# Uniform distribution of polynomially-defined additive function to varying
moduli

Agbolade Akande

##  1 Introduction

Let $q$ be a positive integer and $f$ be an integer-valued arithmetic
function. We say $f$ is uniformly distributed modulo $q$ if for every residue
class $a\ (\mathrm{mod}\ q)$

$\displaystyle\\#\\{n\leq x:f(n)\equiv a\ (\mathrm{mod}\ q)\\}\sim\frac{x}{q}$
$\displaystyle\text{as $x\rightarrow\infty$}.$

One way to prove that $f$ is uniformly distributed is by considering the mean
value of $f$ composed with additive characters modulo $q$ [1, Proposition
1.1]:

$f$ is uniformly distributed modulo $q$ if and only if for all
$r=1,2,\ldots,q-1$, one has
$\lim\limits_{x\rightarrow\infty}\frac{1}{x}\sum\limits_{n\leq
x}\exp\left(\frac{2\pi irf(n)}{q}\right)=0$.

Pillai in 1940[2] used this approach to show $\Omega(n)$ is uniformly
distributed modulo $q$ for every positive integer $q$. $\Omega(n)$ is an
example of an additive function: an integer-valued arithmetic function $f$
with the property $f(mn)=f(m)+f(n)$ for coprime $m$ and $n$. In 1969, Delange
gave a general criterion [3] for when additive functions $f$ are uniformly
distributed modulo $q$ for fixed $q$. Delange’s criterion is as follows:

######  Theorem 1.1.

Define two conditions for every integer $t>1$:

$(A_{t})$ The series $\sum\limits_{\underset{t\nmid f(p)}{p}}p^{-1}$ diverges,

and

$(B_{t})$ For all $k\geq 1$, the number $\frac{2f(2^{k})}{t}$ is an odd
integer.

An additive function $f$ is uniformly distributed modulo $q$ in the following
3 cases:

  * •

$q$ is odd and $A_{p}$ holds for every prime divisor $p$ of $q$.

  * •

$2\mid\mid q$, $A_{p}$ holds for every odd prime divisor $p$ of $q$ and either
$A_{2}$ or $B_{2}$ hold.

  * •

$4\mid q$, $A_{p}$ holds for every odd prime divisor $p$ of $q$, and either
$A_{2}$ holds or both $B_{2}$ and $A_{4}$ hold.

Moreover, these conditions are necessary for $f$ to be uniformly distributed
modulo $q$.

We are interested in proving uniform distribution of a class of additive
functions where the modulus is allowed to vary. To illustrate, consider the
sum of prime divisors function $A(n)=\sum\limits_{p^{k}\mid\mid n}kp$. It was
proved by Pollack and Singha Roy that for a fixed $K\geq 1$ and any residue
class $a\mod{q}$,

$\displaystyle\sum\limits_{\overset{n\leq x}{A(n)\equiv a\ (\mathrm{mod}\
q)}}1\sim\frac{x}{q}$ as $x\rightarrow\infty$,

for $q\leq(\log x)^{K}$. This was a byproduct of their work on the
distribution of multiplicative functions in coprime residue classes[4].

In this paper, we will consider polynomially-defined additive functions $f$:
Additive functions $f$ such that $f(p)=F(p)$ for all primes $p$, where $F(x)$
is a nonconstant polynomial in $\mathbb{Z}[x]$. In the next section, given a
polynomially-defined additive function $f$ satisfying mild conditions, we
explicitly describe the collection of integers $q$ such that $f$ is uniformly
distributed modulo $q$. We will then set up the same framework used in [4] by
Pollack and Singha Roy. (See also the paper of Katai [5], where a broadly
similar approach is taken.) A new ingredient is the use of estimates for the
size of sets of the form:

$\mathcal{V}_{F,q,J}(w)=\left\\{(v_{1},\cdots,v_{J})\mod{q}:(v_{1},\ldots,v_{J},q)=1,\sum\limits_{j=1}^{J}F(v_{j})\equiv
w\ (\mathrm{mod}\ q)\right\\}.$

These estimates require bounds on exponential sums adopted from Cochrane [6]
and Loh [7]. The end result is as follows:

######  Theorem 1.2.

Fix $K\geq 1$ and $\epsilon\in(0,1]$. Let $f$ be a polynomially-defined
additive function such that $f(p)=F(p)$ for all primes $p$ where
$F(x)\in\mathbb{Z}[x]$ with degree $d$. Let $\mathcal{S}_{f}$ be the set of
all positive integers $q$ such that $f$ is uniformly distributed modulo $q$.
Then as $x\rightarrow\infty$,

$\sum\limits_{\overset{n\leq x}{f(n)\equiv a\ (\mathrm{mod}\
q)}}1\sim\frac{x}{q},$

for all residue classes $a\mod{q}$ and $q\in\mathcal{S}_{f}\cap[1,(\log
x)^{K}]$ provided that either

  * •

$d=1$, or

  * •

$q\leq(\log x)^{\left(1-\epsilon\right)\left(1-\frac{1}{d}\right)^{-1}}$.

As we now explain, the extra condition on $q$ in Theorem 1.2 when $d>1$ is
essentially optimal. Fix an integer $d\geq 2$ and define $F(x)=(x-1)^{d}+1$.
Let $f$ be the completely additive function where $f(p)=F(p)$ for all primes
$p$ and let $q=q_{1}^{d}$. By Theorem 1.1, $f$ is uniformly distributed for
all $q$. Notice that $F(p)\equiv 1\ (\mathrm{mod}\ q)$ if and only if $p\equiv
1\ (\mathrm{mod}\ q_{1})$. Hence, by Siegel-Walfisz’s theorem, there are at
least $\frac{1+o(1)}{\varphi(q_{1})}\frac{x}{\log x}$ primes $p\leq x$ with
$F(p)\equiv 1\ (\mathrm{mod}\ q)$. Therefore, the residue class $1\mod q$ is
overrepresented if $\frac{1}{\varphi(q_{1})}\frac{x}{\log x}$ is much bigger
than $\frac{x}{q}$, for example if $q^{1-1/d}>(\log x)^{(1+\delta)}$ for some
fixed $\delta>0$. This means that for uniform distribution we want $q$ to be
no more than $\approx(\log x)^{(1-\frac{1}{d})^{-1}}$. So the condition is
essentially optimal.

##  2 Polynomial-defined arithmetic functions that are uniformly distributed
modulo $q$

Given a polynomially-defined additive function, we will use Delange’s
criterion in Theorem 1.1 to determine the collection of integers $q$ such that
$f$ is uniformly distributed modulo $q$. To check condition $B_{2}$, we will
impose the restriction: $f(2^{k})$ is odd for all $k\geq 2$.

######  Proposition 2.1.

Fix $q\in\mathbb{N}\backslash\\{1\\}$. Let $f$ be a polynomially-defined
additive function, such that $f(p)=F(p)$ for all primes $p$,
$F(x)\in\mathbb{Z}[x]$. Suppose $f(2^{k})$ is an odd integer for all $k\geq
2$. If there exist an odd prime $p$ such that $p\mid q$ and $F(x)\mod{p}$ has
a factor of

$\prod\limits_{i=1}^{p-1}(x-i),$

then $f$ is not uniformly distributed modulo $q$.

If $2\mid\mid q$ and $F$ has an even constant and $F(x)\mod{2}$ has a factor
of $(x-1)$, then $f$ is not uniformly distributed modulo $q$.

If $4\mid q$ and the alternating sum of coefficients of F is either $0$ or
$2\mod{4}$, then $f$ is not uniformly distributed modulo $q$.

If neither of these conditions hold, then $f$ is uniformly distributed modulo
$q$.

###### Proof.

First, let $p$ be an odd prime divisor $p$ of $q$. By Theorem 1.1, $f$ is not
uniformly distributed modulo $q$ if $A_{p}$ does not hold. Now we will use the
fact that, for all $d\geq 2$,

$A_{d}$ holds if and only if there exist $a$ such that $(a,d)=1$ and
$f(a)\not\equiv 0\ (\mathrm{mod}\ d)$

This follows from Dirichlet’s theorem on primes in arithmetic progressions.
Therefore, the only polynomials where $A_{p}$ does not hold are the
polynomials $F$ such that $F\mod{p}$ has a factor of
$\prod\limits_{i=1}^{p-1}(x-i)$.

