Short Character Sums and the P\'{o}lya-Vinogradov Inequality

TL;DR

This paper investigates the relationship between the Pólya-Vinogradov inequality and cancellation in character sums for odd characters of fixed order, establishing conditions under which improvements imply cancellation and vice versa.

Contribution

It provides a quantitative link between improved bounds on the Pólya-Vinogradov inequality and cancellation in short character sums for odd characters of fixed order.

Findings

Improvement in Pólya-Vinogradov inequality implies cancellation in short sums.

Cancellation in short sums implies potential improvement in Pólya-Vinogradov bounds.

Generalizes previous results by Fromm and Goldmakher.

Abstract

We show in a quantitative way that any odd character $\chi$ modulo $q$ of fixed order $g \geq 2$ satisfies the property that if the P\'{o}lya-Vinogradov inequality for $\chi$ can be improved to $$\max_{1 \leq t \leq q} \left|\sum_{n \leq t} \chi(n)\right| = o_{q \rightarrow \infty}(\sqrt{q}\log q)$$ then for any $\epsilon > 0$ one may exhibit cancellation in partial sums of $\chi$ on the interval $[1,t]$ whenever $t > q^{\epsilon}$, i.e.,$$\sum_{n \leq t} \chi(n) = o_{q \rightarrow \infty}(t) \text{ for all $t > q^{\epsilon}$.}$$ This generalizes and extends a result of Fromm and Goldmakher. We also prove a converse implication, to the effect that if all odd primitive characters of fixed order dividing $g$ exhibit cancellation in short sums then the P\'{o}lya-Vinogradov inequality can be improved for all odd primitive characters of order $g$. Some applications are also discussed.

Equations

$$\underset{1\le t\le q}{\max }\left|\sum _{n\le t}\chi (n)\right|=o_{q\to \infty }(\sqrt{q}\log q)$$

$$\sum _{n\le t}\chi (n)=o_{q\to \infty }(t)\text{ for all t>qε.}$$

$$\sum _{n\le t}\chi (n)=o_{q\to \infty }(t)$$

$$\underset{1\le t\le q}{\max }\left|\sum _{n\le t}\chi (n)\right|\ll \sqrt{q}\log q.$$

$$\underset{1\le t\le q}{\max }\left|\sum _{n\le t}\xi (n)\right|\le \sqrt{q}\frac{\log q}{f(q)}$$

$$\underset{1\le t\le q}{\max }\left|\sum _{n\le t}\chi (n)\right|\le \sqrt{q}\frac{\log q}{a(q)}.$$

$$\sum _{n\le t}\chi (n){\ll }_{g}t{\left(\frac{{\log }_{2}a(t)}{\log a(t)}\right)}^{1/(19g^2)}.$$

$$\underset{1\le t\le q}{\max }\left|\sum _{n\le t}\chi (n)\right|=o_{q\to \infty }(\sqrt{q}\log q).$$

$$\sum _{n\le t}\chi (n)=o_{q\to \infty }(t).$$

$$\underset{1\le t\le q}{\max }\left|\sum _{n\le t}\chi (n)\right|>\delta \sqrt{q}\log q.$$

$$\sum _{n\le t}\chi (n)\overline{\psi }(n){\gg }_{\delta }t.$$

$$\underset{1\le t\le p}{\max }\left|\sum _{n\le t}\chi (n)\right|=o_{p\to \infty }(\sqrt{p}\log p).$$

$$\underset{1\le t\le q}{\max }\left|\sum _{n\le t}\chi (n)\right|{\ll }_{\epsilon }\sqrt{q}(\log q{)}^{1-{\delta }_g}({\log }_{2}q{)}^{-1/4+\epsilon }$$

$$\underset{1\le N\le q}{\max }\left|\sum _{1\le n\le N}\frac{\xi (n)}{n}\right|\ll \frac{\log q}{f(q)}.$$

$$S_f(t):=\sum _{p\le t}\frac{1-f(p)}{p}.$$

$$\underset{1\le t\le q}{\max }\left|\sum _{1\le n\le t}\chi (n)\right|=\frac{|\tau (\chi )|}{2\pi }\underset{\alpha \in [0,1]}{\max }\left|\sum _{1\le |n|\le q}\frac{\chi (n)}{n}\left(1-e(n\alpha )\right)\right|+O(\sqrt{q}),$$

$$\underset{1\le N\le q}{\max }\left|\sum _{1\le n\le N}\frac{\chi (n)}{n}\right|=o(\log q).$$

$$\frac{1}{t}\sum _{n\le t}\chi (n)\ll 1/\xi (t).$$

$$\left|\frac{1}{t}\sum _{n\le t}\chi (n)\right|$$

$$\sum _{\genfrac{}{}{0ex}{}{p\le t^{1/C}}{p|q}}\frac{1}{p}\ge \sum _{p|q}\frac{1}{p}-C{t}^{-1/C}\sqrt{\log q}\ge \log \xi (q)-O(1),$$

$$\left|\frac{1}{t}\sum _{n\le t}\chi (n)\right|\ll 1/\xi (q)\le 1/\xi (t),$$

$$\mathcal{Q}(\xi ):=\left\{q\in \mathbb{N}:\sum _{p|q}\frac{1}{p}<\log \xi (q)\right\}.$$

$$|(\sqrt{x},x]\backslash \mathcal{Q}(\xi )|\le \frac{1}{\log \xi (\sqrt{x})}\sum _{\sqrt{x}<q\le x}\left(\sum _{p|q}\frac{1}{p}\right)\ll \frac{1}{\log \xi (x)}\sum _{p\le x}\frac{1}{p}\lfloor \frac{x}{p}\rfloor \ll \frac{x}{\log \xi (x)},$$

$$\xi (t){\ll }_{\epsilon }(\log t{)}^{1/(27k^2)-\epsilon }$$

$$\left|\frac{1}{x}\sum _{n\le x}f(n)\right|>1/\xi (x),$$

$$\underset{\sqrt{x}<y\le x}{\max }\left|\frac{1}{\log y}\sum _{n\le y}\frac{f(n)}{n}\right|+\frac{1}{\log x}{\gg }_k\xi (x{)}^{-38k^2\xi (x{)}^{19k^2}}.$$

$$\underset{\sqrt{x}<y\le x}{\max }\left|\frac{1}{\log y}\sum _{n\le y}\frac{f(n)}{n}\right|+\frac{1}{\log x}{\gg }_k\xi (x{)}^{-72k^2}.$$

$$g(p^k)$$

$$=\sum _{0\le u,v\le k}f(p{)}^{u-v}(k+1-(u+v))=\sum _{|w|\le k}f(p{)}^w\sum _{\max \{0,-w\}\le v\le \min \{k,k-w\}}(k+1-w-2v).$$

$$g(p^k)=\sum _{|w|\le k}f(p{)}^w(k+1-|w|)=(k+1)K_{k+1}({\theta }_p),$$

$$K_N(t):=\sum _{|j|\le N-1}(1-|j|/N)e(jt)=\frac{1}{N}{\left(\frac{\sin (\pi Nt)}{\sin (\pi t)}\right)}^2$$

Full text

[TABLE]

then for any $\varepsilon>0$ one may exhibit cancellation in partial sums of $\chi$ on the interval $[1,t]$ whenever $t>q^{\varepsilon}$, i.e.,

[TABLE]

We also prove a converse implication, to the effect that if all odd primitive characters of fixed order dividing $g$ exhibit cancellation in short sums then the Pólya-Vinogradov inequality can be improved for all odd primitive characters of order $g$. Some applications are also discussed.

