Explicit bounds for small prime nonresidues

TL;DR

This paper provides explicit upper bounds for the smallest prime nonresidues of Dirichlet characters modulo a prime, improving understanding of their distribution with precise bounds depending on the prime and the number of nonresidues.

Contribution

It establishes explicit bounds for the smallest prime nonresidues of Dirichlet characters, with constants depending on given parameters, advancing previous theoretical results.

Findings

Bounds depend on prime p and number of nonresidues n

Explicit constants are provided for practical bounds

Results hold for sufficiently large p and fixed n

Abstract

Let $\chi$ be a Dirichlet character modulo a prime~$p$. We give explicit upper bounds on $q_1<q_2<\dots<q_n$, the $n$ smallest prime nonresidues of $\chi$. More precisely, given $n_0$ and $p_0$ there exists an absolute constant $C=C(n_0,p_0)>0$ such that $q_n\leq Cp^{\frac{1}{4}}(\log p)^{\frac{n+1}{2}}$ whenever $n\leq n_0$ and $p\geq p_0$.

Tables

Table 1. Table 1: Constants for various p 0 subscript 𝑝 0 p_{0} and n 0 subscript 𝑛 0 n_{0}

	${10}^7$	${10}^8$	${10}^9$	${10}^{10}$	${10}^{15}$	${10}^{20}$	${10}^{25}$	${10}^{30}$	${10}^{35}$
$1$	$1.530$	$1.433$	$1.378$	$1.344$	$1.282$	$1.264$	$1.254$	$1.248$	$1.244$
$2$	$2.408$	$2.070$	$1.909$	$1.821$	$1.692$	$1.670$	$1.661$	$1.655$	$1.651$
$3$	$-$	$7.170$	$3.087$	$2.468$	$1.926$	$1.876$	$1.864$	$1.858$	$1.854$
$4$	$-$	$-$	$-$	$-$	$2.230$	$2.014$	$1.987$	$1.980$	$1.976$
$5$	$-$	$-$	$-$	$-$	$6.469$	$2.198$	$2.079$	$2.063$	$2.058$
$6$	$-$	$-$	$-$	$-$	$-$	$3.386$	$2.205$	$2.128$	$2.116$
$7$	$-$	$-$	$-$	$-$	$-$	$-$	$2.916$	$2.222$	$2.165$
$8$	$-$	$-$	$-$	$-$	$-$	$-$	$-$	$2.745$	$2.240$

Equations

$$H\le p^{\frac{1}{4}}(\log p{)}^{\frac{n+1}{2}}g(n,p)$$

$$g(n,p)=\frac{\pi }{3}\sqrt{2e}\left(\frac{n}{n+1}\right){\left(\frac{1+\frac{\sqrt{2}}{2B\log p-3}}{1-\frac{{\pi }^2}{9}\frac{\log X^{\ast }+9}{3X^{\ast }}}\right)}^{\frac{1}{2}}$$

$$X^{\ast }=\frac{\pi }{3}(2e{)}^{-\frac{1}{2}}{\left(\frac{n+1}{n}\right)}^{n-1}\frac{p^{\frac{1}{4}}}{(\log p{)}^{\frac{n-1}{2}}}\cdot \left(1-\frac{n+1}{n}{e}^{-1/n}(\log p{)}^{-1}\right),B=\frac{n}{2(n+1)},$$

$$q_n\le C{p}^{\frac{1}{4}}(\log p{)}^{\frac{n+1}{2}}$$

$$S(\chi ,h,r)=\sum _{x=0}^{p-1}{\left|\sum _{m=0}^{h-1}\chi (x+m)\right|}^{2r}\le \sqrt{2}{\left(\frac{2r}{e}\right)}^{r}p{h}^r+(2r-1)p^{\frac{1}{2}}{h}^{2r}.$$

$$S(\chi ,h,r)\le \frac{(2r)!}{2^{r}r!}p{h}^r+(2r-1)p^{\frac{1}{2}}{h}^{2r}.$$

$$\frac{(2r)!}{2^{r}r!}\le \frac{\sqrt{2\pi }(2r{)}^{2r+\frac{1}{2}}{e}^{-2r}{e}^{\frac{1}{24r}}}{2^r\sqrt{2\pi }{r}^{r+\frac{1}{2}}{e}^{-r}{e}^{\frac{1}{12r+1}}}\le \sqrt{2}{\left(\frac{2r}{e}\right)}^r.$$