Suppose $2\mid\mid q$. By Theorem 1.1, $f$ is not uniformly distributed modulo
$q$ if $A_{2}$ and $B_{2}$ both fail. The $A_{2}$ case is similar to the last
proof: the polynomials where $A_{2}$ does not are the polynomials with the sum
of its coefficients equal to 0 modulo 2. This is equivalent to polynomials $F$
where $F\mod{2}$ has a factor of $(x-1)$. The $B_{2}$ case fails when the
constant term is even.

Suppose $4\mid q$. By Theorem 1.1, $f$ is not uniformly distributed modulo $q$
if $A_{4}$ does not hold. Let $F(x)=\sum\limits_{i=0}^{d}a_{i}x^{i}$ where
$a_{i}\in\mathbb{Z}$. If $F(1)\equiv F(3)\equiv 0\ (\mathrm{mod}\ 4)$. Then
$\sum\limits_{\overset{0\leq i\leq d}{\text{$i$
even}}}a_{i}+\sum\limits_{\overset{0\leq i\leq d}{\text{$i$ odd}}}a_{i}\equiv
0\ (\mathrm{mod}\ 4)$ and $\sum\limits_{\overset{0\leq i\leq d}{\text{$i$
even}}}a_{i}-\sum\limits_{\overset{0\leq i\leq d}{\text{$i$ odd}}}a_{i}\equiv
0\ (\mathrm{mod}\ 4)$. The only solutions to these system of congruences are
$\sum\limits_{\overset{0\leq i\leq d}{\text{$i$
even}}}a_{i}\equiv\sum\limits_{\overset{0\leq i\leq d}{\text{$i$
odd}}}a_{i}\equiv 0\ (\mathrm{mod}\ 4)$ and $\sum\limits_{\overset{0\leq i\leq
d}{\text{$i$ even}}}a_{i}\equiv\sum\limits_{\overset{0\leq i\leq d}{\text{$i$
odd}}}a_{i}\equiv 2\ (\mathrm{mod}\ 4)$.

If $F$ does not satisfy any of those conditions mentioned, the criteria hold
for all the cases in Theorem 1.1. Thus, $f$ is uniformly distributed modulo
$q$. ∎

##  3 Bounds on polynomials composed with additive characters

We will use bounds from Cochrane [6] and Loh [7]. The first bound in [6]
applies to polynomials that are nonconstant modulo $\ell$ for $\ell$ prime; a
polynomial $F$ is nonconstant modulo $\ell$ if and only if the map
$F:\mathbb{Z}/\ell\mathbb{Z}\mapsto\mathbb{Z}/\ell\mathbb{Z}$ is nonconstant.

######  Lemma 3.1.

For all primes $\ell$ and polynomials $F(x)\in\mathbb{Z}[x]$ that are
nonconstant modulo $\ell$,

$\left\lvert\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}{\exp\left(\frac{2\pi
iF(v)}{\ell^{k}}\right)}\right\rvert\leq 8.82\ell^{k(1-\frac{1}{d+1})}.$

Lemma 3.1 cannot be applied to all polynomials we are interested in. For
example, let $F(x)=x^{2}+x+1$ and $f$ be the completely additive function such
that $f(p)=F(p)$ for all primes $p$. By Theorem 1.1, $f$ is uniformly
distributed modulo $q$ for all positive integers $q$ but $F$ is constant
modulo 2. So, in the rest of this section, we will focus on the polynomials
Lemma 3.1 does not apply to.

Firstly, notice that
$\left\lvert\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}{\exp\left(\frac{2\pi
iF(v)}{\ell^{k}}\right)}\right\rvert=\left\lvert\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}{\exp\left(\frac{2\pi
i(F(v)+t)}{\ell^{k}}\right)}\right\rvert$ for all $t\in\mathbb{Z}$ so the
constant term of $F$ does not play a role. Also, if $F$ is constant modulo
$\ell$ with $d\geq 1$, there have to be at least 2 terms in the polynomial.

Next, we will consider the polynomials where $\ell$ does not divide the
content of $F$: We define the content of $F$ to be the greatest common divisor
of the coefficients of $F$. The following lemma, which gives an upper bound on
the exponential sum with a condition on the content of $F$, is stated in [7].

######  Lemma 3.2.

Let $F(x)$ be a polynomial in $\mathbb{Z}[x]$ such that

$F(x)=a_{s}x^{d_{s}}+a_{s-1}x^{d_{s-1}}+\cdots+a_{2}x^{d_{2}}+a_{1}x^{d_{1}}$

with $s\geq 2$ and $d_{s}>d_{s-1}>\cdots>d_{1}\geq 1$, and
$a_{i}\in\mathbb{Z}\backslash\\{0\\}$.

Suppose $\ell$ is prime and $k$ is a positive integer.

If $\ell>d_{s}$ and $\ell$ does not divide the content of $F$, then

$\left\lvert\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}{\exp\left(\frac{2\pi
iF(v)}{\ell^{k}}\right)}\right\rvert\leq(d_{s}-1)\ell^{m(1-\frac{1}{s})}.$

If $\ell\leq d_{s}$ and $\ell$ does not divide the content of $F$, then

$\left\lvert\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}{\exp\left(\frac{2\pi
iF(v)}{\ell^{k}}\right)}\right\rvert\leq
R(\ell^{-\tau}F^{\prime})\ell^{\frac{\tau+1}{s}}\ell^{m(1-\frac{1}{s})}$

where $\tau$ is the largest nonnegative integer such that $\ell^{\tau}$
divides the content of $F^{\prime}(x)$ and $R(F)$ is the number of roots of
the congruence

$F(X)\equiv 0\ (\mathrm{mod}\ \ell).$

Since $\tau$ and $R(\ell^{-\tau}F^{\prime})$ depend only on $F$, they are both
$O_{F}(1)$. Also, if $d$ is the degree of $f$, then $s<d+1$ . Hence, the
inequalities in Lemma 3.2 can be combined into a single inequality so that a
uniform upper bound on the exponential sum is
$O_{F}(\ell^{m(1-\frac{1}{d+1})})$. Still, there are polynomials that Lemma
3.1 and Lemma 3.2 cannot be applied to, the polynomials $F$ where $\ell$ does
divide the content of $F$ without its constant term and $F$ is constant modulo
$\ell$. For instance, $F(x)=2x+1$ is constant modulo 2 and 2 divides the
content of $F(x)-1$ but the completely additive function $f$ such that
$f(p)=F(p)$ for all primes $p$ is uniformly distributed modulo $q$ for all
positive integers $q$.

Let $t$ be the largest nonnegative integer such that $\ell^{t}$ divides the
content of $F(x)$ without its constant term $c_{0}$. Then,
$F(x)=\ell^{t}G(x)+c_{0}$ where $G(x)\in\mathbb{Z}[x]$ and $\ell$ does not
divide the content of $G(x)$. Then for $k>t$,

$\left\lvert\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}{\exp\left(\frac{2\pi
iF(v)}{\ell^{k}}\right)}\right\rvert=\left\lvert\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}{\exp\left(\frac{2\pi
iG(v)}{\ell^{m-t}}\right)}\right\rvert=\ell^{t}\left\lvert\sum\limits_{\overset{v\mod\ell^{m-t}}{(v,\ell)=1}}{\exp\left(\frac{2\pi
iG(v)}{\ell^{m-t}}\right)}\right\rvert.$

We now apply the bounds that we already have on the last sum. Since $\ell^{t}$
divides the content of $F$ without its constant term, $\ell^{t}$ is
$O_{F}(1)$. Combining all the bounds that we have, we can form a corollary:

######  Corollary 3.3.

Let $F(x)$ be a nonconstant polynomial in $\mathbb{Z}[x]$. Then, for all
positive integers $k$ and primes $\ell$,

$\left\lvert\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}{\exp\left(\frac{2\pi
iF(v)}{\ell^{k}}\right)}\right\rvert\leq C_{F}\ell^{k(1-\frac{1}{d+1})}$

where $C_{F}$ is a constant that depends on $F$.

##  4 Proof of Theorem 1.2: the notion of convenient numbers

Fix $K\geq 1$ and $\epsilon>0$. Let $f$ be a polynomially-defined additive
function such that $f(p)=F(p)$ for all primes $p$ and $F(x)\in\mathbb{Z}[x]$.
We will focus on integers $n\leq x$ and varying
$q\in\mathcal{S}_{f}\cap[1,(\log x)^{K}]$. Similar to Pollack and Singha Roy’s
paper[4], we will consider the $n$ with many large non-repeated prime factors
and we will show that the count of everything else is negligible compared to
the count of these special integers.

