1 Introduction and statement of results
For any
$k\geqslant 2$
,
$q\geqslant 3$
, and integers
$(a_{j},q)=1~(1\leqslant j\leqslant k)$
, we consider the number of integers
${\leqslant}x$
which can be written as product of
$k$
distinct primes
$p_{1}p_{2}\cdots p_{k}$
with
$p_{j}\equiv a_{j}\hspace{0.2em}{\rm mod}\hspace{0.2em}q$
$(1\leqslant j\leqslant k)$
. Here when we count the number of such integers, we allow any ordering of the prime factors.
Ford and Sneed [Reference Ford and Sneed5] investigated subtle biases in the distribution of the product of two primes in different arithmetic progressions subject to the generalized Riemann hypothesis (GRH) and the linear independence (LI) conjecture on the imaginary parts of the zeros of Dirichlet L-functions. Recently, the author [Reference Meng12] generalized their results to study the bias of numbers composed of
$k$
(
${\geqslant}2$
) prime factors (either counted with or without multiplicity) in two different arithmetic progressions. For each fixed
$k$
and
$q$
, different arithmetic progressions contain virtually the same number of such integers below
$x$
, indeed, under the GRH, the number equals
$(1/\unicode[STIX]{x1D719}(q))(x/\text{log}\,x)((\log \log x)^{k-1}/(k-1)!)+O(x^{1/2+o(1)})$
.
It is reasonable to expect that these integers break up very evenly, with errors of size
$O(x^{1/2+o(1)})$
, when one specifies further which arithmetic progression modulo
$q$
each prime factor lies in. However, this is not the case. Dummit, Granville and Kisilevsky [Reference Dummit, Granville and Kisilevsky3] showed that there is a very large bias for the odd integers
$p_{1}p_{2}\leqslant x$
with two prime factors satisfying
$p_{1}\equiv p_{2}\equiv 3\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$
. More precisely, they showed that
$$\begin{eqnarray}\frac{\#\{p_{1}p_{2}\leqslant x:p_{1}\equiv p_{2}\equiv 3\hspace{0.2em}{\rm mod}\hspace{0.2em}4\}}{(1/4)\#\{p_{1}p_{2}\leqslant x\}}=1+\frac{c+o(1)}{\log \log x},\end{eqnarray}$$
for some positive constant
$c$
. The authors exhibit a similar bias for products of 2 primes, where
$\unicode[STIX]{x1D712}_{q}(p_{1})=\unicode[STIX]{x1D712}_{q}(p_{2})=\unicode[STIX]{x1D702}$
,
$\unicode[STIX]{x1D712}_{q}$
is a quadratic Dirichlet character with fixed conductor
$q$
, and
$\unicode[STIX]{x1D702}\in \{-1,1\}$
. If
$q$
is allowed to grow with
$x$
, they further conjecture that the bias may be a bit larger. Recently, Hough [Reference Hough7] confirmed their conjecture and showed that, for
$\unicode[STIX]{x1D702}\in \{-1,1\}$
, there exist many
$q\leqslant x$
for which
$$\begin{eqnarray}\displaystyle \frac{\#\{p_{1}p_{2}\leqslant x:\unicode[STIX]{x1D712}_{q}(p_{1})=\unicode[STIX]{x1D712}_{q}(p_{2})=\unicode[STIX]{x1D702}\}}{(1/4)\#\{p_{1}p_{2}\leqslant x:(p_{1}p_{2},q)=1\}} & & \displaystyle \nonumber\end{eqnarray}$$
is at least as large as
$$\begin{eqnarray}\displaystyle 1+\frac{\log \log \log x+O(1)}{\log \log x}. & & \displaystyle \nonumber\end{eqnarray}$$
On the other hand, Moree [Reference Moree15] considered all the integers with every prime factor from the same arithmetic progression
$a\hspace{0.2em}{\rm mod}\hspace{0.2em}q$
, and proved that there is a large bias towards certain residue classes
$a\hspace{0.2em}{\rm mod}\hspace{0.2em}q$
.
In this paper, we generalize the large bias results found in [Reference Dummit, Granville and Kisilevsky3] to products of any
$k\geqslant 2$
primes and any fixed modulus
$q\geqslant 3$
, and prove uniform estimates in a large range of
$k$
. For any fixed
$A>0$
and fixed
$q\geqslant 3$
, we prove an asymptotic formula uniformly for
$2\leqslant k\leqslant A\log \log x$
for the number of integers
$p_{1}\cdots p_{k}\leqslant x$
with
$p_{j}\equiv a_{j}\hspace{0.2em}{\rm mod}\hspace{0.2em}q~(1\leqslant j\leqslant k)$
. We show that, there are large biases for some arithmetic progressions
$(a_{1}\hspace{0.2em}{\rm mod}\hspace{0.2em}q,\ldots ,a_{k}\hspace{0.2em}{\rm mod}\hspace{0.2em}q)$
, and such a phenomenon has connections with Mertens’ theorem and the least prime in arithmetic progressions.
Let
$\mathbf{a}:=(a_{1},a_{2},\ldots ,a_{k})\in (\mathbb{Z}/q\mathbb{Z})^{k}$
with
$(a_{j},q)=1$
for all
$1\leqslant j\leqslant k$
. One may regard the vector
$\mathbf{a}$
as an unordered
$k$
-tuple, or as a multiset. Denote
$$\begin{eqnarray}\displaystyle M_{k}(x;\mathbf{a}) & := & \displaystyle \#\{\!n\leqslant x:n=p_{1}p_{2}\cdots p_{k},p_{j}~\text{distinct primes},\nonumber\\ \displaystyle & & \displaystyle \quad p_{j}\equiv a_{j}\hspace{0.2em}{\rm mod}\hspace{0.2em}q,(a_{j},q)=1,1\leqslant j\leqslant k\!\},\nonumber\end{eqnarray}$$
where the prime factors
$p_{j}$
can be in any order, and
$$\begin{eqnarray}\displaystyle S_{k}(x) & := & \displaystyle \#\{\!n\leqslant x:n=p_{1}p_{2}\cdots p_{k},(p_{1}p_{2}\cdots p_{k},q)=1,\nonumber\\ \displaystyle & & \displaystyle \quad p_{j}~\text{distinct primes},1\leqslant j\leqslant k\!\}.\nonumber\end{eqnarray}$$
Let
$\unicode[STIX]{x1D712}$
be a Dirichlet character modulo
$q$
, and
$\unicode[STIX]{x1D712}_{0}$
be the principal character modulo
$q$
. Denote
$$\begin{eqnarray}C(q,a):=\lim _{x\rightarrow \infty }\Bigg(\unicode[STIX]{x1D719}(q)\mathop{\sum }_{\substack{ p\leqslant x \\ p\equiv a\hspace{0.2em}{\rm mod}\hspace{0.2em}q}}\frac{1}{p}-\mathop{\sum }_{\substack{ p\leqslant x \\ (p,q)=1}}\frac{1}{p}\Bigg)=\mathop{\sum }_{\unicode[STIX]{x1D712}\neq \unicode[STIX]{x1D712}_{0}}\bar{\unicode[STIX]{x1D712}}(a)\mathop{\sum }_{p}\frac{\unicode[STIX]{x1D712}(p)}{p}.\end{eqnarray}$$
We will see in our theorems that this constant
$C(q,a)$
reflects the bias in our problem.
Our first result is for the special case when
$a_{1}=a_{2}=\cdots =a_{k}=a$
. In other words, all the
$k$
prime factors are from the same residue class
$a\hspace{0.2em}{\rm mod}\hspace{0.2em}q$
.
Theorem 1. Let
$q\geqslant 3$
be fixed, and
$\mathbf{a}=(a,a,\ldots ,a)$
,
$(a,q)=1$
. We have the following results.