1. Introduction and Main Results

A central problem in analytic number theory concerns sharply estimating the partial sums of non-principal Dirichlet characters. There are two important types of problems regarding character sums to which a wealth of literature is devoted.

Let $\chi$ be a non-principal Dirichlet character modulo $q$, with $q$ large. A first, typically harder, problem concerns demonstrating estimates for short sums in the form

[TABLE]

for any $t>q^{\varepsilon}$ and any $\varepsilon>0$. Except under exceptional circumstances (e.g., when $q$ is smooth, see [13] and [6]), such estimates have been difficult to demonstrate. The best general result in this direction, due to Burgess, shows that such cancellation occurs (at least for $q$ cube-free) provided that $t>q^{1/4+\varepsilon}$. For this reason, we shall refer to estimates like (1) as being of Burgess-type.

A second type of problem concerns estimating the maximal size of partial sums of Dirichlet characters. A classical result in this direction, proven independently by Pólya and I.M. Vinogradov states that

[TABLE]

This result has subsequently been improved assuming the Generalized Riemann Hypothesis to the best possible upper bound111For $k\geq 1$, we write $\log_{k}q=\log_{k-1}(\log q)$, and $\log_{1}q=\log q$. $O(\sqrt{q}\log_{2}q)$ by Montgomery and Vaughan, and this is best possible according to an old construction due to Paley ([17], in loc. cit.). However, unconditional improvements to (2) have been hard to come by in general (though see Remark 1.5 below). We refer to the problem of improving (2) as one of PV-type.

A priori, these two problems are not directly connected. Nevertheless, the work of Bober and Goldmakher [2] on one hand, and the more recent paper of Fromm and Goldmakher [3] have revealed a connection between hypothetical improvements to the Pólya-Vinogradov inequality and estimates for short quadratic character sums. For instance, Bober and Goldmakher [2] showed that if the Pólya-Vinogradov inequality can be improved for any even222As usual, we call a character $\xi$ even if $\xi(-1)=1$, and odd if $\xi(-1)=-1$. character $\xi$ modulo $q$ in the form

[TABLE]

for some non-decreasing function $f:\mathbb{R}\rightarrow\mathbb{R}$ tending to infinity, then for any odd character $\chi$ modulo $m$ there is an integer $n\ll_{\varepsilon}m^{\varepsilon}$ such that $\chi(n)\neq 0,1$. This result was extended (still in the case of quadratic characters $\xi$ and $\chi$) in [3], where it was shown that a hypothesis like (3) for quadratic even characters implies the bound $\sum_{n\leq t}\chi(n)=o(t)$ for any $t>m^{\varepsilon}$, whenever $\chi$ is a quadratic odd character of prime conductor.

In this paper, we extend the latter results in two ways. First, we show that a hypothesis like (3) for an odd character $\chi$ implies a Burgess-type bound for $\chi$ itself in intervals of length $q^{\varepsilon}$. Second, we extend this result to characters of any fixed order, rather than just for quadratic characters. Moreover, our result quantifies the range $t>q^{\varepsilon}$ as well as the $o(t)$ bound here.

Theorem 1.1**.**

Let $g\geq 2$ and let $q$ be large. Let $a(t)$ be a non-decreasing function such that333A classical result of Schur implies that $\max_{1\leq N\leq q}|\sum_{n\leq N}\chi(n)|\gg\sqrt{q}$, and hence $a(q)\ll\log q$ for any non-principal character $\chi$. We will thus have $\log a(q)\ll\log_{2}q$ in general, and the above assumption is rather mild. For instance, it would easily hold for any function of the form $a(t)=(\log_{k}t)^{A}$, for any $k\geq 1$ and $A>0$. $\log a(q)\asymp\log a(t)$ for all $t>\exp\left(\sqrt{\log q}\right)$. Let $\chi$ be a primitive odd character modulo $q$ with order $g$, and assume that the Pólya-Vinogradov inequality for $\chi$ can be improved in the form

[TABLE]

Then for any $t>\exp\left(\frac{\log q}{a(q)^{1/6}}\right)$, we have

[TABLE]

As a consequence of Theorem 1.1, we have the following corollary regarding the relationship between improvements to the Pólya-Vinogradov inequality and Burgess-type bounds for fixed order characters.

Corollary 1.2**.**

Let $g\geq 2$, and let $\chi$ be an odd primitive character with modulus $q$ and order $g$. Assume that the Pólya-Vinogradov inequality can be improved in the form

[TABLE]

Then for any $\varepsilon>0$ and any $t>q^{\varepsilon}$ we have

[TABLE]

The above results demonstrate that in order to prove cancellation in short sums of a given odd character of fixed order, it is sufficient to improve estimates for maximal sums of that same character. A converse implication to a similar effect also holds for both odd and even characters of a fixed order.

Proposition 1.3**.**

Let $q$ be large and $g\geq 2$ fixed. Let $1>\delta>Cg\left(\frac{\log_{3}q}{\log_{2}q}\right)^{1/2}$ for $C>0$ sufficiently large. Let $\chi$ be a primitive character modulo $q$ of order $g$, and suppose that

[TABLE]

Then there is an absolute constant $c>0$ such that for any $\varepsilon\geq c\delta/\log(1/\delta)$, there is a unique primitive character $\psi$ satisfying $\chi\psi(-1)=-1$ whose order $k$ satisfies $k|g$ and whose conductor $m=O(1)$ is independent of $\delta$, and a $t>q^{\varepsilon}$ such that

[TABLE]

In particular, if for any odd primitive character $\chi^{\prime}$ of conductor $q^{\prime}$ and order dividing $g$ we have $\sum_{n\leq t}\chi^{\prime}(n)=o(t)$ for any $t>(q^{\prime})^{\varepsilon}$ and any $\varepsilon>0$, then for any odd primitive character $\chi$ of order $g$ and conductor $q$ we have $\max_{1\leq t\leq q}\left|\sum_{n\leq t}\chi(n)\right|=o(\sqrt{q}\log q)$.

See Remark 4.3 for an application of this proposition in regards to Burgess-type problems for even characters of a fixed (even) order $g$.

We deduce from Corollary 1.2 the following, which for instance gives a single condition required to verify Vinogradov’s conjecture for any given large prime $p\equiv 3\pmod{4}$.

In the sequel, let $n_{\chi}:=\min\{n\in\mathbb{N}:\chi(n)\neq 0,1\}$, and for $\chi=\left(\frac{\cdot}{p}\right)$, let $n_{p}:=n_{\chi}$.