We will first define parameters $J=\lfloor\log\log\log x\rfloor$ and
$y=\exp((\log x)^{\frac{\epsilon}{2}})$. We call a positive integer $n$
convenient if the following conditions hold:

  * •

$n\leq x$,

  * •

the $J$ largest prime factors of $n$ with multiplicity exceed $y$,

  * •

none of these $J$ primes are repeated in $n$.

To summarize, $n$ is convenient if and only if $n$ admits an expression:
$n=mP_{1}\cdots P_{J}$, where $P_{1},\ldots,P_{J}$ are primes with

$\displaystyle\max\\{P(m),y\\}<P_{J}<\cdots<P_{1},$ $\displaystyle P_{1}\cdots
P_{J}\leq x/m.$

######  Lemma 4.1.

The number of inconvenient $n\leq x$ is at most $o(x)$.

###### Proof.

We follow the proof in [4, Lemma 3.1]. Suppose $n\leq x$ is inconvenient. We
can assume $P(n)>z=x^{1/\log\log x}$ ($P(n)$ is the largest prime factor of
$n$) because the number of exceptional $n\leq x$, by theorems on smooth
numbers [8, Theorem 5.13 and Corollary 5.19, Chapter III.5], is
$\frac{x}{(\log x)^{(1+o(1))\log\log\log x}}=o(x)$. We can also assume that
$n$ has no repeated prime factors exceeding $y$ because the number of
exceptional $n\leq x$ is $O(x/y)=o(x)$.

We will denote $n=PAB$ where $P=P(n)$ and $A$ is the largest divisor of $n/P$
supported on primes exceeding $y$. Then $z<P\leq x/AB$. If $A$ and $B$ are
given, the number of possibilities of $P$ is bounded by
$\pi(x/AB)\ll\frac{x}{AB\log z}\ll\frac{x\log\log x}{AB\log x}$. We sum on $A$
and $B$ to bound the number of inconvenient $n$. As $n$ has no repeated primes
exceeding $y$ and $n$ is inconvenient, $\Omega(A)<J$. Thus
$\sum\frac{1}{A}\leq\left(1+\sum\limits_{p\leq
x}\frac{1}{p}\right)^{J}\leq(2\log\log x)^{J}\leq\exp\left(O((\log\log\log
x)^{2})\right)$. Since $B$ is $y$-smooth, $\sum\frac{1}{B}=\prod\limits_{p\leq
y}\left(\sum\limits_{j=0}^{\infty}\frac{1}{p^{j}}\right)\ll\exp\left(\sum\limits_{p\leq
y}\frac{1}{p}\right)\ll\log y=(\log x)^{\frac{\epsilon}{2}}$. Combining all
the results, the number of the remaining inconvenient $n\leq x$ is at most

$\displaystyle\frac{x\log\log x}{(\log
x)^{1-\frac{\epsilon}{2}}}\exp(O((\log\log\log x)^{2}))=o(x).$

Next, we show that the count of inconvenient integers $n\leq x$ with the
condition $f(n)\equiv a\ (\mathrm{mod}\ q)$, for polynomially-defined additive
function $f$, is negligible compared to the count of convenient integers with
the same condition:

######  Lemma 4.2.

Let $f$ be a polynomially-defined function, such that $f(p)=F(p)$ for all
primes and $F(x)$ is a polynomial in $\mathbb{Z}[x]$ with degree $d$. If $q$
satisfies the conditions in Theorem 1.2, the number of inconvenient $n\leq x$
where $f(n)\equiv a\ (\mathrm{mod}\ q)$ is $o(x/q)$ for all residue classes
$a\mod q$.

###### Proof.

We follow the the verification of Hypothesis B in section 5 of [4]. Suppose
$n\leq x$ is inconvenient and $f(n)\equiv a\ (\mathrm{mod}\ q)$. From the
proof in Lemma 4.1, we can assume $P(n)>z=x^{1/\log\log x}$ and that $n$ has
no repeated prime factors exceeding $y$ because the number of exceptional
$n\leq x$ is $o(x/q)$.

We will denote $n=PAB$ where $P=P(n)$ and $A$ is the largest divisor of $n/P$
supported on primes exceeding $y$. Then $z<P\leq x/AB$ and $f(P)+f(AB)\equiv
a\ (\mathrm{mod}\ q)$. Let $\xi(q)$ be the maximum number of roots $v\mod{q}$
of any congruence $F(v)\equiv a^{\prime}\ (\mathrm{mod}\ q)$, where the
maximum is over all residue classes $a^{\prime}\mod{q}$. By the
Brun–Titchmarsh inequality, given $A$ and $B$ there are
$\ll\xi(q)\frac{x}{\varphi(q)AB\log(z/q)}\ll\xi(q)\frac{x\log\log
x}{\varphi(q)AB\log x}$ corresponding values of $P$. Summing over $A$ and $B$,
using the same process as Lemma 4.1, the number of remaining inconvenient
$n\leq x$ such that $f(n)\equiv a\ (\mathrm{mod}\ q)$ is at most

$\xi(q)\frac{x\log\log x}{\varphi(q)(\log
x)^{1-\frac{\epsilon}{2}}}\exp(O((\log\log\log x)^{2})).$

Since $\frac{1}{\log\log 3q}\ll\frac{\varphi(q)}{q}\ll 1$ for all $q\geq 2$,
in our range of $q$ the ratio $\frac{\varphi(q)}{q}$ is negligible compared to
any power of $\log x$. Hence, it is enough to show $\xi(q)=o\left((\log
x)^{1-\frac{3}{4}\epsilon}\right)$.

When $d=1$, $\xi(q)=O_{F}(1)$. (In fact, it is bounded above by the absolute
value of leading coefficient of $F$.) Hence, let $d\geq 2$. We will first
consider when $q$ is prime. Then, $\xi(q)$ is at most $d$ expect for finitely
many primes. (If $F$ does not reduce to a constant modulo $q$, $\xi(q)$ is at
most $d$). Extending to when $q$ is squarefree, $\xi(q)=O_{F}(d^{\omega(q)})$.
Using the asymptotic formula $\omega(q)\ll\frac{\log q}{\log\log q}$ from [9],
in our range of $q$, $d^{\omega(q)}=\exp\left(O_{F}\left(\frac{\log\log x\log
d}{\log\log\log x}\right)\right)$ which is smaller than the bound we want.
Otherwise, by a result of Konyagin in [10], each congruence $F(v)\equiv
a^{\prime}\ (\mathrm{mod}\ q)$ has $O_{F}(q^{1-1/d})$ roots modulo $q$ for any
residue class $a^{\prime}\mod{q}$. Using the condition in Theorem 1.2,
$q\leq(\log x)^{\left(1-\epsilon\right)\left(1-1/d\right)^{-1}}$ will satisfy
our bound. ∎

Therefore, by Lemmas 4.1 and 4.2, it suffices to show

$\sum\limits_{\overset{\text{convenient $n\leq x$}}{f(n)\equiv a\
(\mathrm{mod}\ q)}}1\sim\frac{1}{q}\sum\limits_{\text{convenient $n\leq
x$}}1.$ (1)

We will now rewrite the number of ways we can count the number of convenient
numbers as follows:

$\sum\limits_{\text{convenient $n\leq x$}}1=\frac{1}{J!}\sum\limits_{m\leq
x}\sum\limits_{\overset{P_{1},\ldots,P_{J}\text{
distinct}}{\overset{P_{1}\cdots P_{J}\leq x/m}{{\text{each }P_{J}>L_{m}}}}}1$
(2)

where $L_{m}=\max\\{y,P(m)\\}$. We will define for an arbitrary class $w\mod
q$

$\mathcal{V}(w)=\\{(v_{1},\cdots,v_{J})\mod{q}:(v_{1},\ldots,v_{J},q)=1,\sum\limits_{j=1}^{J}F(v_{j})\equiv
w\ (\mathrm{mod}\ q)\\}$

so we can write

$\sum\limits_{\overset{\text{convenient $n\leq x$}}{f(n)\equiv a\
(\mathrm{mod}\ q)}}1=\frac{1}{J!}\sum\limits_{m\leq
x}\sum\limits_{v\in\mathcal{V}_{m}}\sum\limits_{\overset{P_{1},\ldots,P_{J}\text{
distinct}}{\overset{P_{1}\cdots P_{J}\leq x/m}{\overset{\text{each
}P_{J}>L_{m}}{\text{each }P_{j}\equiv v_{j}\ (\mathrm{mod}\ q)}}}}1$ (3)

where $L_{m}=\max\\{y,P(m)\\}$ and $\mathcal{V}_{m}=\mathcal{V}(a-f(m))$.