$$\mathcal{I}(a,b):=\left(\frac{bp}{a},\frac{bp+H}{a}\right],$$

$$\mathcal{J}(a,b):=\left[\frac{bp-H}{a},\frac{bp}{a}\right),$$

$$\left|\sum _{m=0}^{h-1}\chi (z+m)\right|\ge h-2j.$$

$$\left|\sum _{m=0}^{h-1}\chi (z+m)\right|\ge \left|\sum _{m=0}^{h-1}\bar{\chi }(a)\right|-\sum _{m=0}^{h-1}\left|\chi (z+m)-\bar{\chi }(a)\right|\ge h-2j.$$

$$2x\sum _{a\le x}\frac{\varphi (a)}{a}-\sum _{a\le x}\varphi (a)\ge \frac{9}{{\pi }^2}{x}^{2}f(x)$$

$$f(x)=1-\frac{{\pi }^2}{9}\frac{\log x+9}{3x}.$$

$$S(\chi ,h,r)\ge \frac{18}{{\pi }^2}h(h-2j{)}^{2r}\frac{\varphi (u_1)}{u_1^2}{X}^{2}f\left(\frac{X}{u_1}\right),$$

$$S(\chi ,h,r)$$

$$\ge \sum _{\begin{array}{c}0\le b<a\le X\\ (a,b)=1\\ u_1|a\end{array}}\sum _{z\in {\mathcal{I}}^{\ast }(a,b)\cup {\mathcal{J}}^{\ast }(a,b)}{\left|\sum _{m=0}^{h-1}\chi (z+m)\right|}^{2r}.$$

$$S(\chi ,h,r)$$

$$=2(h-2j{)}^{2r}\sum _{\begin{array}{c}1\le a\le X\\ u_1|a\end{array}}\left(\frac{H}{a}-h\right)\varphi (a)$$

$$=2(h-2j{)}^{2r}\sum _{t\le \frac{X}{u_1}}\left(\frac{H}{t{u}_1}-h\right)\varphi (t{u}_1)$$

$$\ge 2(h-2j{)}^{2r}\sum _{t\le \frac{X}{u_1}}\left(\frac{H}{t{u}_1}-h\right)\varphi (t)\varphi (u_1)$$

$$=2(h-2j{)}^{2r}\varphi (u_1)\left(\frac{H}{u_1}\sum _{t\le \frac{X}{u_1}}\frac{\varphi (t)}{t}-h\sum _{t\le \frac{X}{u_1}}\varphi (t)\right).$$

$$2(h-2j{)}^{2r}h\varphi (u_1)\left(\frac{2X}{u_1}\sum _{0\le t\le \frac{X}{u_1}}\frac{\varphi (t)}{t}-\sum _{0\le t\le \frac{X}{u_1}}\varphi (t)\right).$$

$$S(\chi ,h,r)\ge \frac{18}{{\pi }^2}h(h-2j{)}^{2r}\frac{\varphi (u_1)}{u_1^2}{X}^{2}f\left(\frac{X}{u_1}\right).$$

$${\left(\frac{h}{h-2j}\right)}^{2r}\le \exp \left(\frac{16rj}{3h}\right).$$

$$\log \left(\frac{h}{h-2j}\right)\le \frac{h}{h-2j}-1=\frac{2j}{h-2j}\le \frac{8j}{3h}$$

$$2r\log \left(\frac{h}{h-2j}\right)\le \frac{16rj}{3h},$$

$${\left(\frac{h}{h-2j}\right)}^{2r}\le \exp \left(\frac{16rj}{3h}\right).$$

$${\left(\frac{2B}{Ae}\right)}^{B\log p}=p^{-\frac{1}{2}};$$

$$\frac{X}{u_1}$$

$$\ge \frac{\frac{\pi }{3}\sqrt{2e}\left(\frac{n}{n+1}\right)p^{\frac{1}{4}}(\log p{)}^{\frac{n+1}{2}}(1-h^{-1})}{2(h-1{)}^{k+1}}$$

$$\ge \frac{\frac{\pi }{3}\sqrt{2e}\left(\frac{n}{n+1}\right)p^{\frac{1}{4}}(\log p{)}^{\frac{n+1}{2}}\left(1-\frac{n+1}{n}{e}^{-1/n}(\log p{)}^{-1}\right)}{2{\left(\frac{n}{n+1}{e}^{1/n}\log p\right)}^n}$$