-
(i) For fixed
$k\geqslant 2$
,
$$\begin{eqnarray}\frac{M_{k}(x;\mathbf{a})}{(1/\unicode[STIX]{x1D719}^{k}(q))S_{k}(x)}=1+\frac{(k-1)C(q,a)}{\log \log x}+O_{q,k}\biggl(\frac{1}{(\log \log x)^{2}}\biggr).\end{eqnarray}$$
-
(ii) If
$k=o(\log \log x)$
, as
$x\rightarrow \infty$
,
$$\begin{eqnarray}\frac{M_{k}(x;\mathbf{a})}{(1/\unicode[STIX]{x1D719}^{k}(q))S_{k}(x)}=1+\frac{(k-1)(C(q,a)+o(1))}{\log \log x}.\end{eqnarray}$$
-
(iii) For fixed
$A>0$
and
$k\sim A\log \log x$
, we have, as
$x\rightarrow \infty$
,
$$\begin{eqnarray}\frac{M_{k}(x;\mathbf{a})}{(1/\unicode[STIX]{x1D719}^{k}(q))S_{k}(x)}\sim \mathop{\prod }_{p}\frac{1+(A\unicode[STIX]{x1D719}(q)\unicode[STIX]{x1D7D9}_{p\equiv a\hspace{0.2em}{\rm mod}\hspace{0.2em}q}(p)/p)}{1+(A\unicode[STIX]{x1D712}_{0}(p)/p)}.\end{eqnarray}$$
Remark 1. If
$k$
is fixed, by Lemmas 9 and 10,
$M_{k}(x;\mathbf{a})$
and
$(1/\unicode[STIX]{x1D719}^{k}(q))S_{k}(x)$
have main terms of the same order which is
$(1/\unicode[STIX]{x1D719}^{k}(q))(x/\text{log}\,x)((\log \log x)^{k-1}/(k-1)!)$
with different secondary terms and hence the bias is determined by the constant
$C(q,a)$
. Thus, we see that, as
$k$
increases, the bias will become larger and larger.
For
$k\sim A\log \log x$
, the main terms of
$M_{k}(x;\mathbf{a})$
and
$(1/\unicode[STIX]{x1D719}^{k}(q))S_{k}(x)$
have the same order of magnitude but with different coefficients. One may compare this with the result of Moree [Reference Moree15] who showed that the counting function
$N(x;q,a):=\#\{n\leqslant x:p|n\Rightarrow p\equiv a\hspace{0.2em}{\rm mod}\hspace{0.2em}q\}$
satisfies
$N(x;q,a)\sim B_{q,a}x/(\log x)^{1-1/\unicode[STIX]{x1D719}(q)}$
for some positive constant
$B_{q,a}$
depending on
$q$
and
$a$
, and in particular,
$N(x;4,3)\geqslant N(x;4,1)$
holds for all
$x$
.
For the general case, assume there are
$l$
distinct values
$b_{1},\ldots ,b_{l}$
in the coordinates of
$\mathbf{a}$
. Fix
$l$
, for each
$1\leqslant j\leqslant l$
, let
$k_{j}$
be the number of prime factors congruent to
$b_{j}\hspace{0.2em}{\rm mod}\hspace{0.2em}q$
. Then
$\sum _{j=1}^{l}k_{j}=k$
.
Theorem 2. Let
$q\geqslant 3$
be fixed. Then, for fixed
$k\geqslant 2$
,
$$\begin{eqnarray}\displaystyle \frac{M_{k}(x;\mathbf{a})}{(1/\unicode[STIX]{x1D719}^{k}(q))(k!/k_{1}!k_{2}!\cdots k_{l}!)S_{k}(x)} & = & \displaystyle 1+\frac{k-1}{\log \log x}\frac{1}{k}\mathop{\sum }_{j=1}^{k}C(q,a_{j})\nonumber\\ \displaystyle & & \displaystyle +\,O_{q,k,l}\biggl(\frac{1}{(\log \log x)^{2}}\biggr).\nonumber\end{eqnarray}$$
Moreover, for fixed
$l$
and fixed
$A>0$
, assume
$k=\sum _{j=1}^{l}k_{j}\sim A\log \log x$
and
$e_{j}:=\lim _{x\rightarrow \infty }(k_{j}/\text{log}\,\log x)$
exists for every
$1\leqslant j\leqslant l$
. Then as
$x\rightarrow \infty$
,
$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{M_{k}(x;\mathbf{a})}{(1/\unicode[STIX]{x1D719}^{k}(q))(k!/k_{1}!k_{2}!\cdots k_{l}!)S_{k}(x)}\nonumber\\ \displaystyle & & \displaystyle \quad \sim \mathop{\prod }_{p}\frac{\mathop{\prod }_{j=1}^{l}(1+\unicode[STIX]{x1D719}(q)e_{j}\unicode[STIX]{x1D7D9}_{p\equiv b_{j}\hspace{0.2em}{\rm mod}\hspace{0.2em}q}(p)/p)}{1+A\unicode[STIX]{x1D712}_{0}(p)/p},\end{eqnarray}$$
where
$\sum _{j=1}^{l}e_{j}=A$
.
Remark 2. In the general case
$\mathbf{a}=(a_{1},\ldots ,a_{k})$
, there are
$k!/k_{1}!k_{2}!\cdots k_{l}!$
orderings of the numbers
$a_{1},\ldots ,a_{k}$
.
Remark 3. For
$k\sim A\log \log x$
, if the coordinates of
$\mathbf{a}$
cover all the reduced residue classes modulo
$q$
and all values of
$e_{j}$
are the same, then the right-hand side of (1.1) is exactly
$1$
.
1.1 Mertens’ theorem and the least prime in arithmetic progressions
The constant
$C(q,a)$
, which affects the biases in our theorems, is related to the classical Mertens’ theorem [Reference Hardy and Wright6, §22.8] and the Mertens’ theorem [Reference Mertens13] for arithmetic progressions, that
$$\begin{eqnarray}\mathop{\sum }_{p\leqslant x}\frac{1}{p}=\log \log x+\unicode[STIX]{x1D6FE}+B+O\biggl(\frac{1}{\log x}\biggr),\end{eqnarray}$$
and if
$(a,q)=1$
,
$$\begin{eqnarray}\mathop{\sum }_{\substack{ p\leqslant x \\ p\equiv a\hspace{0.2em}{\rm mod}\hspace{0.2em}q}}\frac{1}{p}=\frac{\log \log x}{\unicode[STIX]{x1D719}(q)}+M(q,a)+O\biggl(\frac{1}{\log x}\biggr),\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FE}$
is Euler’s constant,
$B:=\sum _{p}(\log (1-1/p)+1/p)$
is Mertens’ constant, and
$M(q,a)$
is a number depending on
$q$
and
$a$
. Languasco and Zaccagnini [Reference Languasco and Zaccagnini10] investigated the value of
$M(q,a)$
and other related constants. By (1.2), (1.3), and the orthogonality of Dirichlet characters, letting
$x\rightarrow \infty$
, we obtain
$$\begin{eqnarray}\displaystyle & \displaystyle C(q,a)=\unicode[STIX]{x1D719}(q)M(q,a)-\unicode[STIX]{x1D6FE}-B+\mathop{\sum }_{p|q}\frac{1}{p}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{\substack{ a\hspace{0.2em}{\rm mod}\hspace{0.2em}q \\ (a,q)=1}}M(q,a)=\unicode[STIX]{x1D6FE}+B-\mathop{\sum }_{p|q}\frac{1}{p}. & \displaystyle\end{eqnarray}$$
Hence, the value of
$M(q,a)$
determines how the bias behaves.
In particular, with the values of
$M(q,a)$
calculated by Languasco and Zaccagnini [Reference Languasco and Zaccagnini11], by (1.4), we have
$$\begin{eqnarray}\displaystyle & \displaystyle C(3,2)\approx 0.641945,\quad C(3,1)\approx -0.641945; & \displaystyle \nonumber\\ \displaystyle & \displaystyle C(4,3)\approx 0.334981,\quad C(4,1)\approx -0.334981; & \displaystyle \nonumber\\ \displaystyle & \displaystyle C(7,2)\approx 1.83747,\quad C(7,5)\approx 0.159006,\quad C(7,6)\approx -0.946269; & \displaystyle \nonumber\\ \displaystyle & \displaystyle C(13,3)\approx 2.68478,\quad C(13,6)\approx -0.846522,\quad C(13,8)\approx -1.31962. & \displaystyle \nonumber\end{eqnarray}$$
Here the interesting phenomenon is that 2 is a quadratic residue modulo 7, while 5 and 6 are quadratic non-residues modulo 7; 3 is a quadratic residue modulo 13, while 6 and 8 are quadratic non-residues modulo 13. There is no consistent preference for either quadratic non-residue classes or quadratic residue classes modulo
$q$
.
The above phenomenon is different from the biases among products of
$k$
primes studied in [Reference Ford and Sneed5] and [Reference Meng12]. Using a method similar to that in [Reference Meng12], one can show that, under the GRH and LI, the integers
$n=p_{1}\cdots p_{k}$
, which are products of exactly
$k$
distinct primes, have preference for either quadratic non-residues or quadratic residues, depending on the parity of
$k$
.