We will proceed as in [4] to remove the congruence in the innermost sum over
the $J$ largest primes in (3). Write

$\sum\limits_{\overset{P_{1},\ldots,P_{J}\text{
distinct}}{\overset{P_{1}\cdots P_{J}\leq x/m}{\overset{\text{each
}P_{j}>L_{m}}{\text{each }P_{j}\equiv v_{j}\ (\mathrm{mod}\
q)}}}}1=\sum\limits_{\overset{P_{2},\ldots,P_{J}\text{
distinct}}{\overset{P_{2}\cdots P_{J}\leq x/mL_{m}}{\overset{\text{each
}P_{j}>L_{m}}{\text{each }P_{j}\equiv v_{j}\ (\mathrm{mod}\
q)}}}}\sum\limits_{\overset{P_{1}\neq P_{2},\ldots,P_{J}\text{
distinct}}{\overset{L_{m}<P_{1}\leq x/(mP_{2}\cdots P_{J})}{P_{1}\equiv v_{1}\
(\mathrm{mod}\ q)}}}1.$

Since $L_{m}\geq y$ and $q\leq(\log x)^{K}=(\log y)^{2K}$, the Siegel-Walfisz
theorem implies that

$\sum\limits_{\overset{P_{1}\neq P_{2},\ldots,P_{J}\text{
distinct}}{\overset{L_{m}<P_{1}\leq x/(mP_{2}\cdots P_{J})}{P_{1}\equiv v_{1}\
(\mathrm{mod}\ q)}}}1=\frac{1}{\varphi(q)}\sum\limits_{\overset{P_{1}\neq
P_{2},\ldots,P_{J}\text{ distinct}}{{L_{m}<P_{1}\leq x/(mP_{2}\cdots
P_{J})}}}1+O\left(\frac{x}{mP_{2}\cdots P_{J}}\exp(-C_{K}\sqrt{\log
y})\right).$

Thus,

$\sum\limits_{\overset{P_{1},\ldots,P_{J}\text{
distinct}}{\overset{P_{1}\cdots P_{J}\leq x/m}{\overset{\text{each
}P_{j}>L_{m}}{\text{each }P_{j}\equiv v_{j}\ (\mathrm{mod}\
q)}}}}1=\frac{1}{\varphi(q)}\sum\limits_{\overset{P_{1},\ldots,P_{J}\text{
distinct}}{\overset{P_{1}\cdots P_{J}\leq x/m}{\overset{\text{each
}P_{j}>L_{m}}{\text{$(\forall j\geq 2)$ }P_{j}\equiv v_{j}\ (\mathrm{mod}\
q)}}}}1+O\left(\frac{x}{m}\exp\left(-\frac{1}{2}C_{K}(\log
x)^{\epsilon/4}\right)\right).$

Repeating the same steps for $P_{2},\ldots,P_{J}$, we have

$\sum\limits_{\overset{P_{1},\ldots,P_{J}\text{
distinct}}{\overset{P_{1}\cdots P_{J}\leq x/m}{\overset{\text{each
}P_{j}>L_{m}}{\text{each }P_{j}\equiv v_{j}\ (\mathrm{mod}\
q)}}}}1=\frac{1}{\varphi(q)^{J}}\sum\limits_{\overset{P_{1},\ldots,P_{J}\text{
distinct}}{\overset{P_{1}\cdots P_{J}\leq x/m}{{\text{each
}P_{j}>L_{m}}}}}1+O\left(\frac{x}{m}\exp\left(-\frac{1}{4}C_{K}(\log
x)^{\epsilon/4}\right)\right).$

Putting this back into (3) and noting that $\\#\mathcal{V}_{m}\leq(\log
x)^{KJ}$,

$\sum\limits_{\overset{\text{convenient $n\leq x$}}{f(n)\equiv a\
(\mathrm{mod}\ q)}}1=\sum\limits_{m\leq
x}\frac{\\#\mathcal{V}_{m}}{\varphi(q)^{J}}\left(\frac{1}{J!}\sum\limits_{\overset{P_{1},\ldots,P_{J}\text{
distinct}}{\overset{P_{1}\cdots P_{J}\leq x/m}{{\text{each
}P_{j}>L_{m}}}}}1\right)+O\left(x\exp\left(-\frac{1}{8}C_{K}(\log
x)^{\epsilon/4}\right)\right).$ (4)

We will find an asymptotic formula for $\\#\mathcal{V}_{m}$ using exponential
sums. We can write
$\\#\mathcal{V}_{m}=\\#\mathcal{V}(w)=\prod\limits_{\ell^{e}\mid\mid
q}\\#V_{\ell^{e}}$ for each prime power $\ell^{e}\mid\mid q$ where
$V_{\ell^{e}}$ can be written as follows:

$V_{\ell^{e}}=\\{(v_{1},\cdots,v_{J})\mod{\ell^{e}}:(v_{1},\ldots,v_{J},\ell)=1,\sum\limits_{j=1}^{J}F(v_{j})\equiv
w\ (\mathrm{mod}\ \ell^{e})\\}.$

By the orthogonality relations for exponential sums,

$\ell^{e}\\#V_{\ell^{e}}=\sum\limits_{r\mod\ell^{e}}\sum\limits_{\overset{v_{1},\ldots,v_{J}}{(v_{1}\cdots
v_{J},\ell)=1}}\exp\left(\frac{2\pi
ir\left[\sum\limits_{j=1}^{J}F(v_{j})-w\right]}{\ell^{e}}\right)$
$=\varphi(\ell^{e})^{J}+\sum\limits_{0<r<\ell^{e}}\exp\left(\frac{-2\pi
irw}{\ell^{e}}\right)\left[\sum\limits_{\overset{v\mod\ell^{e}}{(v,\ell)=1}}\exp\left(\frac{2\pi
irF(v)}{\ell^{e}}\right)\right]^{J}$
$=\varphi(\ell^{e})^{J}+\sum\limits_{k=1}^{e}(\ell^{J})^{e-k}\left(\sum\limits_{\overset{0<r<\ell^{k}}{\ell\nmid
r}}\exp\left(\frac{-2\pi
irw}{\ell^{k}}\right)\left[\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}\exp\left(\frac{2\pi
irF(v)}{\ell^{k}}\right)\right]^{J}\right).$

We can rewrite this as
$\varphi(\ell^{e})^{J}\left[1+\sum\limits_{k=1}^{e}H(\ell,k,J)\right]$, where

$H(\ell,k,J)=\sum\limits_{\overset{0<r<\ell^{k}}{\ell\nmid
r}}\exp\left(\frac{-2\pi
irw}{\ell^{k}}\right)\left[\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}\frac{\exp\left(\frac{2\pi
irF(v)}{\ell^{k}}\right)}{\varphi(\ell^{k})}\right]^{J}.$

Let $\ell$ be a prime dividing $q$ with $e=e(\ell,q)$ satisfying
$\ell^{e}\mid\mid q$. Define $S_{\ell}=\mathbb{N}\cap[1,e]$ and three subsets
$X_{\ell}$, $Y_{\ell}$, and $Z_{\ell}$ such that

  * •

$X_{\ell}=\left\\{k\in
S_{\ell}:\left\lvert\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}\exp\left(\frac{2\pi
irF(v)}{\ell^{k}}\right)\right\rvert={\varphi(\ell^{k})}\right\\}$,

  * •

$Y_{\ell}=\\{k\in S_{\ell}\backslash
X_{\ell}:\frac{C_{F}\ell^{k(1-\frac{1}{d+1})}}{\varphi(\ell^{k})}<1\\}$ where
$C_{F}$ is the constant in Corollary 3.3, and

  * •

$Z_{\ell}=\\{k\in S_{\ell}\backslash(X_{\ell}\cup Y_{\ell})\\}$.

