Pollack 2017 bounds first several prime character nonresidues

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theorem_1_1: Pollack's theorem that for each eps > 0 there are m_0(eps) and kappa(eps) > 0 such that every nontrivial character chi mod m, m > m_0, has more than m^kappa prime chi-nonresidues not exceeding m^(1/(4 sqrt e) + eps).

theorem_1_2: Pollack's theorem that for eps > 0 and k_0 >= 2 there are m_0(eps, k_0) and kappa(eps, k_0) > 0 such that every nontrivial character chi mod m, m > m_0, of order k >= k_0 has more than m^kappa prime chi-nonresidues not exceeding m^(1/(4 u_{k_0}) + eps), where rho(u_k) = 1/k.

theorem_1_3: Pollack's theorem that for eps > 0 and A > 0 there is m_0(eps, A) such that every quadratic character chi modulo m > m_0 has at least (log m)^A primes l <= m^(1/4 + eps) with chi(l) = 1.

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Pollack, Paul, Bounds for the first several prime character nonresidues. Proc. Amer. Math. Soc. 145 (2017), no. 7, 2815--2826, doi:10.1090/proc/13432. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1508.05035), every other right reserved. The copy read for this card is arXiv:1508.05035v2, and the pages cited are its pages.

Pollack proves (Theorem 1.1, p. 1) that for each $\varepsilon>0$ there are $m_0(\varepsilon)$ and $\kappa(\varepsilon)>0$ such that for every $m>m_0$ and every nontrivial Dirichlet character $\chi$ mod $m$, more than $m^\kappa$ prime $\chi$-nonresidues (primes $\ell$ with $\chi(\ell)\notin\{0,1\}$) do not exceed $m^{\frac1{4\sqrt e}+\varepsilon}$: the Burgess--Norton bound for the least nonresidue holds for a power-sized set of prime nonresidues. Theorem 1.2 (p. 2) generalizes this to characters of order $k\ge k_0$, with exponent $\frac1{4u_{k_0}}+\varepsilon$, where $\rho(u_{k_0})=1/k_0$ for Dickman's function $\rho$; Theorem 1.1 is the case $k_0=2$ (p. 3), as $u_2=e^{1/2}$ (p. 2). Theorem 1.3 (p. 3) is a partial analogue for quadratic characters: at least $(\log m)^A$ primes $\ell\le m^{\frac14+\varepsilon}$ with $\chi(\ell)=1$, for $m>m_0(\varepsilon,A)$, a count that falls short of a power of $m$; its proof ends in a contradiction with Siegel's theorem (p. 10). The proof of Theorems 1.1 and 1.2 combines a sieve fundamental lemma, Norton's version of the Burgess character-sum bounds, and a theorem of Tenenbaum on smooth numbers subject to a coprimality condition. The paper reads Theorems 1.1 and 1.3 as statements about quadratic fields: many inert (resp. split) primes below a power of the discriminant (p. 3). A remark on p. 8 states, with the proof only outlined, a version for primes outside any proper subgroup of index at least $k_0$ of $(\mathbf Z/m\mathbf Z)^\times$ (Theorem 2.7). Theorem 1.3 is the input to a negative answer to problem 1141, which asks whether infinitely many $n$ have $n-k^2$ prime for every $k$ coprime to $n$ with $k^2<n$: in the preprint arXiv:2604.06609, Alexeev, Putterman, Sawhney, Sellke and Valiant deduce from it that only finitely many $n$ do.

Source: https://arxiv.org/abs/1508.05035.

Read status: claims checked for Theorems 1.1, 1.2 and 1.3, Theorems 2.3, 2.4 and 2.7, Proposition 3.1 and the remarks of pp. 2, 3, 8 and 10, read clause by clause on the page images; the deduction of Theorem 1.2 (§ 2.3) and the proof of Theorem 1.3 (§ 3) followed, the proof of Theorem 2.4 (§ 2.2) read for structure. Nothing here is independently reviewed. Result pages: theorem_1_1, theorem_1_2 and theorem_1_3.

Bears on. #1141: the paper does not mention the problem; Theorem 1.3 (p. 3) is the input from which the negative answer recorded on the problem's claim page is deduced.

Results.

• Theorem 1.1 (p. 1): for $m>m_0(\varepsilon)$ every nontrivial $\chi$ mod $m$ has more than $m^{\kappa(\varepsilon)}$ prime $\chi$-nonresidues not exceeding $m^{\frac1{4\sqrt e}+\varepsilon}$.
• Theorem 1.2 (p. 2): the same with exponent $\frac1{4u_{k_0}}+\varepsilon$ for characters of order $k\ge k_0\ge2$, with $m_0$ and $\kappa$ depending on $\varepsilon$ and $k_0$; the page also records Theorem 2.7 (p. 8).
• Theorem 1.3 (p. 3): for $m>m_0(\varepsilon,A)$ every quadratic $\chi$ mod $m$ has at least $(\log m)^A$ primes $\ell\le m^{\frac14+\varepsilon}$ with $\chi(\ell)=1$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.