The biases in Theorems 1 and 2 ultimately stem from the fact that
$M(q,a)$
is heavily dependent on the least prime
$p(q,a)$
in the arithmetic progression
$a\hspace{0.2em}{\rm mod}\hspace{0.2em}q$
. Pomerance [Reference Pomerance19] and Norton [Reference Norton18] independently showed that
$$\begin{eqnarray}\mathop{\sum }_{\substack{ p\leqslant x \\ p\equiv a\hspace{0.2em}{\rm mod}\hspace{0.2em}q}}\frac{1}{p}-\frac{\log \log x}{\unicode[STIX]{x1D719}(q)}=\frac{1}{p(q,a)}+O\biggl(\frac{\log 2q}{\unicode[STIX]{x1D719}(q)}\biggr),\end{eqnarray}$$
where the implied constant is uniform for all
$q$
,
$a$
, and
$x\geqslant q$
.
In Theorem 2, we allow any ordering of the primes
$p_{j}$
$(1\leqslant j\leqslant k)$
, and hence the constant
$(1/k)\sum _{j=1}^{k}C(q,a_{j})$
represents the bias. One may ask, for which
$\mathbf{a}=(a_{1},\ldots ,a_{k})$
, this constant is zero? Trivially, by (1.4) and (1.5), if
$\mathbf{a}$
covers every element of the reduced residue class modulo
$q$
the same number of times,
$(1/k)\sum _{j=1}^{k}C(q,a_{j})=0$
. However, we do not know if the converse is true. Alternatively, by (1.4) and (1.5), we may consider the distribution of the values of
$M(q,a_{j})$
$(1\leqslant j\leqslant k)$
. By (1.6), it is reasonable to conjecture that all
$M(q,a_{j})$
are distinct and that, except in the trivial case, they are linearly independent over
$\mathbb{Q}$
. Hence, we propose the following open problem.
Open problem. Is the trivial case the only case for which
$(1/k)\sum _{j=1}^{k}$
$C(q,a_{j})=0$
?
The answer is yes if the numbers
$\sum _{p}(\unicode[STIX]{x1D712}(p)/p)$
are linearly independent over algebraic numbers or the numbers
$\bar{\unicode[STIX]{x1D712}}(a)\sum _{p}(\unicode[STIX]{x1D712}(p)/p)$
are linearly independent over
$\mathbb{Q}$
for all
$\unicode[STIX]{x1D712}\neq \unicode[STIX]{x1D712}_{0}\hspace{0.2em}{\rm mod}\hspace{0.2em}q$
. These values are close to
$\log L(1,\unicode[STIX]{x1D712})$
or
$\bar{\unicode[STIX]{x1D712}}(a)\log L(1,\unicode[STIX]{x1D712})$
. Baker, Birch, and Wirsing [Reference Baker, Birch and Wirsing1] showed that if
$(q,\unicode[STIX]{x1D719}(q))=1$
then the numbers
$L(1,\unicode[STIX]{x1D712})$
are linearly independent over
$\mathbb{Q}$
for all non-principal characters
$\hspace{0.2em}{\rm mod}\hspace{0.2em}~q$
. Moreover, they proved that the numbers
$L(1,\unicode[STIX]{x1D712})$
are linearly independent over algebraic numbers for non-trivial even characters
$\hspace{0.2em}{\rm mod}\hspace{0.2em}~q$
(see also [Reference Murty and Murty16, Corollary 2] or [Reference Murty and Rath17, Corollary 25.6]). For any odd Dirichlet character
$\unicode[STIX]{x1D712}$
, it is known [Reference Murty and Rath17, Lemma 25.7] that the number
$L(1,\unicode[STIX]{x1D712})$
is an algebraic multiple of
$\unicode[STIX]{x1D70B}$
.
2 Lemmas and preparations
Lemma 1 ([Reference Karatsuba8, Ch. IX, §2, Theorem 2] and [Reference Davenport2, p. 96, (12)]).
The Dirichlet
$L$
-function
$L(s,\unicode[STIX]{x1D712})$
has no zeros in the domain
$$\begin{eqnarray}\Re (s)=\unicode[STIX]{x1D70E}\geqslant 1-\frac{c_{1}}{\log q(|t|+2)},\end{eqnarray}$$
for some constant
$c_{1}>0$
, except a possible simple real zero close to 1 when
$\unicode[STIX]{x1D712}$
is real, which is called a Siegel zero. If
$\unicode[STIX]{x1D712}$
is real, there exists an effective constant
$c_{2}>0$
such that
$L(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D712})\neq 0$
in the range
$$\begin{eqnarray}\unicode[STIX]{x1D70E}>1-\frac{c_{2}}{\sqrt{q}\log ^{2}q}.\end{eqnarray}$$
We need the following terminologies [Reference Tenenbaum20, Part II, Ch. 5.2].
Defintion 2. Let
$z\in \mathbb{C}$
,
$c_{0}>0$
,
$0<\unicode[STIX]{x1D6FF}\leqslant 1$
,
$M>0$
. We say that a Dirichlet series
$F(s)$
has the property
${\mathcal{P}}(z;c_{0},\unicode[STIX]{x1D6FF},M)$
if the Dirichlet series
$G(s;z):=F(s)\unicode[STIX]{x1D701}(s)^{-z}$
can be analytically continued to the region
$\unicode[STIX]{x1D70E}\geqslant 1-c_{0}/(\log (|t|+2))$
, and in this region,
$|G(s;z)|\leqslant M(1+|t|)^{1-\unicode[STIX]{x1D6FF}}$
.
Defintion 3. We say
$F(s)$
has type
${\mathcal{T}}(z,w;c_{0},\unicode[STIX]{x1D6FF},M)$
, if
$F(s)=\sum _{n\geqslant 1}a_{n}/n^{s}$
has property
${\mathcal{P}}(z;c_{0},\unicode[STIX]{x1D6FF},M)$
, and there exists a sequence of non-negative real numbers
$\{b_{n}\}_{n=1}^{\infty }$
such that
$|a_{n}|\leqslant b_{n}$
, and the series
$\sum _{n\geqslant 1}b_{n}/n^{s}$
satisfies
${\mathcal{P}}(w;c_{0},\unicode[STIX]{x1D6FF},M)$
for some complex number
$w$
,
Lemma 4 [Reference Tenenbaum20, Part II, Theorem 5.2].
Let
$F(s):=\sum _{n\geqslant 1}a_{n}/n^{s}$
be a Dirichlet series of type
${\mathcal{T}}(z,w;c_{0},\unicode[STIX]{x1D6FF},M)$
. For
$x\geqslant 3$
,
$N\geqslant 0$
,
$A>0$
,
$|z|\leqslant A$
, and
$|w|\leqslant A$
, we have
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}a_{n}=x(\log x)^{z-1}\biggl\{\mathop{\sum }_{0\leqslant n\leqslant N}\frac{u_{n}(z)}{(\log x)^{n}}+O(MR_{n}(x))\biggr\},\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle & u_{n}(z):=\frac{1}{\unicode[STIX]{x1D6E4}(z-n)}\mathop{\sum }_{l+j=n}\frac{1}{l!j!}G^{(l)}(1;z)\unicode[STIX]{x1D6FE}_{j}(z), & \displaystyle \nonumber\\ \displaystyle & G^{l}(s;z):=\frac{\unicode[STIX]{x2202}^{l}}{\unicode[STIX]{x2202}s^{l}}G(s,z),\qquad \unicode[STIX]{x1D6FE}_{j}(z):=\frac{d^{j}}{ds^{j}}\biggl(\frac{\{(s-1)\unicode[STIX]{x1D701}(s)\}^{z}}{s}\biggr), & \displaystyle \nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}R_{N}(x)=\text{e}^{-c_{1}\sqrt{\log x}}+\biggl(\frac{c_{2}N+1}{\log x}\biggr)^{N+1},\end{eqnarray}$$
for some constants
$c_{1}$
and
$c_{2}$
depending at most on
$c_{0}$
,
$\unicode[STIX]{x1D6FF}$
, and
$A$
.