$$=\frac{\pi }{3}\frac{1}{\sqrt{2e}}\frac{{\left(\frac{n+1}{n}\right)}^{n-1}{p}^{\frac{1}{4}}}{(\log p{)}^{\frac{n-1}{2}}}\left(1-\frac{n+1}{n}{e}^{-1/n}(\log p{)}^{-1}\right).$$

$$\frac{18}{{\pi }^2}h(h-2j{)}^{2r}\frac{\varphi (u_1)}{u_1^2}{\left(\frac{H}{2h}\right)}^{2}f\left(\frac{X}{u_1}\right)$$

$$\frac{18}{{\pi }^2}\frac{\varphi (u_1)}{u_1^2}{H}^{2}f\left(\frac{X}{u_1}\right)$$

$$\le 4(h-1)(2r)e^{\frac{16rj}{3h}}{p}^{\frac{1}{2}}\left(1+\frac{\sqrt{2}}{2r-1}{\left(\frac{2r}{he}\right)}^r{p}^{\frac{1}{2}}\right).$$

Full text

1 Introduction

Let $\chi$ be a nonprincipal Dirichlet character modulo a prime $p$. If $\chi(n)\notin\{0,1\}$, then we refer to $n$ as a nonresidue of $\chi$. Let $q_{1}<q_{2}<\dots<q_{\ell}$ denote the $\ell$ smallest prime nonresidues of $\chi$. Giving an upper bound on $q_{1}$ is an important classical problem that has received much attention. Indeed, in the case of the Legendre symbol, $q_{1}$ is the least quadratic nonresidue mod $p$. In 1963, Burgess showed that for each $\varepsilon>0$, one has $q_{1}\ll p^{\frac{1}{4\sqrt{e}}+\varepsilon}$ (see [3, 4]), and this result has stood as the state of the art since this time, save a recent improvement to the “$\varepsilon$” in the quadratic case (see [1]). In 2015, Pollack proved the following result (see [9]): For each $\varepsilon>0$, there are numbers $m_{0}=m_{0}(\varepsilon)$ and $\kappa=\kappa(\varepsilon)>0$ such that for all $m>m_{0}$ and each nonprincipal character $\chi$ modulo $m$, there are more than $m^{\kappa}$ prime nonresidues of $\chi$ not exceeding $m^{\frac{1}{4\sqrt{e}}+\varepsilon}$. In particular, for all $\varepsilon>0$ and all $k$, one has $q_{k}\ll p^{\frac{1}{4\sqrt{e}}+\varepsilon}$, although this hides the dependence on $k$ and $\varepsilon$. As we alluded to a moment ago, Banks and Guo have recently shown that $q_{k}\ll p^{\frac{1}{4\sqrt{e}}}\exp(\sqrt{e^{-1}\log p\log\log p})$ in the case where $\chi$ is the Legendre symbol (see [1]), provided $k\leq p^{\frac{1}{8\sqrt{e}}}\exp(\frac{1}{2}\sqrt{e^{-1}\log p\log\log p}-\frac{1}{2}\log\log p)$.

Often in applications (see, for example, [7, 5, 10, 2]) one requires estimates that are completely explicit, and one is willing to accept a weaker asymptotic in order to obtain constants of a reasonable magnitude. Our goal here is to give an explicit upper bound on $q_{k}$, the $k$th smallest prime nonresidue. Naturally, our upper bounds are asymptotically weaker than those given in [9] and [1]. The following is our main result from which one can easily derive bounds of the desired form.

Theorem 1**.**

Let $\chi$ be a nonprincipal Dirichlet character modulo $p$. Let $u\in\mathbb{Z}^{+}$ be squarefree and assume all its prime factors are less than $p$. Set $n=\omega(u)+1$ where $\omega(u)$ is the number of distinct prime factors of $u$. Suppose that $\chi(m)=1$ whenever $0<m\leq H<p$ and $(m,u)=1$. Then

[TABLE]

where

[TABLE]

using

[TABLE]

provided $X^{*}>3.8$ and $\log p>\max\{\exp(8/3),\,8(n-1)\}$.

Corollary 2**.**

Fix two integer constants $p_{0}$ and $n_{0}$ such that $X^{*}(p_{0},n_{0})>3.8$ and $p_{0}>\max\{2\cdot 10^{6},\exp(8(n-1))\}$. Then there exists an explicit constant $C=g(n_{0},p_{0})$ such that

[TABLE]

for all $p\geq p_{0}$ and $n\leq n_{0}$.

