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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

Keywords:

character sums
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