Lemma 5. Let
$a_{z}(n)$
be an arithmetic function depending on a complex parameter
$z$
and
$a_{z}(n)=\sum _{k=0}^{\infty }c_{k}(n)z^{k}$
in the disk
$|z|\leqslant A$
. Suppose there exists a function
$h(z)$
holomorphic for
$|z|\leqslant A$
, and a quantity
$R(x)$
, independent of
$z$
, such that, for
$x\geqslant 3$
and
$|z|\leqslant A$
, we have
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}a_{z}(n)=x(\log x)^{z-1}\{zh(z)+O_{A}(R(x))\}.\end{eqnarray}$$
If
$|h^{\prime \prime }(z)|\leqslant B_{1}$
for
$|z|\leqslant A$
, then uniformly for
$x\geqslant 3$
,
$1\leqslant k\leqslant A\log \log x$
, we have
$$\begin{eqnarray}\displaystyle C_{k}(x) & = & \displaystyle \frac{x}{\log x}\frac{(\log \log x)^{k-1}}{(k-1)!}\nonumber\\ \displaystyle & & \displaystyle \times \biggl\{h\biggl(\frac{k-1}{\log x}\biggr)+O_{A}\biggl(\frac{B_{1}(k-1)}{(\log \log x)^{2}}+\frac{\log \log x}{k}R(x)\biggr)\biggr\}.\end{eqnarray}$$
If we suppose
$|h^{(4)}(z)|\leqslant B_{2}$
for
$|z|\leqslant A$
, then uniformly for
$x\geqslant 3$
,
$3\leqslant k\leqslant A\log \log x$
, we have
$$\begin{eqnarray}\displaystyle C_{k}(x) & = & \displaystyle \frac{x}{\log x}\frac{(\log \log x)^{k-1}}{(k-1)!}\nonumber\\ \displaystyle & & \displaystyle \times \biggl\{h(0)+\frac{k-1}{\log \log x}h^{\prime }(0)+\frac{(k-1)(k-2)}{(\log \log x)^{2}}g\biggl(\frac{k-3}{\log \log x}\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,O_{A}\biggl(\frac{B_{2}(k-1)(k-2)(k-3)}{(\log \log x)^{4}}+\frac{\log \log x}{k}R(x)\biggr)\biggr\},\end{eqnarray}$$
where
$$\begin{eqnarray}g(z)=\int _{0}^{1}h^{\prime \prime }(tz)(1-t)\,dt.\end{eqnarray}$$
Proof. Formula (2.2) is a special case of [Reference Tenenbaum20, Part II, Theorem 6.3].
For all
$r\leqslant A$
, the main term in (2.3) is from
$$\begin{eqnarray}\displaystyle I & := & \displaystyle \frac{x}{\log x}\frac{1}{2\unicode[STIX]{x1D70B}\text{i}}\oint _{|z|=r}h(z)\frac{\text{e}^{z\log \log x}}{z^{k}}\,dz\nonumber\\ \displaystyle & = & \displaystyle \frac{x}{\log x}\frac{1}{2\unicode[STIX]{x1D70B}\text{i}}\oint _{|z|=r}(h(0)+zh^{\prime }(0)+z^{2}g(z))\frac{\text{e}^{z\log \log x}}{z^{k}}\,dz,\nonumber\end{eqnarray}$$
where
$g(z)=\int _{0}^{1}h^{\prime \prime }(tz)(1-t)\,dt$
. When
$k\leqslant A\log \log x$
, choose
$r_{j}=((k-j)/\text{log}\,\log x)$
(
$1\leqslant j\leqslant 3$
), we see that
$$\begin{eqnarray}\displaystyle I & = & \displaystyle \frac{x}{\log x}\frac{1}{2\unicode[STIX]{x1D70B}\text{i}}\oint _{|z|=r_{1}}h(0)\frac{\text{e}^{z\log \log x}}{z^{k}}\,dz+\frac{x}{\log x}\frac{1}{2\unicode[STIX]{x1D70B}\text{i}}\oint _{|z|=r_{2}}h^{\prime }(0)\frac{\text{e}^{z\log \log x}}{z^{k-1}}\,dz\nonumber\\ \displaystyle & & \displaystyle +\,\frac{x}{\log x}\frac{1}{2\unicode[STIX]{x1D70B}\text{i}}\oint _{|z|=r_{3}}g(z)\frac{\text{e}^{z\log \log x}}{z^{k-2}}\,dz\nonumber\\ \displaystyle & = & \displaystyle \frac{x}{\log x}\frac{(\log \log x)^{k-1}}{(k-1)!}\bigg\{h(0)+\frac{k-1}{\log \log x}h^{\prime }(0)\bigg\}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{x}{\log x}\frac{1}{2\unicode[STIX]{x1D70B}\text{i}}\oint _{|z|=r_{3}}g(z)\frac{\text{e}^{z\log \log x}}{z^{k-2}}\,dz.\end{eqnarray}$$
Next, we examine the last integral in (2.4). Since we assume
$|h^{(4)}(z)|\leqslant B_{2}$
for
$|z|\leqslant A$
, we have
$$\begin{eqnarray}\displaystyle g(z) & = & \displaystyle g(r_{3})+(z-r_{3})g^{\prime }(r_{3})+(z-r_{3})^{2}\int _{0}^{1}(1-t)g^{\prime \prime }(r_{3}+t(z-r_{3}))\,dt\nonumber\\ \displaystyle & = & \displaystyle g(r_{3})+(z-r_{3})g^{\prime }(r_{3})+O(B_{2}|z-r_{3}|^{2}).\nonumber\end{eqnarray}$$
Thus, the last integral in (2.4) equals
$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{x}{\log x}\biggl\{\frac{g(r_{3})}{2\unicode[STIX]{x1D70B}\text{i}}\oint _{|z|=r_{3}}\frac{\text{e}^{z\log \log x}}{z^{k-2}}dz+\frac{1}{2\unicode[STIX]{x1D70B}\text{i}}\oint _{|z|=r_{3}}(z-r_{3})\frac{\text{e}^{z\log \log x}}{z^{k-2}}\,dz\nonumber\\ \displaystyle & & \displaystyle \qquad +\,O\biggl(B_{2}\int _{0}^{2\unicode[STIX]{x1D70B}}|\text{e}^{\text{i}\unicode[STIX]{x1D6FC}}-1|^{2}\text{e}^{r_{3}\log \log x\cos \unicode[STIX]{x1D6FC}}r_{3}^{5-k}\,d\unicode[STIX]{x1D6FC}\biggr)\biggr\}\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{x}{\log x}\biggl\{g(r_{3})\frac{(\log \log x)^{k-3}}{(k-3)!}+\frac{(\log \log x)^{k-4}}{(k-4)!}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,r_{3}\frac{(\log \log x)^{k-3}}{(k-3)!}+O\biggl(B_{2}\frac{(\log \log x)^{k-5}}{(k-4)!}\biggr)\biggr\}\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{x}{\log x}\frac{(\log \log x)^{k-3}}{(k-3)!}\biggl\{g\biggl(\frac{k-3}{\log \log x}\biggr)+O\biggl(\frac{B_{2}(k-3)}{(\log \log x)^{2}}\biggr)\biggr\}.\end{eqnarray}$$
The error term
$O(R(x)\log \log x/k)$
is the same as that in the proof of (2.2). Combing (2.4) and (2.5), we obtain the desired result.◻
We need some results for holomorphic functions of several variables [Reference Ebeling4].
Defintion 6. Let
$\mathbb{R}_{{>}0}^{l}:=\{\mathbf{y}=(y_{1},\ldots ,y_{l})\in \mathbb{R}^{l}~|~y_{j}>0\text{ for all }j\}$
,
$\mathbf{r}=(r_{1},\ldots ,r_{l})\in \mathbb{R}_{{>}0}^{l}$
,
$\mathbf{a}\in \mathbb{C}^{l}$
. Then,
$\unicode[STIX]{x1D6E5}_{\mathbf{r}}(\mathbf{a}):=\{\mathbf{z}\in \mathbb{C}^{l}~|~|z_{j}-r_{j}|<r_{j},1\leqslant j\leqslant l\}$
is called the polycylinder around
$\mathbf{a}$
with (poly-)radius
$\mathbf{r}$
. The boundary of the closure of
$\unicode[STIX]{x1D6E5}_{\mathbf{r}}(\mathbf{a})$
contains an
$n$
-dimensional torus
$T_{\mathbf{r}}(\mathbf{a}):=\{\mathbf{z}\in \mathbb{C}^{l}~|~|z_{j}-a_{j}|=r_{j},1\leqslant j\leqslant l\}$
.
In order to simplify the expressions in our proof, we introduce multi-indices. Let
$v_{j}$
,
$1\leqslant j\leqslant l$
, be non-negative integers,
$\mathbf{z}=(z_{1},\ldots ,z_{l})\in \mathbb{C}^{l}$
. Denote
$\mathbf{v}:=(v_{1},\ldots ,v_{l})$
,
$|\mathbf{v}|=v_{1}+\cdots +v_{l}$
,
$\mathbf{v}!:=v_{1}!\cdots v_{l}!$
,
$\mathbf{z}^{\mathbf{v}}:=z_{1}^{v_{1}}\cdots z_{l}^{v_{l}}$
, and
$$\begin{eqnarray}D^{\mathbf{v}}f=\frac{\unicode[STIX]{x2202}^{|\mathbf{v}|}}{\unicode[STIX]{x2202}z_{1}^{v_{1}}\cdots \unicode[STIX]{x2202}z_{l}^{v_{l}}}.\end{eqnarray}$$
We have the following result.