To our knowledge, the previous corollary constitutes the first explicit upper bound on $q_{k}$ when $k\geq 3$. When $k=1$, there is the work of Norton (see [8]) that was later superceded by Treviño (see [13]) and when $k=2$ there is a paper by McGown (see [6]). The proof of our result involves a modification of McGown’s work (see [6]), which is based on the method of Burgess (see [3, 4]), and the adoption of Treviño’s results (see [12]).

2 Preparations

Lemma 3**.**

Let $\chi$ be a nonprincipal Dirichlet character to a prime modulus $p$. Let $h,r$ be positive integers satisfying $h<p$ and $r\leq 9h$. Then

[TABLE]

Proof.

From Theorem 1.1 of [13] we have

[TABLE]

Using the explicit version of Stirling’s formula given in [11], we have

[TABLE]

∎

Definition 1**.**

Let $0\leq b<a\leq X$ with $(a,b)=1$. For constants $H$, $h$, define the following intervals:

[TABLE]

Lemma 4**.**

Let $X$ be a real number such that $2XH<p$. Then the intervals $\mathcal{I}(a,b)$, $\mathcal{J}(a,b)$ where $0\leq b<a\leq X$ and $(a,b)=1$ are disjoint subintervals of $(0,p-H)$, except for $\mathcal{J}(1,0)=[-H,0)$.

Proof.

This is Lemma 2 of [6]. ∎

Lemma 5**.**

Let $h,u_{1},u_{2}\in\mathbb{Z}^{+}$ where $u_{1}=q_{1,1}q_{1,2}\ldots q_{1,k}$ with each prime $q_{1,i}<h$ and $u_{2}=q_{2,1}q_{2,2}\ldots q_{2,j}$ with each prime $h\leq q_{2,i}<p$. Suppose $\chi$ is Dirichlet character modulo $p$ such that $\chi(n)=1$ for all $n\in(0,H]$ with $(n,u_{1}u_{2})=1$. If $z\in\mathcal{I}^{*}(a,b)\cup\mathcal{J}^{*}(a,b)$ with $(a,b)=1$ and $u_{1}|a$, then

[TABLE]

Proof.

Suppose $z\in\mathcal{I}^{*}(a,b)$ with $(a,b)=1$ and $u_{1}|a$. Then for all $0\leq n\leq h-1$, $z+n\in\mathcal{I}(a,b)$. Therefore $a(z+n)-bp\in(0,H]$. Since $\chi(z+n)=\bar{\chi}(a)\chi(a(z+n)-bp)$ and $a(z+n)-bp\in(0,H]$ and $(u_{1},a(z+n)-bp)=1$, this will equal $\chi(z+n)=\bar{\chi}(a)$ if $(u_{2},a(z+n)-bp)=1$. This can only fail once for each divisor $q_{2,i}$ of $u_{2}$.

To see this, suppose $q_{2,i}\mid a(z+n_{1})-bp$ and $q_{2,i}\mid a(z+n_{2})-bp$ for two different values $n_{1},n_{2}\in[0,h-1]$. Since $(a,b)=1$ we also know $q_{2,i}\nmid a$. We have $q_{2,i}\mid a(z+n_{1})-bp-[a(z+n_{2})-bp]$ and thus $q_{2,i}\mid(n_{1}-n_{2})$. Now we have $h\leq q_{2,i}\leq n_{1}-n_{2}\leq h-1$, which is a contradiction.

Application of the triangle inequality now gives the result; indeed, we have

[TABLE]

The proof when $z\in\mathcal{J}^{*}(a,b)$ is similar. In this case we have $-a(z+n)+bp\in(0,H]$ which implies $\chi(z+n)=\bar{\chi}(-a)$ if $(u_{2},a(z+n)-bp)=1$. The result follows as before. ∎

Lemma 6**.**

Let $x>1$ be a real number. Then

[TABLE]

where

[TABLE]

Proof.

This is Lemma 3.2 of [13]. ∎

Proposition 7**.**

Let $h,r,u_{1},u_{2}\in\mathbb{Z}^{+}$ with $u_{1}=q_{1,1}q_{1,2}\ldots q_{1,k}$ with each prime $q_{1,i}<h$ and $u_{2}=q_{2,1}q_{2,2}\ldots q_{2,j}$ with each $h\leq q_{2,i}<p$. Suppose $\chi$ is a Dirichlet character modulo a prime $p$ such that $\chi(n)=1$ for all $n\in[1,H]$ satisfying $(n,u_{1}u_{2})=1$. Assume $2h<H<(hp)^{\frac{1}{2}}$ and set $X:=\frac{H}{2h}>1$. Then

[TABLE]

provided $X/u_{1}>1$.