Corollary 1.4**.**

Let $p$ be a large prime and $\chi$ be a primitive odd character modulo $p$ such that

[TABLE]

Then $\log n_{\chi}=o_{p\rightarrow\infty}(\log p)$.

The above corollary should be compared to the main result of [2], in which an a priori stronger assumption about improvements in estimates of maximal character sums of all even quadratic characters is used to derive the above conclusion (albeit with a nice quantitative relationship between the $n_{\chi}$ and the amount of savings in Pólya-Vinogradov inequality).

Remark 1.5**.**

It would be nice to be able to apply a result like Theorem 1.1 in cases where an improvement to the Pólya-Vinogradov inequality is known to be possible. Such is the case for instance when $\chi$ has odd order, as was demonstrated in a breakthrough result of Granville and Soundararajan [9]. In light of refinements by Goldmakher [4] and by Lamzouri and the author [16], the Pólya-Vinogradov inequality for a character $\chi$ modulo $q$ of odd order $g\geq 3$ can be improved unconditionally in the form

[TABLE]

for any $\varepsilon>0$, where $\delta_{g}:=1-\frac{g}{\pi}\sin(\pi/g).$

Unfortunately, the method of proof here relies crucially on the parity of $\chi$ being odd, and odd order characters are necessarily of even parity. At best, our proof allows us to show that if $\chi$ is a character of fixed odd order and $\psi$ is any odd character of small conductor (e.g., the Legendre symbol modulo $3$) then we have cancellation in the partial sums of $\chi\psi$ on the scale $t>q^{\varepsilon}$ for any $\varepsilon>0$ (see Remark 3.7 below). However, such a result follows directly from Corollary 1.7 of [8], which admits a relatively simple proof.

1.1. Strategy of Proof

The idea of the proof of Theorem 1.1 is inspired by the work of Fromm and Goldmakher [3], and we explain it here. Let $\xi$ be a quadratic character with prime modulus $p$ and let $\psi$ be some auxiliary quadratic character of opposite parity (and small conductor). Assuming an “improved Pólya-Vinogradov” hypothesis like (3) for $\xi\psi$, it can be shown that

[TABLE]

In [3], the strategy implemented is to show that if the Césaro partial sum $\sum_{n\leq t}\xi(n)$ is $\gg t$ for some $t>q^{\varepsilon}$ then this implies that the logarithmically-averaged partial sum $\left|\sum_{1\leq n\leq t^{\prime}}\frac{\xi(n)}{n}\right|$ is prohibitively large in light of (5), with $t^{\prime}$ of size commensurate with $t$ (in logarithmic scale). To do this they use a fact about real-valued multiplicative functions, namely that the sizes of the logarithmic and Césaro means of a real-valued function $f$ are both controlled (in terms of both upper and lower bounds) by

[TABLE]

This is implied directly by a result of Hall and Tenenbaum [11]. Crucial use in this connection is made of the fact that the function $1\ast f$ is non-negative, and a lower bound for this non-negative function is used crucially in the proof.

In this paper we shall deal with characters that are not necessarily real-valued, preventing us from directly employing the above techniques.444While the main result in [11] is applicable to complex-valued functions as well, they are for instance not well-suited to functions taking non-zero values in roots of unity of sufficiently large order. For instance, in the notation there, the best-possible constant $K$ appearing in the estimate $t^{-1}\left|\sum_{n\leq t}g(n)\right|\ll e^{-K\text{Re}(S_{g}(t))}$ is, for those functions $g$ considered here, necessarily [math] (apply (5) there with $\delta=1$ and $\phi=0$). However, by invoking ideas from pretentious number theory we will be follow a more elaborate proof scheme that is similar to the one described above to arrive at the desired ends.

To be precise, we derive a quantitative relationship between the Césaro partial sums and the logarithmic partial sums of a multiplicative function $f$ whose non-zero values are roots of unity of some fixed order $k$; see Proposition 2.2 below. Upper bounds on the log-averaged sums of $\chi$ will therefore imply, according to this relationship, that the Césaro partial sums of $\chi$ cannot be too large, even when $t\asymp_{\varepsilon}q^{\varepsilon}$ (and in fact, with $t=q^{o(1)}$, as $q\rightarrow\infty$).

These improved upper bounds for log-averaged sums of $\chi$ are established in Proposition 3.1, which invokes the hypothesized improvement in the Pólya-Vinogradov inequality. Pólya’s Fourier expansion (see Lemma 3.3 below) implies that555Here and elsewhere, given a character $\xi$ modulo $\ell$, we write the Gauss sum of $\xi$ as $\tau(\xi):=\sum_{a\pmod{\ell}}\xi(a)e^{2\pi ia/\ell}.$

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which demonstrates that the log-averaged exponential sums of $\chi$ must accordingly be small. Specializing to $\alpha=a/m$ with $m$ small in some precise sense and writing $e(na/m)$ in the basis of Dirichlet characters modulo $m$, it follows first that the log-averaged sum of $\chi\psi$ is small, where $\psi$ is some non-principal character with $\psi(-1)=-\chi(-1)$ (provided $m$ is chosen suitably). With some additional work (see e.g., Lemmata 3.3 and 3.5) this can then be used to establish the required savings666This is where the assumption in Theorem 1.1 that $\chi$ is odd is used crucially, as we must pass from a sum of $\chi(n)/n$ over $1\leq|n|\leq q$ to a sum over $1\leq n\leq q$. If $\chi$ were even then the two-sided sum vanishes, and we can deduce nothing about the 1-sided sum.

[TABLE]

It should be noted that neither the bounds nor the range in $t$ in Theorem 1.1 are expected to be optimal; it is not our main objective here to give the best such result.

2. Relating the Logarithmic and Césaro Partial Sums of Multiplicative Functions

In this section, we show how Lemma B of [3] regarding large Césaro and logarithmic sums of real-valued $1$-bounded mutliplicative functions can be adapted to deal with complex-valued Dirichlet characters of any fixed order.

Our first observation towards Theorem 1.1 is the following, which will allow us to reduce matters to bounding short sums of characters whose moduli have relatively few small prime factors.

Lemma 2.1**.**

Let $\xi(t)$ be a non-decreasing function. Suppose $\chi$ is a primitive character modulo $q$, and that $\sum_{p|q}1/p\geq\log\xi(q)$. Then for any $t>\exp(\sqrt{\log q})$ we have

[TABLE]

Proof.

By the fundamental lemma of the sieve, there is a large constant $C>0$ such that

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Note that the number of primes $p|q$ with $p>t^{1/C}$ is at most $C\frac{\log q}{\log t}\leq C\sqrt{\log q}$, so that

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by hypothesis. Inserting this into the above yields

[TABLE]

as claimed. ∎

In light of Lemma 2.1, we shall mostly focus on characters whose moduli have relatively few small prime factors. To be precise, consider the collection $\mathcal{S}$ of non-decreasing functions $\xi(t)$ tending to infinity with $t$, such that for all large $x$, $\log\xi(x)\asymp\log\xi(t)$ for all $t\in(\exp\left(\sqrt{\log x}\right),x]$. Given a map $\xi\in\mathcal{S}$, we define

[TABLE]

Note that for any unbounded function $\xi$, the set of all large primes belong to $\mathcal{Q}(\xi)$. Moreover, by Markov’s inequality, for any large $x$,

[TABLE]

so that anyway $\mathcal{Q}(\xi)$ is a set of density 1 for all $\xi\in\mathcal{S}$. Moreover, we have a well-known uniform bound $\sum_{p|q}1/p\leq\log\log\log q+O(1)$, which shows that if $\xi(q)\geq\log\log q$ then $\mathcal{Q}(\xi)$ consists of all sufficiently large moduli.