Lemma 7 [Reference Ebeling4, Ch. 2, Propositions 2.7 and 2.11].
Let
$U\subset \mathbb{C}^{l}$
be open and
$f:U\rightarrow \mathbb{C}$
holomorphic. Furthermore, let
$\mathbf{w}\in U$
and
$\unicode[STIX]{x1D6E5}:=\unicode[STIX]{x1D6E5}_{\mathbf{r}}(\mathbf{w})$
be a polycylinder around
$\mathbf{w}$
with
$\bar{\unicode[STIX]{x1D6E5}}\subset U$
,
$T=T_{\mathbf{r}}(\mathbf{w})$
. Then
$f$
can be expanded as a power series
$$\begin{eqnarray}f(\mathbf{z})=\mathop{\sum }_{\mathbf{v}=0}^{\infty }a_{\boldsymbol{ v}}(\mathbf{z}-\mathbf{w})^{\mathbf{v}}=\mathop{\sum }_{v_{1}\geqslant 0,\ldots ,v_{l}\geqslant 0}a_{\mathbf{v}}(z_{1}-w_{1})^{v_{1}}\cdots (z_{l}-w_{l})^{v_{l}}\end{eqnarray}$$
in a neighborhood of
$\mathbf{w}$
, with coefficients
$$\begin{eqnarray}\displaystyle a_{\mathbf{v}} & = & \displaystyle \frac{D^{\mathbf{v}}f(\mathbf{w})}{\mathbf{v}!}=\frac{1}{\mathbf{v}!}\frac{\unicode[STIX]{x2202}^{v_{1}+\cdots +v_{l}}f}{\unicode[STIX]{x2202}z_{1}^{v_{1}}\cdots \unicode[STIX]{x2202}z_{l}^{v_{l}}}(\mathbf{w})\nonumber\\ \displaystyle & = & \displaystyle \biggl(\frac{1}{2\unicode[STIX]{x1D70B}\text{i}}\biggr)^{l}\int _{T}\frac{f(\boldsymbol{\unicode[STIX]{x1D701}})}{(\unicode[STIX]{x1D701}_{1}-w_{1})^{v_{1}+1}\cdots (\unicode[STIX]{x1D701}_{l}-w_{l})^{v_{l}+1}}\,d\boldsymbol{\unicode[STIX]{x1D701}}.\nonumber\end{eqnarray}$$
3 Proof of theorems
3.1 Associated Dirichlet series
Let
$(a,q)=1$
. We define a function
$\unicode[STIX]{x1D706}_{a}(n)$
in the following way,
$$\begin{eqnarray}\unicode[STIX]{x1D706}_{a}(n)=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }n~\text{square-free},~p|n\Rightarrow p\equiv a\hspace{0.2em}{\rm mod}\hspace{0.2em}q,\\ 0\quad & \text{otherwise}.\end{array}\right.\end{eqnarray}$$
We consider the Dirichlet series
$$\begin{eqnarray}F(s;a,z):=\mathop{\sum }_{n=1}^{\infty }\frac{(z\unicode[STIX]{x1D706}_{a}(n))^{\unicode[STIX]{x1D714}(n)}}{n^{s}}=\mathop{\prod }_{p}\biggl(1+\frac{z\unicode[STIX]{x1D706}_{a}(p)}{p^{s}}\biggr)\quad (\Re (s)>1),\end{eqnarray}$$
where
$\unicode[STIX]{x1D714}(n)$
is the number of distinct prime factors of
$n$
. Let
$\unicode[STIX]{x1D712}_{0}$
be the principal character modulo
$q$
, denote
$$\begin{eqnarray}\displaystyle F(s;z) & := & \displaystyle \mathop{\sum }_{n=1}^{\infty }\frac{\unicode[STIX]{x1D707}^{2}(n)(z\unicode[STIX]{x1D712}_{0}(n))^{\unicode[STIX]{x1D714}(n)}}{n^{s}}=\mathop{\prod }_{p}\biggl(1+\frac{z\unicode[STIX]{x1D712}_{0}(p)}{p^{s}}\biggr)=\mathop{\prod }_{p\nmid q}\biggl(1+\frac{z}{p^{s}}\biggr)\quad \nonumber\\ \displaystyle & & \displaystyle (\Re (s)>1),\nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D707}(n)$
is the Möbius function.
Then we have the following lemma.
Lemma 8. For any
$A>0$
,
$|z|\leqslant A$
, and
$\Re (s)>1$
,
$$\begin{eqnarray}F(s;a,z)=(L(s,\unicode[STIX]{x1D712}_{0}))^{z/\unicode[STIX]{x1D719}(q)}\mathop{\prod }_{\unicode[STIX]{x1D712}\neq \unicode[STIX]{x1D712}_{0}}(L(s,\unicode[STIX]{x1D712}))^{\bar{\unicode[STIX]{x1D712}}(a)z/\unicode[STIX]{x1D719}(q)}G_{1}(s;a,z),\end{eqnarray}$$
and
$$\begin{eqnarray}F(s;z)=(L(s,\unicode[STIX]{x1D712}_{0}))^{z}G_{2}(s;z),\end{eqnarray}$$
where
$\unicode[STIX]{x1D712}$
is a Dirichlet character modulo
$q$
, and
$G_{1}(s;a,z)$
and
$G_{2}(s;z)$
are absolutely convergent for
$\Re (s)>\frac{1}{2}$
.
Given
$\mathbf{a}=(a_{1},a_{2},\ldots ,a_{k})$
, assume there are
$l$
distinct values
$b_{1},\ldots ,b_{l}$
in the coordinates of
$\mathbf{a}$
. We assume
$b_{i}$
(
$1\leqslant i\leqslant l$
) appears
$k_{i}({>}0)$
times in
$\mathbf{a}$
with
$k_{1}+k_{2}+\cdots +k_{l}=k$
. Let
$\mathbf{k}(\mathbf{a}):=(k_{1},k_{2},\ldots ,k_{l})$
,
$\mathbf{b}(\mathbf{a}):=(b_{1},\ldots ,b_{l})$
, and
$\mathbf{z}=(z_{1},z_{2},\ldots ,z_{l})$
. Denote
$$\begin{eqnarray}F(s;\mathbf{a},\mathbf{z}):=\mathop{\prod }_{j=1}^{l}F(s;b_{j},z_{j})=\mathop{\prod }_{j=1}^{l}\mathop{\prod }_{p}\biggl(1+\frac{z_{j}\unicode[STIX]{x1D706}_{b_{j}}(p)}{p^{s}}\biggr).\end{eqnarray}$$
Let
$\mathbf{n}=(n_{1},\ldots ,n_{l})\in \mathbb{Z}^{l}$
(
$n_{j}>0,1\leqslant j\leqslant l$
). We write the Dirichlet series
$F(s;\mathbf{a},\mathbf{z})=\sum _{\mathbf{n}>\mathbf{0}}(a(\mathbf{n};\mathbf{z})/P^{s}(\mathbf{n}))$
with
$P(\mathbf{n})=\prod _{1\leqslant j\leqslant l}n_{j}$
. Then,
$$\begin{eqnarray}a(\mathbf{n};\mathbf{z})=\mathop{\sum }_{\substack{ \mathbf{k}=(k_{1},\ldots ,k_{l}) \\ k_{j}\geqslant 0}}c(\mathbf{k},\mathbf{n})z_{1}^{k_{1}}\cdots z_{l}^{k_{l}},\end{eqnarray}$$
for some
$c(\mathbf{k},\mathbf{n})\in \mathbb{Z}^{+}$
. Thus, for given
$\mathbf{a}$
, by Lemma 7,
$$\begin{eqnarray}\displaystyle M_{k}(x;\mathbf{a}) & = & \displaystyle \mathop{\sum }_{P(\mathbf{n})\leqslant x}c(\mathbf{k}(\mathbf{a}),\mathbf{n})\nonumber\\ \displaystyle & = & \displaystyle \biggl(\frac{1}{2\unicode[STIX]{x1D70B}\text{i}}\biggr)^{l}\oint _{|z_{l}|=r_{l}}\cdots \oint _{|z_{1}|=r_{1}}\biggl(\mathop{\sum }_{P(\mathbf{n})\leqslant x}a(\mathbf{n};\mathbf{z})\biggr)\frac{dz_{1}}{z_{1}^{k_{1}+1}}\cdots \frac{dz_{l}}{z_{l}^{k_{l}+1}}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
3.2 A uniform result
First, we prove the following result.
