delta)$, $B(\delta)$ depend only on $\delta$ and not on $p$. The proof of Burgess' theorem is very complicated; it is based on the Hasse–Weil theorem on the number of solutions of the hyper-elliptic congruence

$$y^2\equiv f(x)\pmod{p},$$

the proof of which requires techniques of abstract algebraic geometry. For a simple account of Burgess' theorem see [St], [Ka].

References

[Bu]	D.A. Burgess, "The distribution of quadratic residues and non-residues" Mathematika, 4 : 8 (1957) pp. 106–112 MR0093504 Zbl 0081.27101
[Ga]	C.F. Gauss, "Untersuchungen über höhere Arithmetik", A. Maser (1889) (Translated from Latin) MR0188045 Zbl 21.0166.04
[Ka]	A.A. Karatsuba, "Character sums and primitive roots in finite fields" Soviet Math.-Dokl., 9 : 3 (1968) pp. 755–757 Dokl. Akad. Nauk SSSR, 180 : 6 (1968) pp. 1287–1289 Zbl 0182.37501
[Li]	Yu.V. Linnik, Dokl. Akad. Nauk SSSR, 36 (1942) pp. 131
[St]	S.A. Stepanov, "Constructive methods in the theory of equations over finite fields" Proc. Steklov Inst. Math., 132 (1975) pp. 271–281 Trudy Mat. Inst. Steklov., 132 (1973) pp. 237–246 MR0337976
[Vi]	I.M. Vinogradov, "Selected works", Springer (1985) (Translated from Russian) MR0807530 Zbl 0577.01049