The main result of this section is the following. Here and elsewhere, $\mathbb{U}$ denotes the closed unit disc.

Proposition 2.2**.**

Let $k\geq 2$ and let $q$ be large. Let $\xi\in\mathcal{S}$ be such that $q\in\mathcal{Q}(\xi)$, and that

[TABLE]

*for all $t\geq\exp\left(\sqrt{\log q}\right)$. Let $f:\mathbb{N}\rightarrow\mathbb{U}$ be a completely multiplicative function vanishing at precisely those primes dividing $q$, such that for each prime $p$ where $f(p)\neq 0$, we have $f(p)^{k}=1$.

Then there is a constant $x_{0}=x_{0}(\xi,k)$ such that the following is true. Suppose that*

[TABLE]

*for some $x\geq\max\left\{x_{0},e^{\sqrt{\log q}}\right\}$.

a) In general, we have*

[TABLE]

b) If $k$ is odd then

[TABLE]

The proof proceeds by showing that the logarithmic partial sums of $f$ can be minorized by the Césaro average of a certain non-negative multiplicative function $g$ to be constructed below. Some work will then be done to show that when $\xi(x)$ is not too large the partial sums of $g$ are sufficiently large for this lower bound to be useful.

Our first lemma shows how to construct this function $g$ for the complex-valued functions that we shall consider.

Lemma 2.3**.**

*Let $f:\mathbb{N}\rightarrow\mathbb{U}$ be completely multiplicative, such that $f(p)\in S^{1}\cup\{0\}$ for all primes $p$, and let $g:=1\ast 1\ast f\ast\overline{f}$. Then $g$ is a non-negative multiplicative function. *

Proof.

That $g$ is multiplicative is immediate, given that it is defined from convolutions of multiplicative functions. We shall thus prove that for any prime power $p^{k}$, we have $g(p^{k})\geq 0$.

Assume first that $f(p)\neq 0$, so that $|f(p)|=1$. Write $h:=1\ast f$. Then $g=h\ast\overline{h}$, and so we have

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It is easily checked that the inner sum is $k+1-|w|$, for all $|w|\leq k$. Upon writing777As is standard, given $t\in\mathbb{R}$ we write $e(t):=e^{2\pi it}$. $f(p)=e(\theta_{p})$, this gives

[TABLE]

where for $N\in\mathbb{N}$,

[TABLE]

denotes the order $N$ Féjer kernel. This establishes that $g(p^{k})\geq 0$ for all $k$ whenever $|f(p)|=1$.

Now, suppose that $f(p)=0$. Then $h(p^{k})=1$ for all $k\geq 0$, and thus

[TABLE]

This completes the proof in the case $f(p)=0$. The claim follows. ∎

Lemma 2.4**.**

Let $f:\mathbb{N}\rightarrow\mathbb{U}$ be completely multiplicative and such that for all primes $p$ we have $f(p)\in S^{1}\cup\{0\}$. Then there is an absolute constant $t_{0}\geq 2$ such that for any $t\geq t_{0}$,

[TABLE]

where $g=1\ast 1\ast f\ast\overline{f}$.

Proof.

As in the proof of the previous lemma, set $h:=1\ast f$, so that $g=h\ast\overline{h}$. We will bound the partial sums of $h$ in two ways. Let $t$ be sufficiently large. On one hand, for any $\sqrt{t}<x\leq t$ we have

[TABLE]

Now, the previous lemma shows that $g$ is multiplicative and non-negative. Thus, applying the hyperbola method, we get on the other hand that

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Since $|h|\leq\tau$, we easily get

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As such, we get that

[TABLE]

Combining this with (8), and dividing both sides by $\log t$, proves the claim. ∎

In order to establish Proposition 2.2, we will need to be able to show that the partial sums of $g$ are not too small. To this end, we will make use of the notion of pretentiousness of multiplicative functions, due originally to Granville and Soundararajan (for a detailed account of this theory, see [10]). A key object in this connection is the so-called pretentious distance, which we define as follows. For $f_{1},f_{2}:\mathbb{N}\rightarrow\mathbb{U}$ multiplicative and $x\geq 2$, set

[TABLE]

Moreover, for $T\geq 0$, we define888Henceforth, for convenience, given $t\in\mathbb{R}$ we will write $n^{it}$ to denote the multiplicative function mapping $n\mapsto n^{it}$.

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Roughly speaking, a hypothesis like (7) implies (via Halász’ theorem, see e.g., (9) below) that $\mathcal{D}_{f}(x;(\log x)^{2})$ is small (in a manner depending on $\xi$), and thus $f$ is $n^{it_{0}}$-pretentious for some $t_{0}\in[-(\log x)^{2},(\log x)^{2}]$. An important feature of multiplicative functions whose values are bounded order roots of unity is that we can guarantee that $t_{0}=0$. The following lemma gives a quantitative relation in this direction.

Lemma 2.5**.**

Let $k\in\mathbb{N}$ and let $1\leq k,T\leq x$. For any multiplicative function999Given a positive integer $m$, we write $\mu_{m}$ to denote the set of $m$th order roots of unity. $f:\mathbb{N}\rightarrow\mu_{k}$,

[TABLE]

Proof.

This is Lemma 3.1 in [15]. In the statement there, the $O(1)$ term depends on $k$, but following the proof it is easy to verify that it can in fact be taken absolute. ∎

Lemma 2.6**.**

Let $k\geq 2$, let $\xi$ be as in the statement of Proposition 2.2, and suppose that $q\in\mathcal{Q}(\xi)$. Let $f:\mathbb{N}\rightarrow\mu_{k}\cup\{0\}$ be a completely multiplicative function such that $f(p)=0$ if, and only if, $p|q$. Then there is an $x_{0}=x_{0}(\xi,k)$ such that the following holds. If $x\geq\max\{x_{0},e^{\sqrt{\log q}}\}$ and

[TABLE]

then we have

[TABLE]

Proof.

The result is immediate if $\mathbb{D}(f,1;x)^{2}\leq 200k^{2}$, so in the sequel we assume that $\mathbb{D}(f,1;x)^{2}>200k^{2}$.