Theorem 3. For any
$A>0$
, fixed
$q\geqslant 3$
and fixed
$l\geqslant 1$
, uniformly for
$2\leqslant k\leqslant A\log \log x$
, we have
$$\begin{eqnarray}\displaystyle M_{k}(x;\mathbf{a})=\frac{x}{\log x}\biggl\{\frac{1}{\unicode[STIX]{x1D719}(q)}Q_{\mathbf{k}}\biggl(\frac{\log \log x}{\unicode[STIX]{x1D719}(q)}\biggr)+O_{A,q,l}\biggl(\frac{1}{\unicode[STIX]{x1D719}^{k}(q)}\frac{(\log \log x)^{k}}{k_{1}!\cdots k_{l}!\log x}\biggr)\biggr\}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$Q_{\mathbf{k}}(X)$
is a polynomial of degree at most
$k-1$
(
$k=k_{1}+\cdots +k_{l}$
). In particular, the coefficient of the term
$(x/\text{log}\,x)(\log \log x)^{k-1}$
is
$(1/\unicode[STIX]{x1D719}^{k}(q))(k/k_{1}!k_{2}!\cdots k_{l}!),$
and the coefficient of
$(x/\text{log}\,x)(\log \log x)^{k-2}$
is
$$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D719}^{k}(q)}\frac{k(k-1)}{k_{1}!k_{2}!\cdots k_{l}!}\biggl(\unicode[STIX]{x1D6FE}+B+\frac{1}{k}\mathop{\sum }_{j=1}^{k}C(q,a_{j})-\mathop{\sum }_{p|q}\frac{1}{p}\biggr),\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FE}$
is Euler’s constant, and
$B:=\sum _{p}(\log (1-1/p)+1/p)$
is Mertens’ constant.
Proof of Theorem 3.
$$\begin{eqnarray}\displaystyle F(s;\mathbf{a},\mathbf{z}) & = & \displaystyle (L(s,\unicode[STIX]{x1D712}_{0}))^{(z_{1}+\cdots +z_{l})/\unicode[STIX]{x1D719}(q)}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\prod }_{\unicode[STIX]{x1D712}\neq \unicode[STIX]{x1D712}_{0}}(L(s,\unicode[STIX]{x1D712}))^{(\bar{\unicode[STIX]{x1D712}}(b_{1})z_{1}+\cdots +\bar{\unicode[STIX]{x1D712}}(b_{l})z_{l})/\unicode[STIX]{x1D719}(q)}\mathop{\prod }_{j=1}^{l}G_{1}(s;b_{j},z_{j})\nonumber\\ \displaystyle & = & \displaystyle (\unicode[STIX]{x1D701}(s))^{(z_{1}+\cdots +z_{l})/\unicode[STIX]{x1D719}(q)}H(s;\mathbf{a},\mathbf{z}),\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle H(s;\mathbf{a},\mathbf{z}) & = & \displaystyle \mathop{\prod }_{p|q}\biggl(1-\frac{1}{p^{s}}\biggr)^{(z_{1}+\cdots +z_{l})/\unicode[STIX]{x1D719}(q)}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\prod }_{\unicode[STIX]{x1D712}\neq \unicode[STIX]{x1D712}_{0}}(L(s,\unicode[STIX]{x1D712}))^{(\bar{\unicode[STIX]{x1D712}}(b_{1})z_{1}+\cdots +\bar{\unicode[STIX]{x1D712}}(b_{l})z_{l})/\unicode[STIX]{x1D719}(q)}\mathop{\prod }_{j=1}^{l}G_{1}(s;b_{j},z_{j})\nonumber\\ \displaystyle & = & \displaystyle \mathop{\prod }_{p}\biggl(1-\frac{1}{p^{s}}\biggr)^{(z_{1}+\cdots +z_{l})/\unicode[STIX]{x1D719}(q)}\mathop{\prod }_{j=1}^{l}\biggl(1+\frac{z_{j}\unicode[STIX]{x1D706}_{b_{j}}(p)}{p^{s}}\biggr).\nonumber\end{eqnarray}$$
Let
$\unicode[STIX]{x1D70E}=\Re (s)$
. Kolesnik [Reference Kolesnik9] showed that, for
$\frac{1}{2}\leqslant \unicode[STIX]{x1D70E}\leqslant 1$
,
$$\begin{eqnarray}|L(s,\unicode[STIX]{x1D712})|\ll (|t|+2)^{(35/108)(1-\unicode[STIX]{x1D70E})}q^{1-\unicode[STIX]{x1D70E}}\log ^{3}(q(|t|+2)).\end{eqnarray}$$
Let
$q$
be fixed. By Lemma 1 and (3.6), for any
$A>0$
,
$|z_{j}|\leqslant A$
(
$1\leqslant j\leqslant l$
), and
$0<\unicode[STIX]{x1D6FF}<1$
, we can choose
$c_{0}=c_{0}(A,\unicode[STIX]{x1D6FF})$
such that,
$L(s,\unicode[STIX]{x1D712})$
has no zeros in the region
$\unicode[STIX]{x1D70E}\geqslant 1-c_{0}/(\log (|t|+2))$
, and by [Reference Montgomery and Vaughan14, Theorem 11.4], in this region,
$|H(s;\mathbf{a},\mathbf{z})|\ll _{q,A,\unicode[STIX]{x1D6FF}}(|t|+2)^{1-\unicode[STIX]{x1D6FF}}.$
Thus, by Definitions 2 and 3,
$F(s;\mathbf{a},\mathbf{z})$
is in
${\mathcal{T}}((z_{1}+\cdots +z_{l})/\unicode[STIX]{x1D719}(q),w;c_{0},\unicode[STIX]{x1D6FF},M)$
. By (3.5) and following the same method given in the proof of Lemma 4 ([Reference Tenenbaum20, Part II, Theorem 5.2], the only difference in the proof is the expansion of
$H(s;\mathbf{a},\mathbf{z})$
), we deduce that
$$\begin{eqnarray}\mathop{\sum }_{n_{1}\cdots n_{l}\leqslant x}a(\mathbf{n};\mathbf{z})=x(\log x)^{(z_{1}+\cdots +z_{l})/\unicode[STIX]{x1D719}(q)-1}\biggl\{u_{0}(\mathbf{a};\mathbf{z})+O_{A}\biggl(\frac{1}{\log x}\biggr)\biggr\},\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle u_{0}(\mathbf{a};\mathbf{z})=\frac{z_{1}+\cdots +z_{l}}{\unicode[STIX]{x1D719}(q)}u(\mathbf{a};\mathbf{z}), & & \displaystyle \nonumber\end{eqnarray}$$
with
$$\begin{eqnarray}\displaystyle u(\mathbf{a};\mathbf{z}):=\frac{H(1;\mathbf{a},\mathbf{z})}{\unicode[STIX]{x1D6E4}((z_{1}+\cdots +z_{l})/\unicode[STIX]{x1D719}(q)+1)}. & & \displaystyle\end{eqnarray}$$
By (3.4), (3.7), and Lemma 7, we have
$$\begin{eqnarray}\displaystyle M_{k}(x;\mathbf{a})=\frac{x}{\log x}\biggl\{\frac{1}{\unicode[STIX]{x1D719}(q)}Q_{\mathbf{k}}\biggl(\frac{\log \log x}{\unicode[STIX]{x1D719}(q)}\biggr)+\widetilde{R}(x)\biggr\}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$Q_{\mathbf{k}}(X)$
is a polynomial of degree at most
$k-1$
(
$k=k_{1}+\cdots +k_{l}$
),
$$\begin{eqnarray}\displaystyle Q_{\mathbf{k}}(X) & := & \displaystyle \biggl\{\mathop{\sum }_{m_{1}+j_{1}=k_{1}-1}\mathop{\sum }_{m_{2}+j_{2}=k_{2}}\cdots \mathop{\sum }_{m_{l}+j_{l}=k_{l}}+\mathop{\sum }_{m_{1}+j_{1}=k_{1}}\mathop{\sum }_{m_{2}+j_{2}=k_{2}-1}\cdots \mathop{\sum }_{m_{l}+j_{l}=k_{l}}\nonumber\\ \displaystyle & & \displaystyle +\cdots +\mathop{\sum }_{m_{1}+j_{1}=k_{1}}\cdots \mathop{\sum }_{m_{l-1}+j_{l-1}=k_{l-1}}\mathop{\sum }_{m_{l}+j_{l}=k_{l}-1}\biggr\}\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{1}{m_{1}!j_{1}!\cdots m_{l}!j_{l}!}\frac{\unicode[STIX]{x2202}^{m_{1}+\cdots +m_{l}}}{\unicode[STIX]{x2202}z_{1}^{m_{1}}\cdots \unicode[STIX]{x2202}z_{l}^{m_{l}}}u(\mathbf{a};(0,\ldots ,0))X^{j_{1}+\cdots +j_{l}},\end{eqnarray}$$