We will show that for each $\ell$, the sums over $Y_{\ell}$ and $Z_{\ell}$ are
vanishing as $J\rightarrow\infty$, that $X_{\ell}$ and $Z_{\ell}$ are empty
for sufficiently large $\ell$, and that the asymptotic bounds on
$\\#\mathcal{V}(w)$ come from $X_{\ell}$ for small $\ell$.

Define $M=M(F,q)$ to be the largest positive integer dividing $q$ such that
$F(v)\equiv F(1)\ (\mathrm{mod}\ M)$ for all $v$ coprime to $M$. $M$ is well
defined and $M\ll_{F}1$ because $F$ is a nonconstant polynomial. Let
$D_{\ell}$ be the largest positive integer such that $\ell^{D_{\ell}}\mid M$.

######  Lemma 4.3.

A closed form of the sum over $X_{\ell}$ is as follows:

$\sum\limits_{k\in X_{\ell}}H(\ell,k,J)=\begin{cases}0&\text{if $\ell\nmid
M$},\\\ \ell^{D_{\ell}}-1&\text{if $\ell\mid M$ and $F(1)J\equiv w\
(\mathrm{mod}\ \ell^{D_{\ell}})$},\\\ -1&\text{otherwise}.\end{cases}$

###### Proof.

If $k\in X_{\ell}$, $\exp\left(\frac{2\pi
irF(v)}{\ell^{k}}\right)=\exp\left(\frac{2\pi irF(1)}{\ell^{k}}\right)$ for
all $v$ coprime to $\ell$. Thus, $F(v)\equiv F(1)\ (\mathrm{mod}\ \ell^{k})$
and $\ell^{k}\mid M$. It follows that $X_{\ell}=\\{k\in\mathbb{N}:k\leq
D_{\ell}\\}$; in particular, $X_{\ell}=\emptyset$ if $D_{\ell}=0$. This gives
the evaluation of the sum when $\ell\nmid M$. Otherwise,

$\sum\limits_{k\in
X_{\ell}}H(\ell,k,J)=\sum\limits_{k=1}^{D_{\ell}}\sum\limits_{\overset{0<r<\ell^{k}}{\ell\nmid
r}}\exp\left(\frac{-2\pi ir(w-F(1)J)}{\ell^{k}}\right)$
$=\sum\limits_{k=1}^{D_{\ell}}\left[\sum\limits_{r\mod{\ell^{k}}}\exp\left(\frac{-2\pi
ir(w-F(1)J)}{\ell^{k}}\right)-\sum\limits_{r\mod{\ell^{k-1}}}\exp\left(\frac{-2\pi
ir(w-F(1)J)}{\ell^{k-1}}\right)\right]$
$=-1+\left[\sum\limits_{r\mod{\ell^{D_{\ell}}}}\exp\left(\frac{-2\pi
ir(w-F(1)J)}{\ell^{D_{\ell}}}\right)\right].$

By the orthogonality relation for additive characters, the sum on $r$ is
$\ell^{D_{\ell}}$ if $F(1)J\equiv w\ (\mathrm{mod}\ \ell^{D_{\ell}})$ and 0
otherwise. ∎

To study the sum over $Y_{\ell}$ and $Z_{\ell}$, we consider
$B(\ell,k)=\frac{C_{F}\ell^{k(1-\frac{1}{d+1})}}{\varphi(\ell^{k})}$ as a
continuous function on the domain $\\{(\ell,k):\text{$\ell\geq 2$ and $k\geq
1$}\\}$. It is easy to see that $B(\ell,k)$ is a monotonically decreasing
function of both $\ell$ and $k$. Thus, for every fixed $\ell$, $B(\ell,k)<1$
for all large $k$. We record the following consequences:

  * •

For every fixed $\ell$, there exist a smallest positive integer $E_{\ell}$ not
in $X_{\ell}$ such that $B(\ell,E_{\ell})<1$. Then
$Y_{\ell}=\\{k\in\mathbb{N}:E_{\ell}\leq k\leq e\\}$.

  * •

There exists a value of $k$, bounded in terms of $F$, with $B(\ell,k)<1$ for
all $\ell\geq 2$. Thus, $E_{\ell}\ll_{F}1$.

  * •

For large $\ell$, we have $B(\ell,1)<1$. Since $X_{\ell}$ is empty when
$\ell>M$, $E_{\ell}=1$ for all large $\ell$. For these $\ell$, we have
$Y_{\ell}=S_{\ell}$ and $Z_{\ell}=\emptyset$. We fix $\ell_{0}=\ell_{0}(F)$
such that $Y_{\ell}=S_{\ell}$ and $2C_{F}\leq\ell^{\frac{1}{2(d+1)}}$ for all
$\ell>\ell_{0}$.

If $k\not\in X_{\ell}$,
$\left\lvert\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}\frac{\exp\left(\frac{2\pi
irF(v)}{\ell^{k}}\right)}{\varphi(\ell^{k})}\right\rvert<1$. So we may find a
positive $\eta\gg_{F}1$, satisfying for all $\ell\leq\ell_{0}$,

$(1-\eta)\geq\begin{cases}B(\ell,k)&\text{if $k\geq E_{\ell}$},\\\
\left\lvert\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}\frac{\exp\left(\frac{2\pi
irF(v)}{\ell^{k}}\right)}{\varphi(\ell^{k})}\right\rvert&\text{if
$D_{\ell}<k<E_{\ell}$}.\end{cases}$

######  Lemma 4.4.

As $x\rightarrow\infty$ (which coincides with $J\rightarrow\infty$),

$\sum\limits_{\ell\mid q}\left\lvert\sum\limits_{k\in
Y_{\ell}}H(\ell,k,J)\right\rvert=o(1).$

###### Proof.

Using Corollary 3.3, as $J\rightarrow\infty$,

$\left\lvert\sum\limits_{k\in
Y_{\ell}}H(\ell,k,J)\right\rvert\leq\sum\limits_{k=E_{\ell}}^{\infty}\left[\frac{C_{F}\ell^{k(1-\frac{1}{d+1})}}{\varphi(\ell^{k})}\right]^{J}\varphi(\ell^{k})\ll\left[\frac{C_{F}\ell^{E_{\ell}(1-\frac{1}{d+1})}}{\varphi(\ell^{E_{\ell}})}\right]^{J}\ell^{E_{\ell}}\leq\left[\frac{C_{F}\ell^{(1-\frac{1}{d+1})}}{\ell-1}\right]^{J}\ell^{E_{\ell}}.$

We used the fact that
$\sum\limits_{k=E_{\ell}}^{\infty}\left[\frac{C_{F}\ell^{k(1-\frac{1}{d+1})}}{\varphi(\ell^{k})}\right]^{J}\varphi(\ell^{k})$
is a geometric series with ratio $\ell^{(1-\frac{J}{d+1})}$. As $J$ gets
large, $\ell^{(1-\frac{J}{d+1})}\leq 2^{(1-\frac{J}{d+1})}\leq\frac{1}{2}$ so
the sum of the geometric series is at most twice the first term.

If we assume that $\ell>\ell_{0}$, then $E_{\ell}=1$ and
$\frac{C_{F}\ell^{(1-\frac{1}{d+1})}}{\ell-1}\leq\ell^{-\frac{1}{2(d+1)}}$ so
that

$\left\lvert\sum\limits_{k\in
Y_{\ell}}H(\ell,k,J)\right\rvert=O\left(\ell^{1-\frac{J}{2(d+1)}}\right)=O(\ell^{-\frac{J}{4(d+1)}}).$
(5)

For the exceptional $\ell\leq\ell_{0}$, as $J\rightarrow\infty$,

$\left\lvert\sum\limits_{k\in
Y_{\ell}}H(\ell,k,J)\right\rvert\ll\left[\frac{C_{F}\ell^{E_{\ell}(1-\frac{1}{d+1})}}{\varphi(\ell^{E_{\ell}})}\right]^{J}\ell^{E_{\ell}}=O_{F}((1-\eta)^{J})$
(6)

recalling that $E_{\ell}\ll_{F}1$. It follows from (5) and (6) that
$\sum\limits_{\ell\mid q}\left\lvert\sum\limits_{k\in
Y_{\ell}}H(\ell,k,J)\right\rvert=o(1)$ as $J\rightarrow\infty$. ∎