By the Halász-Montgomery-Tenenbaum inequality (see Theorem III.4.6 in [20]), for any $T\geq 1$ we have

[TABLE]

Let $T=\log x$. Let $\tilde{f}:\mathbb{N}\rightarrow S^{1}$ be the completely multiplicative function defined on primes by

[TABLE]

note that $\tilde{f}^{k}=1$. For any fixed $t\in[-T,T]$, the triangle inequality for $\mathbb{D}$ gives

[TABLE]

while we trivially have

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By Lemma 2.5, we have

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Using (10) with $t=t_{0}$ the minimizer of $t\mapsto\mathbb{D}(f,n^{it};x)$ in $[-T,T]$ and (11) with $t=0$, we deduce that

[TABLE]

and hence by the Cauchy-Schwarz inequality (and using $k\geq 2$),

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Plugging this into (9) and using $1/T<(2\xi(x))^{-2}$, we get upon rearranging that

[TABLE]

Note that the arithmetic function $d\mapsto\sum_{p|d}1/p$ has maximal order101010This is witnessed by a primorial $d=\prod_{p\leq\log d^{\prime}}p$, with $d^{\prime}=(1+o(1))\log d$ (by the prime number theorem). $\log_{3}x+O(1)$, for $d\leq x$. Thus, since $\xi(x)\ll(\log x)^{\frac{1}{10k^{2}}}$, it follows from (12) that $\min\{\log_{2}x,\mathbb{D}(f,1;x)^{2}\}=\mathbb{D}(f,1;x)^{2}$. Hence,

[TABLE]

Taking logarithms, and noting that $\log X<X/17$ for $X=\mathbb{D}(f,1;x)^{2}/(8k^{2})>25$, this implies that

[TABLE]

using $\sum_{p|q}1/p<\log\xi(x)$, on account of our assumption $q\in\mathcal{Q}(\xi)$. ∎

Proof of Proposition 2.2.

In light of Lemma 2.4, it suffices to derive a lower bound for $\sum_{n\leq x}g(n)$ in both parts a) and b) (recalling that $g=1\ast 1\ast f\ast\overline{f}$).

a) By Theorem 1.2 of [12], we get

[TABLE]

for some absolute constant $\beta>0$. Here, we have $\sigma_{-}(u):=u\rho(u)$, where $\rho$ is the Dickman-de Bruijn function, which satisfies $\rho(u)\gg u^{-3u/2}$ for large enough $u$ (see, for instance, equation (1.12) in [7]). Now, observe that $g(p)=2(1+\text{Re}(f(p)))$ for all primes $p$. Thus, by Lemma 2.6, we get

[TABLE]

Note that $\max\{0,1-g(p)\}\neq 0$ if, and only if, $\text{Re}(f(p))<-1/2$, in which case the numerator is bounded by 1. Hence, for $q$ (and thus $x$) sufficiently large, we get

[TABLE]

From (13), it follows that for $x$ sufficiently large (in terms of $\xi$),

[TABLE]

By Lemma 2.4, this results in the estimate

[TABLE]

as claimed.

b) We assume throughout the proof that $x>\exp(\sqrt{\log q})$. We would like to apply part ii) of Théorème 1.1 in [19], which gives a better lower bound for partial sums of $g$ compared to Hildebrand’s estimate provided $g$ satisfies various conditions. To this end we verify the conditions i)-v) stipulated there. Since $|f|\leq 1$, it follows that $g(p)\leq 4$ for all $p$, and moreover $g(n)\ll_{\varepsilon}n^{\varepsilon}$ for all $n$. Thus, conditions i) and ii) for that theorem, namely

[TABLE]

both hold with $A=5$ and $B=O(1)$ by the prime number theorem. Moreover, condition iii) holds since for each $\delta>0$, setting $\varepsilon=\delta/5$ and taking $y$ sufficiently large in terms of $\delta$ (in line with the prime number theorem), we get

[TABLE]

for all $y\geq y_{0}$. Turning to the first condition in v) (in light of its application in the proof of the theorem), it is enough to check that for any $\lambda>0$ there is $\eta>0$ such that for any choice of $0<\sigma<\tau<1-\sigma$, we have

[TABLE]

To see this, we note that since $k$ is odd, writing $k=2m+1$, it follows that

[TABLE]

for any prime $p$. Hence, selecting $\eta=\frac{\pi^{2}}{3k^{2}}$, we see that since

[TABLE]

we find $g(p)>\eta$ for all $p$. Hence, condition v) there holds with this $\eta$ for any $\lambda>0$ and any choice of $\sigma,\tau$ as above.

It remains to demonstrate iv) in the notation there. We pick $\sigma=\xi(x)^{-c_{0}}$, for $c_{0}>0$ a constant to be chosen, and $\tau=1/46$. Putting $h:=(1-\tau)/A\tau=9$, it follows that $Q(h):=h\log h-h+1\geq 1/2$. It now suffices to check that

[TABLE]

with the above choices made. To see this, note that

[TABLE]

the error term arising from $g(p)\leq 4$ for all primes $p$, and thus

[TABLE]

For the above sum, noting that $|f(p)-1|^{2}=2(1-\text{Re}(f(p)))$ whenever $p\nmid q$, we have

[TABLE]

In light of Lemma 2.6, choosing $c_{0}=27k^{2}$ gives

[TABLE]

As (6) gives $\sigma\log x\gg_{\varepsilon}(\log x)^{\varepsilon}$ and thus $x^{-\sigma}\log q\leq x^{-\sigma}(\log x)^{2}\ll 1$, we get that if $x$ is sufficiently large in terms of $k$ then

[TABLE]

Hence, iv) holds in the theorem in [19], provided that $x$ is sufficiently large (in terms of $\xi$ and $k$ alone). Théorème 1.1 there then implies111111The result appearing there is given with a constant that is not made implicit in terms of the parameters $\sigma$ and $\eta$. However, the whole argument in Section 2 of that paper can be made to depend explicitly on those two parameters, which may then be allowed to depend on $x$. Following the calculation there, one readily finds that, provided that $\sigma\log x\geq C\log\log x$ for some sufficiently large $C>0$,

$\sum_{n\leq x}g(n)\gg\frac{x}{\log x}\exp\left(\sum_{p\leq x}g(p)/p\right)\left(\frac{\eta^{2}\sigma}{1+\tau}e^{-2A\tau}\left(1-e^{-AQ(h)}\right)-O_{h,A}\left(\frac{\eta}{\log x}+\tau(\log x)e^{-\sqrt{\tau\log x}}\right)\right),$

whenever $g:\mathbb{N}\rightarrow\mathbb{C}$ is non-negative, multiplicative and satisfies conditions i)-v) above with $A>0$, $\eta>0$, $0<\sigma<\tau<\min\{1-\sigma,1/(1+A)\}$ and $h=(1-\tau)/\tau A$. that for any $x\geq\max\{x_{0}(\xi,k),e^{\sqrt{\log q}}\}$ (recalling that $\eta\asymp 1/k^{2}$)

[TABLE]

in light of Lemma 2.6 and our assumption on $\xi(x)$. Combining this with Lemma 2.4, we get that

[TABLE]

and the claim follows. ∎

3. Proof of Theorem 1.1

Let $q$ be large and let $\chi$ be a primitive odd character modulo $q$. Throughout this section, we shall assume that there is a non-decreasing function $a(t)\rightarrow\infty$ as $t\rightarrow\infty$, satisfying $\log a(t)\asymp\log a(q)$ and $a(t)\leq\log t$ for all $t>\exp\left(\sqrt{\log q}\right)$, and such that the Pólya-Vinogradov inequality for $\chi$ can be improved in the form

[TABLE]

Our goal is to show that such a hypothesis implies Theorem 1.1, regarding bounds for $\sum_{n\leq t}\chi(n)$ whenever $t>q^{\varepsilon}$.

