## On Dirichlet characters of polynomials.(English)Zbl 0118.04704

Let $$q$$ be a fixed positive integer and $$f(x)$$ a nonlinear product of rational linear polynomials which is not a perfect $$q$$-th power. Let $$A\ll B$$ mean $$A < k\vert B\vert$$ for some positive constant $$k$$.
Theorem 1. For $$\varepsilon > 0$$ if $$p\equiv 1\pmod q$$ is a sufficiently large prime and $$\chi$$ is a $$q$$-th order character $$\pmod p$$, then for all integers $$H, N$$ with $$p^{\frac14 + \varepsilon}\le H\le p^{\frac12}$$ we have $H - \left\vert \sum_{x=N+1}^{N+H} \chi(f(x))\right\vert \gg H^2p^{-\frac14},$ the constant in the notation depending on $$\varepsilon$$, $$q$$ and $$f$$.
Theorem 2: The maximum number $$H$$ of consecutive integers $$x$$ having the same value for $$\chi(f(x))$$ is $$\ll p^{\frac14} \log p$$. (The exponent is erroneously printed as $$\frac12$$ in the paper.)
Theorem 3: If $$p$$ is a sufficiently large prime and if $$H\gg p^{11/24} (\log p)^{3/2}$$ then the sequence $$N+1, N+2,\ldots, N+H$$ includes a pair of consecutive quadratic residues and a pair of consecutive nonresidues $$\pmod p$$.
The author had previously proved estimates in the case $$f(x) = x$$ [Proc. Lond. Math. Soc. (3) 12, 179–192 (1962; Zbl 0106.04003); ibid. 12. 193–206 (1962; Zbl 0106.04004); ibid. 13, 524–536 (1963; Zbl 0118.04703), reviewed above].
The weaker estimate $\sum_{x=N+1}^{N+H} \chi(f(x)) \ll p^{\frac12} \log p,$ for an arbitrary polynomial $$f(x)$$ not a perfect $$q$$-th power is a known consequence of the Riemann Hypothesis for an algebraic function field over a finite field proved by A. Weil [Sur les courbes algébriques et les variétés qui s’en déduisent. Paris: Hermann (1948; Zbl 0036.16001); deuxième partie, §1 V].

### MSC:

 11L40 Estimates on character sums

character sums

### Citations:

Zbl 0106.04003; Zbl 0106.04004; Zbl 0118.04703; Zbl 0036.16001
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