and
$$\begin{eqnarray}\widetilde{R}(x)\ll _{A}\frac{1}{(2\unicode[STIX]{x1D70B})^{l}\log x}\mathop{\prod }_{j=1}^{l}\oint _{|z_{j}|=r_{j}}(\log x)^{\Re (z_{j})/\unicode[STIX]{x1D719}(q)}\frac{|dz_{j}|}{|z_{j}|^{k_{j}+1}}.\end{eqnarray}$$
Taking
$r_{j}=\unicode[STIX]{x1D719}(q)k_{j}/\text{log}\,\log x$
, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \oint _{|z_{j}|=r_{j}}(\log x)^{\Re (z_{j})/\unicode[STIX]{x1D719}(q)}\frac{|dz_{j}|}{|z_{j}|^{k_{j}+1}}=\biggl(\frac{\log \log x}{\unicode[STIX]{x1D719}(q)k_{j}}\biggr)^{k_{j}}\int _{0}^{2\unicode[STIX]{x1D70B}}\text{e}^{k_{j}\cos \unicode[STIX]{x1D703}}d\unicode[STIX]{x1D703}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \biggl(\frac{\log \log x}{\unicode[STIX]{x1D719}(q)k_{j}}\biggr)^{k_{j}}\biggl(2\int _{0}^{\unicode[STIX]{x1D70B}/2}\text{e}^{k_{j}\cos \unicode[STIX]{x1D703}}d\unicode[STIX]{x1D703}+\unicode[STIX]{x1D70B}\biggr)\nonumber\\ \displaystyle & & \displaystyle \quad =\biggl(\frac{\log \log x}{\unicode[STIX]{x1D719}(q)k_{j}}\biggr)^{k_{j}}\biggl(2\int _{0}^{1}\text{e}^{k_{j}t}\frac{dt}{\sqrt{1-t^{2}}}+\unicode[STIX]{x1D70B}\biggr)\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \biggl(\frac{\log \log x}{\unicode[STIX]{x1D719}(q)k_{j}}\biggr)^{k_{j}}\biggl(2\text{e}^{k_{j}}\int _{0}^{1}\text{e}^{-k_{j}(1-t)}\frac{dt}{\sqrt{1-t}}+\unicode[STIX]{x1D70B}\biggr)\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \biggl(\frac{\log \log x}{\unicode[STIX]{x1D719}(q)k_{j}}\biggr)^{k_{j}}\biggl(2\unicode[STIX]{x1D6E4}\biggl(\frac{1}{2}\biggr)\text{e}^{k_{j}}k_{j}^{-1/2}+\unicode[STIX]{x1D70B}\biggr).\end{eqnarray}$$
Substituting (3.11) into (3.10), we obtain
$$\begin{eqnarray}\widetilde{R}_{N}(x)\ll _{A,l}\frac{1}{\unicode[STIX]{x1D719}^{k}(q)}\frac{(\log \log x)^{k}}{k_{1}!\cdots k_{l}!\log x}.\end{eqnarray}$$
Theorem 3 follows. ◻
Remark 4. Similar to the proof of Lemma 5, we write
$$\begin{eqnarray}u(\mathbf{a},\mathbf{z})=u(\mathbf{a},\mathbf{r})+\mathop{\sum }_{|\mathbf{v}|=1}D^{\mathbf{v}}u(\mathbf{a},\mathbf{r})+\mathop{\sum }_{|\boldsymbol{\unicode[STIX]{x1D6FD}}|=2}(\mathbf{z}-\mathbf{r})^{\boldsymbol{\unicode[STIX]{x1D6FD}}}R_{\boldsymbol{\unicode[STIX]{x1D6FD}}}(\mathbf{z}),\end{eqnarray}$$
where
$$\begin{eqnarray}R_{\boldsymbol{\unicode[STIX]{x1D6FD}}}(\mathbf{z})=\frac{|\boldsymbol{\unicode[STIX]{x1D6FD}}|}{\boldsymbol{\unicode[STIX]{x1D6FD}}!}\int _{0}^{1}(1-t)D^{\boldsymbol{\unicode[STIX]{x1D6FD}}}u(\mathbf{a},\mathbf{r}+t(\mathbf{z}-\mathbf{r}))\,dt.\end{eqnarray}$$
Then, by (3.4) and (3.7), using a similar proof to that of Lemma 5, we have
$$\begin{eqnarray}\displaystyle M_{k}(x;\mathbf{a}) & = & \displaystyle \frac{1}{\unicode[STIX]{x1D719}^{k}(q)}\frac{k}{k_{1}!k_{2}!\cdots k_{l}!}\frac{x(\log \log x)^{k-1}}{\log x}\nonumber\\ \displaystyle & & \displaystyle \times \,\biggl\{g\biggl(\frac{\unicode[STIX]{x1D719}(q)}{\log \log x};\mathbf{k}\biggr)+O_{A,q,l}\biggl(\frac{k}{(\log \log x)^{2}}\biggr)\biggr\},\end{eqnarray}$$
where
$g(z;\mathbf{k}):=\sum _{j=1}^{l}(k_{j}/k)u(\mathbf{a};(k_{1}z,\ldots ,k_{j-1}z,k_{j}^{\prime }z,k_{j+1}z,\ldots ,k_{l}z))$
with
$k_{j}^{\prime }=k_{j}-1$
. Moreover, if
$|kz|\leqslant A$
, then
$|g(z,\mathbf{k})|=O_{A,q,l}(1)$
.
3.3 Proof of Theorems 1 and 2
For
$\mathbf{a}=(a,\ldots ,a)$
,
$(a,q)=1$
, this is a special case of Theorem 3. Denote
$$\begin{eqnarray}H(s;a,z):=F(s;a,z)(\unicode[STIX]{x1D701}(s))^{-z/\unicode[STIX]{x1D719}(q)}=\mathop{\prod }_{p}\biggl(1-\frac{1}{p^{s}}\biggr)^{z/\unicode[STIX]{x1D719}(q)}\biggl(1+\frac{z\unicode[STIX]{x1D706}_{a}(p)}{p^{s}}\biggr),\end{eqnarray}$$
and
$$\begin{eqnarray}h(a;z):=\frac{H(1;a,z)}{\unicode[STIX]{x1D6E4}(z/\unicode[STIX]{x1D719}(q)+1)}.\end{eqnarray}$$
Hence, for this special case, by (3.7) and Lemma 5, we obtain the following result.
Lemma 9. For
$\mathbf{a}=(a,\ldots ,a)$
and any
$A>0$
, uniformly for
$2\leqslant k\leqslant A\log \log x$
, we have
$$\begin{eqnarray}\displaystyle M_{k}(x;\mathbf{a}) & = & \displaystyle \frac{1}{\unicode[STIX]{x1D719}^{k}(q)}\frac{x}{\log x}\frac{(\log \log x)^{k-1}}{(k-1)!}\biggl\{1+\frac{k-1}{\log \log x}C_{a,q}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{(k-1)(k-2)\unicode[STIX]{x1D719}^{2}(q)}{(\log \log x)^{2}}\widetilde{h}\biggl(a;\frac{(k-3)\unicode[STIX]{x1D719}(q)}{\log \log x}\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,O_{A,q}\biggl(\frac{k^{3}}{(\log \log x)^{4}}\biggr)\biggr\},\nonumber\end{eqnarray}$$
where
$$\begin{eqnarray}C_{a,q}:=\unicode[STIX]{x1D719}(q)h^{\prime }(a,0)=\unicode[STIX]{x1D6FE}+\mathop{\sum }_{p}\biggl(\log \biggl(1-\frac{1}{p}\biggr)+\frac{\unicode[STIX]{x1D719}(q)\unicode[STIX]{x1D706}_{a}(p)}{p}\biggr),\end{eqnarray}$$
$\unicode[STIX]{x1D6FE}\approx 0.57722$
is Euler’s constant, and
$$\begin{eqnarray}\widetilde{h}(a,z)=\int _{0}^{1}h^{\prime \prime }(a,tz)(1-t)\,dt.\end{eqnarray}$$
Remark 5. Note that, for
$|z|\leqslant A$
, the function
$|h^{\prime \prime }(a,z)|=O_{q,A}(1)$
and
$|h^{(4)}(a,z)|=O_{q,A}(1)$
.