######  Lemma 4.5.

As $x\rightarrow\infty$,

$\sum\limits_{\ell\mid q}\left\lvert\sum\limits_{k\in
Z_{\ell}}H(\ell,k,J)\right\rvert=o(1).$

###### Proof.

Since $Z_{\ell}$ is empty for $\ell>\ell_{0}$, we may restrict attention to
$\ell\leq\ell_{0}$. By definition of $D_{\ell}$ and $E_{\ell}$, we can write
$Z_{\ell}=\\{k\in\mathbb{N}:D_{\ell}<k<E_{\ell}\\}$. Thus, recalling that
$E_{\ell}\ll_{F}1$,

$\left\lvert\sum\limits_{k\in
Z_{\ell}}H(\ell,k,J)\right\rvert\leq(E_{\ell}-D_{\ell})\max\limits_{k\in
Z_{\ell}}\left\\{\left\lvert\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}\frac{\exp\left(\frac{2\pi
irf(v)}{\ell^{k}}\right)}{\varphi(\ell^{k})}\right\rvert^{J}\varphi(\ell^{k})\right\\}$
$<(\ell^{E_{\ell}}\cdot
E_{\ell})\cdot\max\limits_{D_{\ell}<k<E_{\ell}}\left\\{\left\lvert\sum\limits_{\overset{v\mod\ell^{k}}{(v,\ell)=1}}\frac{\exp\left(\frac{2\pi
irf(v)}{\ell^{k}}\right)}{\varphi(\ell^{k})}\right\rvert\right\\}^{J}.$
$=O\left(\ell^{E_{\ell}}\cdot
E_{\ell}\cdot(1-\eta)^{J}\right)=O_{F}((1-\eta)^{J}).$

Hence, $\sum\limits_{\ell\mid q}\left\lvert\sum\limits_{k\in
Z_{\ell}}H(\ell,k,J)\right\rvert=o(1)$ as $J\rightarrow\infty$. ∎

We can write,

$\\#V_{\ell^{e}}=\frac{\varphi(\ell^{e})^{J}}{\ell^{e}}\Big{[}1+\sum\limits_{k\in
X_{\ell}}H(\ell,k,J)+\sum\limits_{k\in Y_{\ell}}H(\ell,k,J)+\sum\limits_{k\in
Z_{\ell}}H(\ell,k,J)\Big{]}.$

Multiplying over all $\ell$ dividing $q$,

$\\#\mathcal{V}(w)=\frac{\varphi(q)^{J}}{q}\prod\limits_{\ell\mid
q}\left[1+\sum\limits_{k\in X_{\ell}}H(\ell,k,J)+\sum\limits_{k\in
Y_{\ell}}H(\ell,k,J)+\sum\limits_{k\in Z_{\ell}}H(\ell,k,J)\right].$

To proceed, we observe that $\\#\mathcal{V}(w)=0$ unless $F(1)J\equiv w\
(\mathrm{mod}\ \ell^{D_{\ell}})$ for all $\ell$ dividing $M$. Indeed, since
$\ell^{D_{\ell}}$ divides $M$, we have $F(v)\equiv F(1)\ (\mathrm{mod}\
\ell^{D_{\ell}})$ for all $v$ coprime to $\ell$. So if $(v_{1},\ldots,v_{J})$
is any element of $V_{\ell^{e}}$, then (keeping in mind that $D_{\ell}\leq e$)
it must be that $F(1)J\equiv\sum\limits_{j=1}^{J}F(v_{j})\equiv w\
(\mathrm{mod}\ \ell^{D_{\ell}})$. Thus, using Lemma 4.3,

$\\#\mathcal{V}(w)=\mathbbm{1}_{F(1)J\equiv w\ (\mathrm{mod}\
M)}\frac{\varphi(q)^{J}}{q}\prod\limits_{\overset{\ell\mid q}{\ell\mid
M}}\left[\ell^{D_{\ell}}+\sum\limits_{k\in Y_{\ell}\cup
Z_{\ell}}H(\ell,k,J)\right]\prod\limits_{\overset{\ell\mid q}{\ell\nmid
M}}\left[1+\sum\limits_{k\in Y_{\ell}\cup Z_{\ell}}H(\ell,k,J)\right]$
$=\mathbbm{1}_{F(1)J\equiv w\ (\mathrm{mod}\
M)}\frac{M\varphi(q)^{J}}{q}\prod\limits_{\overset{\ell\mid q}{\ell\mid
M}}\left[1+\frac{1}{\ell^{D_{\ell}}}\sum\limits_{k\in Y_{\ell}\cup
Z_{\ell}}H(\ell,k,J)\right]\prod\limits_{\overset{\ell\mid q}{\ell\nmid
M}}\left[1+\sum\limits_{k\in Y_{\ell}\cup Z_{\ell}}H(\ell,k,J)\right].$

By Lemmas 4.4 and 4.5, $\sum\limits_{\ell\mid q}\left\lvert\sum\limits_{k\in
Y_{\ell}\cup Z_{\ell}}H(\ell,k,J)\right\rvert=o(1)$ as $x\rightarrow\infty$.
Therefore,

$\prod\limits_{\overset{\ell\mid q}{\ell\mid
M}}\left[1+\frac{1}{\ell^{D_{\ell}}}\sum\limits_{k\in Y_{\ell}\cup
Z_{\ell}}H(\ell,k,J)\right]\prod\limits_{\overset{\ell\mid q}{\ell\nmid
M}}\left[1+\sum\limits_{k\in Y_{\ell}\cup Z_{\ell}}H(\ell,k,J)\right]=1+o(1),$

and

$\\#\mathcal{V}(w)=(1+o(1))\mathbbm{1}_{F(1)J\equiv w\ (\mathrm{mod}\
M)}\frac{M\varphi(q)^{J}}{q}.$

Hence, using this asymptotic in (4),

$\sum\limits_{\overset{\text{convenient $n\leq x$}}{f(n)\equiv a\
(\mathrm{mod}\ q)}}1=\sum\limits_{m\leq
x}\frac{\\#\mathcal{V}_{m}}{\varphi(q)^{J}}\left(\frac{1}{J!}\sum\limits_{\overset{P_{1},\ldots,P_{J}\text{
distinct}}{\overset{P_{1}\cdots P_{J}\leq x/m}{\overset{\text{each
}P_{J}>L_{m}}{F(1)J\equiv
w\mod{M}}}}}1\right)+O\left(x\exp\left(-\frac{1}{8}C_{K}(\log
x)^{\epsilon/4}\right)\right)$
$=(1+o(1))\frac{M}{q}\left(\frac{1}{J!}\sum\limits_{m\leq
x}\sum\limits_{\overset{P_{1},\ldots,P_{J}\text{
distinct}}{\overset{P_{1}\cdots P_{J}\leq x/m}{\overset{\text{each
}P_{J}>L_{m}}{f(P_{1}\cdots P_{J})\equiv
a-f(m)\mod{M}}}}}1\right)+O\left(x\exp\left(-\frac{1}{8}C_{K}(\log
x)^{\epsilon/4}\right)\right).$

Here we used the fact that $F(1)J\equiv w\mod{M}$ can be rewritten as
$f(P_{1}\cdots P_{J})\equiv a-f(m)\ (\mathrm{mod}\ M)$ because $F(v)\equiv
F(1)\ (\mathrm{mod}\ M)$ for all $v$ coprime to $M$. Since $M\ll_{F}1$,
$P_{j}$ is coprime to $M$ for all $1\leq j\leq J$ (for $y$ large). By (2),
using the fact that $M\ll_{F}1$,

$\sum\limits_{\overset{\text{convenient $n\leq x$}}{f(n)\equiv a\
(\mathrm{mod}\ q)}}1=\frac{M}{q}\sum\limits_{\overset{\text{convenient }n\leq
x}{f(n)\equiv a\ (\mathrm{mod}\ M)}}1+o\left(\frac{x}{q}\right).$ (7)

Since $q\in\mathcal{S}_{f}$, $M\mid q$ and $M\ll_{F}1$, Theorem 1.1 implies

$\sum\limits_{\overset{n\leq x}{f(n)\equiv a\ (\mathrm{mod}\
M)}}1\sim\frac{1}{M}\sum\limits_{n\leq x}1.$ (8)

Using Lemma 4.1 and the fact that $1\leq M\ll_{F}1$,

$\displaystyle\sum\limits_{\overset{n\leq x}{f(n)\equiv a\ (\mathrm{mod}\
M)}}1=\sum\limits_{\overset{\text{convenient }n\leq x}{f(n)\equiv a\
(\mathrm{mod}\ M)}}1+o\left(\frac{x}{M}\right)$ and
$\displaystyle\frac{1}{M}\sum\limits_{n\leq
x}1=\frac{1}{M}\sum\limits_{\text{convenient $n\leq
x$}}1+o\left(\frac{x}{M}\right).$

Thus, by (8), $\sum\limits_{\overset{\text{convenient }n\leq x}{f(n)\equiv a\
(\mathrm{mod}\ M)}}1=\frac{1}{M}\sum\limits_{\text{convenient $n\leq
x$}}1+o\left(\frac{x}{M}\right)$, which along with (7) proves (1) and Theorem
1.2 follows.