Proposition 3.1**.**

Let $q\geq 3$ and let $\chi$ be an odd primitive character modulo $q$. Assume that $\chi$ satisfies (14). Then for any $\varepsilon>0$,

[TABLE]

To prove Proposition 3.1, we first recall the following simple harmonic analysis device which is essentially due to Goldmakher and Lamzouri (based on work of Paley).

Lemma 3.2** (cf. Lemma 2.2 of [5]).**

Let $A>0$. Let $a:\mathbb{Z}\rightarrow\mathbb{C}$ be a sequence with $\sup_{n\geq 1}|a(n)|\leq A$. Then

[TABLE]

Proof.

Writing $a_{n}=Aa^{\prime}_{n}$ for all $n\in\mathbb{Z}$, we find that $\{a_{n}^{\prime}\}_{n}\subset\mathbb{U}$, and the claim then follows immediately from Lemma 2.2 of [5]. ∎

The next lemma allows us to bound maximal Césaro partial sums of $\chi$ from below in terms of maximal logarithmically averaged partial sums of the twist $\chi\psi$, provided that $\chi$ and $\psi$ have opposite parity. This is a variant of a result that has appeared before in several places (including, among others [2], [16]) but with no fixed restrictions on the parity of $\psi$ in particular (previous variants used $\psi$ odd, though we shall require $\psi$ even).

Lemma 3.3**.**

Let $\chi$ and $\psi$ be primitive characters to respective moduli $q,\ell\geq 2$, such that $\chi\psi$ is odd. Then

[TABLE]

Proof.

By the Pólya Fourier expansion (see, e.g., (2.1) in [16]), we have

[TABLE]

where the second equality follows since if the sum is maximized at $\theta_{0}\in[0,1]$, we may take $t_{0}:=\left\lfloor q\theta_{0}\right\rfloor$ and thus

[TABLE]

By the triangle inequality, we observe that

[TABLE]

and by Lemma 3.2 (with $a(n):=\chi(n)(1-e(n\theta)$), the RHS of this last expression is

[TABLE]

Now, on one hand, we have

[TABLE]

On the other, as $\chi\xi(-1)=-1$ and $\psi$ is non-principal, orthogonality modulo $\ell$ gives

[TABLE]

for each $1\leq N\leq q$. It follows that

[TABLE]

and the claim follows. ∎

Corollary 3.4**.**

Let $q\geq 3$ and let $\ell\geq 5$ be prime. Let $\chi$ be an odd primitive character modulo $q$, and let $\psi$ be an even primitive character modulo $\ell$. Then

[TABLE]

Proof.

Following the proof of Lemma 3.3, particularly (15) and (16), we deduce that

[TABLE]

using the fact that $\chi(-1)=-1$ in the last line. Now, since $\ell$ is prime, we observe that

[TABLE]

Combined with the previous estimate, this completes the proof. ∎

Lemma 3.5**.**

Let $m\geq 3$. Let $\psi$ be an even non-principal character modulo $m$. As above, assume that $\chi$ is an odd primitive character modulo $q$ satisfying (14). Then

[TABLE]

Proof.

Factor $\psi=\psi^{\ast}\psi_{0}$, where $\psi^{\ast}$ is a primitive character modulo $m^{\ast}$ and $\psi_{0}$ is principal modulo $m/m^{\ast}$. We then have that

[TABLE]

Since $\psi^{\ast}$ has the same parity as $\psi$, Lemma 3.3 and (14) give

[TABLE]

and the claim follows upon combining these two statements. ∎

Lemma 3.6**.**

Let $\chi$ be as above, and let $\psi$ be a primitive even character modulo $\ell$. Then

[TABLE]

Proof.

By Lemma 3.2, we see that the LHS here is

[TABLE]

Let $R=(\log q)^{5}$, and let $1\leq M\leq R$ be a parameter to be chosen. By the Dirichlet approximation theorem, for any $\alpha\in[0,1]$ we can find a rational $b/r$ such that $|\alpha-b/r|\leq 1/(rR)$. We will say that $\alpha$ is minor arc if $M<r\leq R$, and major arc if $1\leq r\leq M$.

Suppose first that $\alpha$ is minor arc. By Corollary 2.2 of [4], we get

[TABLE]

Now, suppose $|\alpha-b/r|\leq 1/(rR)$, with $r\leq M$. Then applying equation (7.2) in [16] with $N^{\prime}:=\min\{q,|r\alpha-b|^{-1}\}$, we have

[TABLE]

We now invoke the Granville-Soundararajan identity (see Proposition 2.3 of [9]), which reads

[TABLE]

as $\chi\psi(-1)=-1$.

Now, $\psi\overline{\psi^{\prime}}$ has modulus $\leq\ell r$, and $\psi\overline{\psi^{\prime}}$ is even, thus of opposite parity to $\chi$. Hence, applying Lemma 3.5 to the inner sums in (18) gives

[TABLE]

It follows that when $\alpha$ is major arc, we get that

[TABLE]

Combining this with the minor arc case yields

[TABLE]

Choosing $M=a(q)^{2/3}$, the upper bound above becomes

[TABLE]

This completes the proof. ∎

Proof of Proposition 3.1.

Let $\ell=5$ and let $\psi=\left(\frac{\cdot}{\ell}\right)$. Then $\psi$ is an even, primitive character with prime conductor. Combining Corollary 3.4 with Lemma 3.6, we get that

[TABLE]

This completes the proof. ∎

Proof of Theorem 1.1.

The result is immediate (by the Pólya-Vinogradov inequality) if $t>q$, so in what follows, we shall assume that $t\leq q$.

In keeping with previous notation, set

[TABLE]

and $f(t):=a(t)^{1/6}$. Note that this is non-decreasing since $a$ is. By Lemma 2.1, the claim of the proposition is trivial if $\sum_{p|q}1/p\geq\log\xi(q)$. Thus, we suppose henceforth that $q\in\mathcal{Q}(\xi)$.

Assume that

[TABLE]

for some $t\in(q^{1/f(q)},q]$, noting that $q^{1/f(q)}>\exp(\sqrt{\log q})$ since $a(t)\leq\log t$ for all large $t$. By Proposition 2.2 a) (with $k=g$), it follows that

[TABLE]

Let $N$ be the point in $(\sqrt{t},t]$ maximizing the LHS in this last expression. By Proposition 3.1, the above is

[TABLE]

Combining this with the above, rearranging and taking logarithms, we get

[TABLE]

On the other hand, as $\xi(q)^{19g^{2}}=\frac{\log a(q)}{13\log\log a(q)}$, the upper bound here is at most $(2/13)\log a(q)$ for large $q$, a contradiction for $\varepsilon$ sufficiently small.