We also require a formula for
$S_{k}(x)$
. By Lemma 8, and Definitions 2 and 3,
$F(s;z)$
is in
${\mathcal{T}}(z,w;c_{0},\unicode[STIX]{x1D6FF},M)$
. We use the notation
$$\begin{eqnarray}G(s;z):=F(s;z)(\unicode[STIX]{x1D701}(s))^{-z}=\mathop{\prod }_{p}\biggl(1-\frac{1}{p^{s}}\biggr)^{z}\biggl(1+\frac{z\unicode[STIX]{x1D712}_{0}(p)}{p^{s}}\biggr),\end{eqnarray}$$
and
$$\begin{eqnarray}g(z):=\frac{G(1;z)}{\unicode[STIX]{x1D6E4}(z+1)}.\end{eqnarray}$$
Then, applying Lemmas 4 and 5 successively, we obtain the following lemma.
Lemma 10. For any
$A>0$
, uniformly for
$2\leqslant k\leqslant A\log \log x$
, we have
$$\begin{eqnarray}\displaystyle S_{k}(x) & = & \displaystyle \frac{x}{\log x}\frac{(\log \log x)^{k}}{(k-1)!}\biggl\{1+\frac{k-1}{\log \log x}g^{\prime }(0)+\frac{(k-1)(k-2)}{(\log \log x)^{2}}\widetilde{g}\biggl(\frac{k-3}{\log \log x}\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,O_{A,q}\biggl(\frac{k^{3}}{(\log \log x)^{4}}\biggr)\biggr\},\nonumber\end{eqnarray}$$
where
$g^{\prime }(0)=\unicode[STIX]{x1D6FE}+B-\sum _{p|q}(1/p)$
,
$\unicode[STIX]{x1D6FE}$
is Euler’s constant,
$B=\sum _{p}(\log (1-1/p)+1/p)$
is Mertens’ constant in (1.2), and
$$\begin{eqnarray}\widetilde{g}(z)=\int _{0}^{1}g^{\prime \prime }(tz)(1-t)\,dt.\end{eqnarray}$$
Remark 6. Here for
$|z|\leqslant A$
, the function
$|g^{\prime \prime }(z)|=O_{q,A}(1)$
and
$|g^{(4)}(z)|=O_{q,A}(1)$
.
Proof of Theorem 1.
$$\begin{eqnarray}\displaystyle M_{k}(x,\mathbf{a})-\frac{1}{\unicode[STIX]{x1D719}^{k}(q)}S_{k}(x) & = & \displaystyle \frac{1}{\unicode[STIX]{x1D719}^{k}(q)}\frac{x}{\log x}\frac{(\log \log x)^{k-2}}{(k-2)!}\nonumber\\ \displaystyle & & \displaystyle \times \,\biggl\{C(q,a)+\frac{k-2}{\log \log x}\unicode[STIX]{x1D719}^{2}(q)\widetilde{h}\biggl(a;\frac{(k-3)\unicode[STIX]{x1D719}(q)}{\log \log x}\biggr)\nonumber\\ \displaystyle & & \displaystyle -\,\frac{k-2}{\log \log x}\widetilde{g}\biggl(\frac{k-3}{\log \log x}\biggr)+O_{A,q}\biggl(\frac{k}{(\log \log x)^{2}}\biggr)\biggr\}.\nonumber\end{eqnarray}$$
For the cases of fixed
$k$
and
$k=o(\log \log x)$
, by Remarks 5 and 6, and Lemma 10, we immediately obtain the conclusions in Theorem 1 using the equality
$$\begin{eqnarray}\frac{M_{k}(x,\mathbf{a})}{(1/\unicode[STIX]{x1D719}^{k}(q))S_{k}(x)}=1+\frac{M_{k}(x,\mathbf{a})-(1/\unicode[STIX]{x1D719}^{k}(q))S_{k}(x)}{(1/\unicode[STIX]{x1D719}^{k}(q))S_{k}(x)}.\end{eqnarray}$$
For any fixed
$A>0$
, if
$k\sim A\log \log x$
, by Lemmas 9 and 10, and (3.12), as
$x\rightarrow \infty$
, the above quotient will approach
$$\begin{eqnarray}\frac{h(a,A\unicode[STIX]{x1D719}(q))}{g(A)}=\mathop{\prod }_{p}\frac{1+A\unicode[STIX]{x1D719}(q)\unicode[STIX]{x1D7D9}_{p\equiv a\hspace{0.2em}{\rm mod}\hspace{0.2em}q}(p)/p}{1+A\unicode[STIX]{x1D712}_{0}(p)/p},\end{eqnarray}$$
where
$h(a,z)$
and
$g(z)$
are defined in (3.13) and (3.14), respectively.◻
Proof of Theorem 2.
For fixed
$k$
, by Theorem 3 and Lemma 10, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle M_{k}(x;\mathbf{a})-\frac{1}{\unicode[STIX]{x1D719}^{k}(q)}\frac{k!}{k_{1}!k_{2}!\cdots k_{l}!}S_{k}(x)\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{1}{\unicode[STIX]{x1D719}^{k}(q)}\frac{k(k-1)}{k_{1}!k_{2}!\cdots k_{l}!}\frac{x}{\log x}(\log \log x)^{k-2}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\biggl\{\frac{1}{k}\mathop{\sum }_{j=1}^{k}C(q,a_{j})+O_{k,q,l}\biggl(\frac{1}{\log \log x}\biggr)\biggr\}.\nonumber\end{eqnarray}$$
Thus,
$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{M_{k}(x;\mathbf{a})}{(1/\unicode[STIX]{x1D719}^{k}(q))(k!/k_{1}!k_{2}!\cdots k_{l}!)S_{k}(x)}\nonumber\\ \displaystyle & & \displaystyle \quad =1+\frac{M_{k}(x;\mathbf{a})-(1/\unicode[STIX]{x1D719}^{k}(q))(k!/k_{1}!k_{2}!\cdots k_{l}!)S_{k}(x)}{(1/\unicode[STIX]{x1D719}^{k}(q))(k!/k_{1}!k_{2}!\cdots k_{l}!)S_{k}(x)}\nonumber\\ \displaystyle & & \displaystyle \quad =1+\frac{k-1}{\log \log x}\frac{1}{k}\mathop{\sum }_{j=1}^{k}C(q,a_{j})+O_{q,k,l}\biggl(\frac{1}{(\log \log x)^{2}}\biggr).\nonumber\end{eqnarray}$$
For any fixed
$A>0$
, if
$k\sim A\log \log x$
and
$e_{j}:=\lim _{x\rightarrow \infty }(k_{j}/\text{log}\,\log x)$
exists, by (3.12) and Lemma 10, as
$x\rightarrow \infty$
, the above quotient will approach
$$\begin{eqnarray}\frac{u(\mathbf{a};(\unicode[STIX]{x1D719}(q)e_{1},\ldots ,\unicode[STIX]{x1D719}(q)e_{l}))}{g(A)}=\mathop{\prod }_{p}\frac{\mathop{\prod }_{j=1}^{l}(1+\unicode[STIX]{x1D719}(q)e_{j}\unicode[STIX]{x1D7D9}_{p\equiv b_{j}\hspace{0.2em}{\rm mod}\hspace{0.2em}q}(p)/p)}{1+A\unicode[STIX]{x1D712}_{0}(p)/p},\end{eqnarray}$$
where
$u(\mathbf{a};\mathbf{z})$
and
$g(z)$
are defined in (3.8) and (3.14) respectively.◻
Acknowledgements
This research is partially supported by NSF grant DMS-1501982. I would like to thank my advisor, Professor Kevin Ford, for his useful comments and financial support to finish this project. I am grateful for the helpful comments and advice of Professor Andrew Granville. I would like to thank Dr Nathan McNew and Professor Carl Pomerance for the helpful discussions on the least prime in arithmetic progressions. The author thanks Peter Humphries for his comments and for pointing out the reference [Reference Norton18]. The author would like to thank the referee for his/her careful reading and helpful comments, and for pointing out the reference [Reference Baker, Birch and Wirsing1].