## References

  * [1] W. Narkiewicz.  Uniform distribution of sequences of integers in residue classes, volume 1087 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 1984.
  * [2] S. S. Pillai.  Generalisation of a theorem of Mangoldt.  Proc. Indian Acad. Sci., Sect. A., 11:13–20, 1940.
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  * [6] T. Cochrane and Z. Zheng.  A survey on pure and mixed exponential sums modulo prime powers.  In Number theory for the millennium, I (Urbana, IL, 2000), pages 273–300. A K Peters, Natick, MA, 2002.
  * [7] W. K. A. Loh.  Exponential sums on reduced residue systems.  Canad. Math. Bull., 41(2):187–195, 1998.
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  * [10] S. V. Konyagin.  Letter to the editors: “The number of solutions of congruences of the $n$th degree with one unknown” [Mat. Sb. (N.S.) 109(151) (1979), no. 2, 171–187, 327; MR 80k:10013a].  Mat. Sb. (N.S.), 110(152)(1):158, 1979.


JLAB-THY-21-3465

CJ15 global PDF analysis with new electroweak data
from the STAR and SeaQuest experiments

Sanghwa Park1$\star$, Alberto Accardi2,3, Xiaoxian Jing3,4, J.F. Owens5

1 Stony Brook University, Stony Brook, NY, USA

2 Hampton University, Hampton, VA, USA

3 Jefferson Lab, Newport News, VA, USA

4 Southern Methodist University, Dallas, TX, USA

5 Florida State University, Tallahassee, FL, USA

*<EMAIL_ADDRESS>
Proceedings for the XXVIII International Workshop
on Deep-Inelastic Scattering and Related Subjects,
Stony Brook University, New York, USA, 12-16 April 2021
10.21468/SciPostPhysProc.?

## Abstract

We present updates to a recent CTEQ-Jefferson Lab (CJ) global analysis of
parton distribution functions with a new set of electroweak data that provide
unique access to quark flavor separation in the proton. In particular, recent
$\boldsymbol{W}$ and $\boldsymbol{Z}$ boson measurements from the STAR
experiment at RHIC put additional constraints on light quarks and antiquarks
near the valence regime. The new measurement of the Drell-Yan lepton pair
production ratio in $\boldsymbol{p+p}$ and $\boldsymbol{p+d}$ collisions by
the SeaQuest experiment at Fermilab extends the large-$\boldsymbol{x}$
coverage of the previous E866 experiment and sheds new light on the light
antiquarks distribution. In this report, the impact of these new data sets on
parton distribution functions will be presented with emphasis given to the
flavor asymmetry of the light antiquark sea at large values of the parton
momentum $\boldsymbol{x}$.

##  1 Introduction

The flavor asymmetry of the light antiquark sea inside the proton is of great
importance to understand the structure of the nucleon, see, for example, [1,
2] for a review. In particular, an excess of $\bar{d}$ over $\bar{u}$
antiquarks has been observed at moderate values of the parton momentum
fraction $x$ in Deep Inelastic Scattering and Drell-Yan experiments on proton
and deuteron targets. In particular, the measurement of the Drell-Yan
production ratio in $p+p$ and $p+d$ collisions by E866 [3, 4] suggested a
potential sign change of the flavor asymmetry ($\bar{d}-\bar{u}$) at $x\approx
0.25$ region, albeit with large uncertainty – a behavior that is difficult to
understand theoretically and has generated a lot of discussion in the past 20
years.

New data released this year will afford us a fresh look into this issue.
Recent measurements by the SeaQuest collaboration [5] provide new insight into
the light antiquark distributions by extending the large $x$ coverage of
earlier experiments. At the same time new data on $W$ boson production in
$p+p$ collisions by the STAR collaboration [6] offer an alternative way to
probe the light antiquark sea in the proton at $x\approx 0.2$. Indeed, due to
the $V-A$ structure of the weak interaction the $W$ boson only couples to
left-handed quarks and right-handed antiquarks, and correlates the charge of
the produced $W$ with initial state quark flavors. The cross sections and
charge ratio of $W^{+}$ and $W^{-}$ bosons production can thus provide direct
sensitivity to the flavor separated quark distribution functions. Unlike the
Drell-Yan and DIS measurements which use deuteron targets to extract
$\bar{d}/\bar{u}$, the STAR measurement is free from nuclear effects and
naturally probes the proton quarks at a larger scale $Q^{2}=M_{W}^{2}$ set by
the $W$ boson mass – but at slightly smaller values of $x$ than the SeaQuest
experiment.

Here we report recent studies of light antiquark sea distributions performed
in the CTEQ-JLab (CJ) collaboration’s global QCD analysis framework using, for
the first time, the new $W$ boson cross section measurements by STAR and
Drell-Yan measurements by SeaQuest.

##  2 Weak boson production in proton-proton collisions

At leading order in perturbative QCD (pQCD), the charge ratio of $W$ cross
sections is sensitive to light quarks and antiquarks,

$\frac{\sigma_{W^{+}}}{\sigma_{W^{-}}}\ \approx\
\frac{u(x_{1})\bar{d}(x_{2})+\bar{d}(x_{1})u(x_{2})}{d(x_{1})\bar{u}(x_{2})+\bar{u}(x_{1})d(x_{2})}\
\underset{y_{{}_{W}}\approx 0}{\approx}\ \frac{\bar{d}/\bar{u}}{d/u}$ (1)

where $x_{1(2)}=(M_{{}_{W}}/\sqrt{s})\exp(\pm y_{{}_{W}})$ is the fractional
momentum carried by the scattering partons, with $y_{{}_{W}}$ the rapidity of
the produced boson. At midrapidity, where $x_{1}=x_{2}=M_{{}_{W}}/\sqrt{s}$,
the cross section ratio directly accesses the antiquark and quark ratios.

At the Relativistic Heavy Ion Collider (RHIC), where measurements have been
performed at $\sqrt{s}$ = 500 and 510 GeV [6], $W$ bosons are identified via
their lepton $W^{\pm}\rightarrow e^{\pm}\nu$ decays, without detecting the
accompanying (anti)neutrino. Combined with decay kinematics, the observed $W$
cross sections at lepton pseudorapidity $-1\leq\eta_{e}\leq 1.5$ are sensitive
to the unpolarized quark distributions in the region $0.05<x<0.3$. We will
also discuss $Z$ boson production, for which the boson rapidity can be fully
reconstructed via the dilepton $Z\rightarrow e^{+}e^{-}$ decay at
$|y_{{}_{Z}}|\leq 1$.

##  3 Drell-Yan lepton pair production in proton-hadron collisions

At leading order, the Drell-Yan lepton pair production in proton-hadron
collisions can be described as the annihilation of a quark-antiquark pair into
a virtual photon which then decays into a pair of leptons. Assuming charge
symmetry and small deuteron nuclear corrections, the cross section ratio for
lepton pair production in $p+p$ and $p+d$ collisions in the forward region
($x_{F}=x_{b}-x_{t}\gg 0$) can be approximated by

$\frac{\sigma^{pd}}{\sigma^{pp}}\approx
1+\frac{\bar{d}(x_{t})}{\bar{u}(x_{t})}$ (2)

where $x_{b}$ and $x_{t}$ are the fractions of the dilepton pair momentum
relative to, respectively, the momentum of the beam and of the target hadrons.