This contradiction implies that for all $t>q^{1/f(q)}$, we get

[TABLE]

which implies the claim for all $q\in\mathcal{Q}(\xi)$ as well. The theorem is thus proved. ∎

Remark 3.7**.**

Note that if $\chi$ is assumed to be of odd order $g\geq 3$ then (14) holds with $a(t)=(\log t)^{\delta_{g}}$ (see Remark 1.5 above). The above proof can be modified to show cancellation in short sums of characters of the form $\chi(n)\psi(n)$, where $\psi$ is an odd character of small conductor. To do this, we combine Lemma 3.5 with Proposition 2.2 instead, which shows that if $t>\exp((\log t)^{1-\delta_{g}/2})$ and the conductor $\ell$ of $\psi$ is bounded then

[TABLE]

In fact, as $g$ is odd we can use part b) of Proposition 2.2 instead of a) to replace the above bound by one of the shape $O_{g}\left(t(\log t)^{-c/(g\ell)^{2}}\right)$, with $c>0$ sufficiently small in this same range of $t$. However, as mentioned in the introduction, cancellation in character sums of this kind can be shown to follow from Corollary 1.7 of [8]. For instance, assume that $\psi$ is a quadratic character. As $g$ is odd we have $(\chi\psi)^{g}=\psi 1_{(n,q)=1}$, which is a character of very small conductor relative to $q$, and thus necessarily witnessing cancellation along intervals $[1,t]$ at scales $t>q^{\varepsilon}$. Corollary 1.7 of [8] then shows that this cancellation implies cancellation in the partial sums of $\chi\left(\frac{\cdot}{\ell}\right)$ itself.

4. A Converse of Theorem 1.1

In this section, we show that a converse of Theorem 1.1, namely Proposition 1.3 is true. That is, we show that cancellation in short sums of primitive characters of a fixed order imply improvements in the Pólya-Vinogradov inequality for characters of that same fixed order.

To do this, we need two lemmata. The first is a very slight generalization of a result of Granville-Soundarajan [8].

Lemma 4.1**.**

Let $t\in\mathbb{R}$ and let $x$ be large. Let $f:\mathbb{N}\rightarrow\mathbb{U}$ be multiplicative. Put $\lambda:=\mathbb{D}(f,n^{it};x)^{2}+\log(1+|t|)+c$ for some large enough absolute constant, and $\eta=(\lambda e^{\lambda})^{-1}$. Then there is a $x^{\eta}\leq y\leq x$ such that

[TABLE]

Proof.

The result in [8] was stated for $t$ the minimizer of the distance $\mathbb{D}(f,n\mapsto n^{it};x)$ in the interval $[-\log x,\log x]$. However, the proof follows identically for any choice of $t\in\mathbb{R}$. ∎

To glean a bit more information about characters that correlate with one another, we establish the following simple result.

Lemma 4.2**.**

*Let $\chi$ be a primitive character of order $g$ to a large modulus $q$, and suppose $\psi$ is a primitive character of order $k$ modulo $m$, with $m\leq\log q$. Then for any $0\leq T\leq\log q$ at most one of the following holds:

i) $\mathcal{D}_{\chi\overline{\psi}}(q;T)\leq\frac{1}{3(gk)^{2}}\log\log q$, or else

ii) $k\nmid g$.

Moreover, i) can hold for at most one primitive character $\psi$ with conductor of size $\leq\log q$.*

In the proof to follow, given $x,r\geq 2$ and 1-bounded multiplicative functions $f_{1},f_{2}:\mathbb{N}\rightarrow\mathbb{U}$, we define the distance function

[TABLE]

Proof.

Assume instead that both i) and ii) hold. Let $t_{0}\in[-T,T]$ minimize the distance here. Set $k^{\ast}:=k/(k,g)$ and $g^{\ast}:=g/(k,g)$, noting that $k^{\ast}>1$ by assumption. As $k^{\ast}$ and $g^{\ast}$ are coprime, we can choose $r,s\in\mathbb{Z}$ with $0<s<k^{\ast}$ such that $rk^{\ast}+sg^{\ast}=1$. Now, set $t_{0}^{\prime}:=sgt_{0}$. We then have

[TABLE]

By the triangle inequality, the bound $s<k$ and i), we have

[TABLE]

On the other hand, as $(k,g)<k$, $\psi^{(k,g)}$ is a non-principal character modulo $m$ so that the effective version of Siegel’s theorem (see, e.g., Theorem 8.21 of [19]) gives

[TABLE]

This yields the contradiction

[TABLE]

and proves the first claim.

For the uniqueness of $\psi$ we apply Lemma 3.1 of [1] (with $j=2$), which shows that for any two characters $\psi_{1},\psi_{2}$ to moduli of size $O(\log q)$, we have

[TABLE]

for $q$ large enough. Thus, since $g\geq 2$, at most one character $\psi$ of conductor $\leq\log q$ is such that

[TABLE]

as required. ∎

Proof of Proposition 1.3.

We begin by considering the first claim of the proposition, and assume the contrary. Then we can find a primitive character $\chi$ such that

[TABLE]

where we have put $\delta:=\varepsilon\log(1/\varepsilon)$. By the Pólya Fourier expansion (see Lemma 3.3) and primitivity, it follows that

[TABLE]

By the triangle inequality,

[TABLE]

Conversely, equation (7.1) in [16] (with $y=Q=q$) shows that

[TABLE]

where $\psi$ is a primitive character to a conductor $m\leq\log q$ that is of opposite parity from $\chi$ and that minimizes the distance $\mathcal{D}_{\chi\overline{\psi}}(q;(\log q)^{-7/11})$. As $\delta\gg(\log q)^{-1/11}$, it follows that there must be a primitive character $\psi$ modulo $m$ with $m\ll\delta^{-2}$ of opposite parity from $\chi$ such that

[TABLE]

whence we get

[TABLE]

in light of our assumed lower bound $\delta\geq Cg\left(\frac{\log_{3}q}{\log_{2}q}\right)^{1/2}$, with $C>0$ sufficiently large. Lemma 4.2 then implies that $\text{ord}(\psi)|\text{ord}(\chi)$.

Let $t_{0}$ be the choice of $t\in[-(\log q)^{-7/11},(\log q)^{-7/11}]$ that minimizes $t\mapsto\mathbb{D}(\chi\overline{\psi},n^{it};q)^{2}$. By Lemma 4.1, we can set $\lambda=\mathbb{D}(\chi\overline{\psi},n^{it_{0}};q)^{2}+c$ for some large $c>0$ to get that for some $y\in(q^{(\lambda e^{\lambda})^{-1}},q]$,

[TABLE]

As $\lambda\leq\log(1/\delta)+O(1)$, we have $y>q^{\varepsilon}$ with $\varepsilon>\lambda^{-1}e^{-\lambda}\gg\delta/\log(1/\delta)$.