Proof.

First, observe that by Lemma 4 we have

[TABLE]

Applying Lemma 5, and noting that $\mathcal{I}^{*}(a,b)\cup\mathcal{J}^{*}(a,b)$ has at least $2(\frac{H}{a}-h)$ elements, we obtain

[TABLE]

Replacing $H$ with $2Xh$ the above is equal to

[TABLE]

Now we may apply Lemma 6 to conclude

[TABLE]

∎

Lemma 8**.**

Let $h$ and $r$ be positive integers with $j\leq h/8$. Then

[TABLE]

Proof.

By the convexity of the logarithm, we know $\log t\leq t-1$ for all $t$. It follows that, for $j\leq h/8$,

[TABLE]

which implies

[TABLE]

and therefore

[TABLE]

∎

3 Proof of the main result

Proof of Theorem 1.

The conditions $X^{*}>3.8$ and $p>403$ guarantee that all the denominators in the expression for $g(n,p)$ are positive. Notice also that the condition $\log p>8(n-1)$ implies that, in particular, $n\leq\frac{1}{4}\log p$.

Let $h=\lceil A\log p\rceil$ and $r=\lfloor B\log p\rfloor$ with $A=\frac{n}{n+1}\exp\left(\frac{1}{n}\right)$ and $B=\frac{n}{2(n+1)}$. One verifies that $\log{p}>e^{\frac{16B}{3A}}$ is also satisfied. Indeed, $\frac{2B}{A}=e^{-1/n}\leq 1$ and therefore the condition $\log p>\exp(8/3)$ suffices.

Write $X=H/(2h)$. Let $u_{1}$ be composed of the $k$ distinct prime factors of $u$ less than $h$ and let $u_{2}$ be composed of the $j$ prime factors greater than or equal to $h$ as in Proposition 7, so $j+k=n-1$. Note that our choices of $r$, $h$ will allow us to apply Lemma 8, and moreover, we have $A=\frac{2B}{e}\exp\left(\frac{1}{2B}\right)$ from which it follows that

[TABLE]

both of these facts will be employed forthwith.

We may assume that $H>\frac{\pi}{3}\sqrt{2e}\left(\frac{n}{n+1}\right)p^{\frac{1}{4}}(\log{p})^{\frac{n+1}{2}}$ or there would be nothing to prove. Using this and $u_{1}\leq(h-1)^{k}$, we get $X^{*}$ as a lower bound for $\frac{X}{u_{1}}$; indeed,

[TABLE]

Using the upper and lower bounds for $S(\chi,h,r)$ given in Lemma 3 and Proposition 7 respectively, we find

[TABLE]

which implies

[TABLE]

Substituting our values for $h$ and $r$ and using the fact $\log{p}\geq e^{\frac{16B}{3A}}$, together with (1), gives us

[TABLE]

Since $u_{1}$ is composed of $k$ prime factors strictly less than $h$, we have

[TABLE]

Using this, and substituting our values of $A$ and $B$, we find

[TABLE]

Since $n\leq\frac{1}{4}\log{p}$, we can show

[TABLE]

This is less than 1 whenever $\log{p}>4$. Since $k\leq n-1$ we can use this to drop some terms from the product, which yields

[TABLE]

Isolating $H^{2}$, and noting that $f(\frac{X}{u_{1}})\geq f(X^{*})$, gives

[TABLE]

Taking the square root of both sides and rearranging gives the desired result. ∎

Proof of Corollary 2.

This follows immediately from Theorem 1, letting $u$ be the product of the first $n-1$ prime nonresidues. The fact that this holds for all $p\geq p_{0}$ and $n\leq n_{0}$ can be verified by showing that $g(n,p)$ is decreasing with $p$ and increasing with $n$ under the conditions given. This is not hard to verify. Indeed, calculus can be used to show that for $X^{*}>3.8$, the expression

[TABLE]

is increasing with $X^{*}$, and that $X^{*}$ increases with $p$ and decreases with $n$. Similarly, the term

[TABLE]

is increasing with $n$ and decreasing with $p$. ∎

Acknowledgements

This research was completed as part of the Research Experience for Undergraduates and Teachers program at California State University, Chico funded by the National Science Foundation (DMS-1559788). We would also like to thank the anonymous referee for their helpful suggestions which improved the quality of this paper.