The E866 and SeaQuest experiments both used a proton beam against LD2 and LH2
targets. E866 measured the ratio in the kinematic range of $0.015<x_{t}<0.3$
and an average $Q^{2}$ of 54 GeV2. The recently released SeaQuest data covers
the range of $0.1<x_{t}<0.45$ and $Q^{2}$ of 22–40 GeV2, with improved
precision and an evaluation of correlated systematic uncertainties.

##  4 Light Antiquarks in the CJ15 Global Analysis

The latest CJ15 global fit (CJ15) has been performed using the world DIS data
set, as well as a variety of jet and electroweak boson production measurements
[7]. Among these, Drell-Yan measurements by the Fermilab’s E866 experiment
provide the strongest constraints on the light antiquark sea, covering the
$0.015\lesssim x<\lesssim 0.3$ parton momentum region.

At the input scale of $Q^{2}_{0}=3$ GeV2, a standard five-parameter functional
form is used for most of parton species, including the $\bar{u}+\bar{d}$
combination:

$xf(x,Q^{2}_{0})=a_{0}x^{a_{1}}(1-x)^{a_{2}}(1+a_{3}\sqrt{x}+a_{4}x)\ .$ (3)

In the original CJ15 fit, due to the limited $x$ coverage of the E866 data and
the sharp downturn these required of the $\bar{d}/\bar{u}$ ratio, this was
directly parametrized as

$\bar{d}/\bar{u}=a_{0}x^{a_{1}}(1-x)^{a_{2}}+1+a_{3}x(1-x)^{a_{4}}\ ,$ (4)

enforcing the theoretical expectation from most modeling efforts that
$\bar{d}/\bar{u}\to 1$ as $x\to 1$. This assumption was however revisited in
Ref. [8], where the $\bar{d}-\bar{u}$ difference was considered instead of
$\bar{d}/\bar{u}$ and parametrized as in Eq. (3), and the resulting fit was
called CJ15-a. Even if the new parametrization allowed for it, no strong
indication of a sign change in the $\bar{d}-\bar{u}$ asymmetry in the
$x\lesssim 0.3$ region measured by E866 was, however, found.

With the new STAR data sensitive to the smaller-$x$ rise of the
$\bar{d}-\bar{u}$ asymmetry, and the new SeaQuest data constraining the
$\bar{d}/\bar{u}$ ratio at $0.15\lesssim x\lesssim 0.4$, well across the
region where E866 indicated this would drop below 1, we can now revisit this
whole issue. We therefore performed a new fit tentatively called CJ15-a+ that
utilized the more flexible parametrization of CJ15-a but included the STAR
$W^{+}/W^{-}$ cross section ratio and the SeaQuest Drell-Yan proton to
deuteron target cross section ratio on top of the standard CJ15 data set. (The
impact of the charge separated $W$ cross section and of the $Z$ cross section
on the determination of light quark PDF was found to be minimal, and these
data excluded from the fit.) The results are discussed in the next Section.

##  5 Results

Figure 1: A comparison of the CJ15-a+ NLO calculations to the measured
$W^{+}\rightarrow e^{+}$ (a), $W^{-}\rightarrow e^{-}$ (b), $Z\rightarrow
e^{+}e^{-}$ (c) cross sections, and $(W^{+}\rightarrow
e^{+})/(W^{-}\rightarrow e^{-})$ charge ratio (d) by STAR. Note that only the
latter was included in the PDF fit.

Figure 1 compares the CJ15-a+ NLO calculation of $W^{\pm}\rightarrow e^{\pm}$
and $Z/\gamma^{*}\rightarrow e^{+}e^{-}$ differential cross sections and the
lepton charge ratio, $\sigma_{W^{+}\rightarrow e^{+}}/\sigma_{W^{-}\rightarrow
e^{-}}$, to the STAR measurements. We used the MCFM Monte Carlo event
generator [9] interfaced with LHAPDF [10] and APPLgrid [11] to include NLO QCD
corrections. The experimental cut of $25<E^{e}_{T}<50$ GeV for $W$ decay
electrons and transverse momentum requirement of decay electrons $p^{e}_{T}>$
15 GeV/c as well as the invariant mass range of 70 GeV $<M_{e^{+}e^{-}}<$ 110
GeV for $Z$ decay electron pairs are applied respectively. As a result, a fast
NLO interpolation grid was produced which was then incorporated into the CJ
framework.

The new CJ15-a+ fit describes reasonably well the $W$ charge ratio data, which
was included in the fitted data set, as well as the data for the $W^{+}\to
e^{+}$ and $Z\to e^{+}e^{-}$ differential cross sections. However, the
measured $W^{-}\to e^{-}$ cross section shows a more pronounced oscillatory
shape around midrapidity than the perturbative calculation is allowed to
display by the fitted global data set. Furthermore, the highest pseudorapidity
point in the $W^{+}$ cross section (and consequently in the lepton charge
ratio) falls way below the theoretical calculation.

Figure 2: Cross section ratio of Drell-Yan lepton pair production in $p+d$ and
$p+p$ collisions by E866 and SeaQuest experiments compared with CJ15 NLO
calculations before (left) and after (right) including the SeaQuest and STAR
data.

The comparison of calculations and data for Drell-Yan lepton pair production
is shown in Fig. 2 both before (left) and after (right) including the SeaQuest
and STAR data into our fits. When only E866 data are fitted, the
$\sigma^{pd}/\sigma^{pp}$ cross section ratio exhibits a steeper downturn than
allowed by the SeaQuest data, see the left panel. As the right panel shows,
however, including the SeaQuest data convincingly brings the ratio above 1 and
substantially reduces the calculation’s PDF uncertainty.

Figure 3: $\bar{d}$ over $\bar{u}$ ratio at $Q^{2}$ = 10 GeV2 extracted from
the CJ15 PDF fits before (CJ15-a, black solid line) and after (CJ15-a+, red
solid line) adding the STAR $W$ lepton charge ratio data in $p+p$ collisons
and the SeaQuest Drell-Yan lepton pair production ratio in $p+p$ and $p+d$
collisions on top of the data sets used for the standard CJ15 PDF analysis.

Finally, in Figure 3 we plot the $\bar{d}/\bar{u}$ distribution obtained after
including the STAR lepton charge ratio and the new Drell-Yan lepton pair
production ratio by SeaQuest. Compared to the CJ15-a fit, which only included
the E866 data, one can see that $\bar{d}/\bar{u}$ ratio remains above 1 at all
values of $x$ and relaxes to 1 as the parton momentum fraction increases
towards its maximum value. At large $x$ the antiquark ratio is pulled up by
the SeaQuest data, that display a moderate $\approx 2$–$3\sigma$ tension with
respect to E866. On the contrary, STAR data are compatible with the E866
measurement around $x\approx 0.2$, where they induce a marginal suppression of
the previously fitted antiquark ratio. Both experiments clearly reduce the PDF
uncertainty of the $\bar{d}/\bar{u}$ ratio, with STAR data also impacting the
$d/u$ ratio at the $\sim 10\%$ level (not shown here).

##  6 Conclusion

We have performed a global PDF analysis with new weak boson data by the STAR
collaboration at RHIC and Drell-Yan lepton pair production ratio data by the
SeaQuest experiment at Fermilab, and studied the impact of these data sets on
the $\bar{d}/\bar{u}$ distribution. This is the first time the RHIC weak boson
data were included in a global unpolarized PDF analysis. Overall the
sensitivity of the $W$ data is smaller than the Drell-Yan data due to the
limited coverage in $x$ and relatively large experimental uncertainty, however
it provides additional constraints on the light quark distributions in the
intermediate $0.1\lesssim x\lesssim 0.3$ region. With the addition of the
large-$x$ SeaQuest data, the new CJ PDF fit shows that the $\bar{d}/\bar{u}$
ratio stays above unity over the whole measured region, and is compatible with
$\bar{d}/\bar{u}\to 1$ as $x\to 1$.

## Acknowledgements

We would like to thank J. Bane, S. Fazio, M. Posik, and A. Tadepalli for
useful discussions on their experimental measurements.

### Funding information.

This work was supported in part by the U.S. Department of Energy (DOE)
contract DE-AC05-06OR23177, under which Jefferson Science Associates LLC
manages and operates Jefferson Lab. AA also acknowledges support from DOE
contract DE-SC0008791. X. Jing was partially supported by DOE Grant No. DE-
SC0010129. S. Park acknowledges support from DOE contract DE-FG02-05ER41372
and the Center for Frontiers in Nuclear Science.

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