To complete the proof of the first claim, it remains to show that $\psi$ can be selected independently of $\delta$. Indeed, suppose $\psi_{1}$ and $\psi_{2}$ are primitive characters to respective conductors $m_{1},m_{2}$ that satisfy the conclusion of the first claim for parameters $\delta_{1},\delta_{2}>(\log_{2}q)^{-1/2}$. As mentioned above, we have $m_{j}\ll\delta_{j}^{-2}$ for $j=1,2$. Let $t_{1}$ and $t_{2}$ be the corresponding scales at which the partial sums of $\chi\overline{\psi}_{1}$ and $\chi\overline{\psi}_{2}$ are large in the above sense, and let $y:=\max\{t_{1},t_{2}\}$. By (9) we have that

[TABLE]

for $j=1,2$. The triangle inequality for $\mathbb{D}$ thus gives that for some $u\in[-2(\log y)^{2},2(\log y)^{2}]$,

[TABLE]

where $\chi_{0}$ is the principal character modulo $q$. On the other hand, as in the proof of Lemma 4.2, if $\psi_{1}\neq\psi_{2}$ then we have

[TABLE]

This is a contradiction, given $\delta_{1},\delta_{2}>(\log_{2}q)^{-1/2}$. Hence, $\psi_{1}=\psi_{2}$, and we can take $\psi$ independent of $\delta$ in this range.

We turn to the second claim of the proposition. Assume that the conclusion is false for an odd primitive character $\chi$ modulo $q$ of order $g$. Thus, we can find a $\delta>0$ such that

[TABLE]

The first claim of the proposition implies that there is an even primitive character $\psi$ of conductor $m\ll 1$ and a $t>q^{\varepsilon}$ (with $\varepsilon\gg\delta/\log(1/\delta)$) such that

[TABLE]

Put $\tilde{q}:=[q,m]$. Now, write $\chi\overline{\psi}=\xi^{\ast}\xi_{0}$, where $\xi^{\ast}$ is primitive modulo $\tilde{q}^{\ast}$ and $\xi_{0}$ is a principal character modulo $\tilde{q}_{0}$, with $\tilde{q}=\tilde{q}^{\ast}\tilde{q}_{0}$. We note that since $\chi$ and $\psi$ are both primitive, $\tilde{q}_{0}\leq\min\{q,m\}=m\ll 1$, and moreover since $\text{ord}(\psi)|g$, $\text{ord}(\xi^{\ast})=\text{ord}(\chi\overline{\psi})$ must divide $g$.

Clearly, $\xi^{\ast}$ is an odd character since $\chi\overline{\psi}$ is. By hypothesis, we thus know that, uniformly in $x>q^{\varepsilon}\geq(\tilde{q}^{\ast})^{\varepsilon}$,

[TABLE]

We apply this to the estimation of partial sums of $\chi\overline{\psi}$. Indeed,

[TABLE]

To the first double sum we apply (20), giving the contribution

[TABLE]

in light of the bound $\tilde{q}_{0}\ll 1$. Estimating the second sum trivially, we get

[TABLE]

Thus, on the whole, we find that

[TABLE]

at the scale $t$. This contradicts (19) for $\delta>0$ given, so the second claim of the proposition also follows. ∎

Remark 4.3**.**

Since the first claim of the proposition applies to even characters $\chi$ of fixed order as well, the proof of the second claim above can also be used to show that cancellation in short sums for all odd characters of order dividing $g$ implies cancellation in short sums for all even characters of order $g$. Together with our Theorem 1.1, this shows that if the Pólya-Vinogradov inequality is improved for all odd primitive characters of orders dividing $g$ then all even primitive characters of order $g$ exhibit cancellation in their short sums. This complements the work of [3], which showed that in the case $g=2$ improvements in Pólya-Vinogradov for all even primitive quadratic characters implies cancellation in short sums of all odd primitive quadratic characters.

5. Obstructions in Extending Theorem 1.1 to all Odd Primitive Characters

In this section, we explain the shortcoming in our argument that prevents us from extending Theorem 1.1 to odd characters, irrespective of their order.

The key simplifying feature of the bounded order case is the fact that if a character $\chi$ has large Césaro partial sums, $\chi$ must correlate heavily with 1, i.e., that $\mathbb{D}(\chi,1;x)^{2}$ is small. In general, this is not a condition furnished by Halász’ theorem, failing when $k$ is exceedingly large (e.g., on the order of $\phi(q)$, where $q$ is the modulus of $\chi$).

Nevertheless, Halász’ theorem does provide a $t_{0}\in[-T,T]$ such that $\mathbb{D}(\chi,n^{it_{0}};x)$ (for suitably chosen $T$), and one might try to exploit the argument in Section 2 with $h:=\chi n^{-it_{0}}$ instead of $\chi$ itself. The following result of Granville and Soundararajan suggests that this strategy might be effective.

Lemma 5.1** (Theorem 12.1 in [10]).**

Let $f:\mathbb{N}\rightarrow\mathbb{U}$ be multiplicative. Let $t_{0}\in[-(\log x)^{2},(\log x)^{2}]$ be such that $\mathbb{D}(f,n^{it_{0}};x)^{2}$ is minimal. Then

[TABLE]

Now, suppose we wanted to prove a version of Proposition 2.2 for arbitrary multiplicative functions taking values in $S^{1}\cup\{0\}$. Assume for simplicity that $\xi(x)=o\left(\frac{(\log x)^{2-\sqrt{3}}}{\log_{2}x}\right).$ Then the lemma suggests that a hypothesis like $\frac{1}{x}\sum_{n\leq x}f(n)\gg 1/\xi(x)$ implies, in fact, that

[TABLE]

where $\tilde{f}(n):=f(n)n^{-it_{0}}$. The crucial point here is that $\mathbb{D}(\tilde{f},1;x)^{2}$ is now small, and this was of importance in the proof of Proposition 2.2 (in that the lower bound arising from Hildebrand’s theorem for the Césaro partial sums of $1\ast 1\ast\tilde{f}\ast\overline{\tilde{f}}$ depends on $\mathbb{D}(\tilde{f},1;x)^{2}$ precisely). Running through the machinery of Lemma 2.6 and Proposition 2.2, we may conclude that

[TABLE]

is also large in a manner depending on $\xi(x)$ and $|t_{0}|$.

Now in order to proceed as in Section 3, where we selected $f=\chi\psi$, we must find a way of replacing the logarithmic mean of $\tilde{f}$ by that of $f$ in some way, since Lemma 3.3 relies heavily on the periodicity of $f$, and $\tilde{f}$ is not periodic. However, unlike in the case of Césaro partial sums, we do not expect that the logarithmically averaged partial sums of $\tilde{f}$ and $f$ are of the same size. Indeed, if we take a simple example like $f\equiv 1$, the logarithmic partial sums of $\tilde{f}$ behaves like $\zeta(1+it_{0})$, which is of size $1/|t_{0}|$ for small $|t_{0}|$ and of size $O(\log(2+|t_{0}|))$ for larger $t_{0}$. If $|t_{0}|\log y$ is large enough, in particular, then this is small compared to $\log y$.

It would be interesting to determine how an appeal to the function $\tilde{f}$ might be circumvented here.

Acknowledgments

We would like to thank Andrew Granville, Oleksiy Klurman and Maksym Radziwiłł for their helpful advice and encouragement.