On an Additive Prime Divisor Function of Alladi and Erdős

Abstract

This paper discusses the additive prime divisor function $A(n) := \sum \limits _{p^\alpha || n} \alpha \, p$ which was introduced by Alladi and Erdős in 1977. It is shown that A(n) is uniformly distributed (mod q) for any fixed integer $q > 1$ with an explicit bound for the error.

This paper is dedicated to Krishna Alladi on the occasion of his 60th birthday

This research was partially supported by NSA grant H98230-15-1-0035.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Let $n = \prod \limits _{i=1}^r p_i^{a_i}$ be the unique prime decomposition of a positive integer n. In 1977, Alladi and Erdős [1] introduced the additive function

$$A(n) := \sum _{i=1}^r a_i \cdot p_i.$$

Among several other things they proved that A(n) is uniformly distributed modulo 2. This was obtained from the identity

$$\begin{aligned} \sum _{n=1}^\infty \frac{(-1)^{A(n)}}{n^s} \; = \; \frac{2^s+1}{2^s-1} \cdot \frac{\zeta (2s)}{\zeta (s)} \end{aligned}$$

(1)

together with the known zero-free region for the Riemann zeta function. As a consequence they proved that there exists a constant $c > 0$ such that

$$ \sum _{n\le x} (-1)^{A(n)} = \mathscr {O}\left( x \, e^{-c \sqrt{\log x \log \log x}} \right) , $$

for $x \rightarrow \infty .$

In 1969 Delange [3] gave a necessary and sufficient condition for uniform distribution in progressions for integral valued additive functions which easily implies that A(n) is uniformly distributed (mod q) for all $q \ge 2$ (although without a bound for the error in the asymptotic formula). The main goal of this paper is to show that A(n) is uniformly distributed modulo q for any integer $q \ge 2$ with an explicit bound for the error.

Unfortunately, it is not possible to obtain such a simple identity as in (1) for the Dirichlet series

$$\sum _{n=1}^\infty \; \frac{e^{2\pi i \frac{h A(n)}{q}}}{ n^{s}}$$

when $q > 2$ and h, q are coprime. Instead we require a representation involving a product of rational powers of Dirichlet L-functions which will have branch points at the zeros of the L-functions.

The uniform distribution of A(n) is a consequence of the following theorem (1.1) which is proved in §3. To state the theorem we require some standard notation. Let $\mu$ denote the Mobius function and let $\phi$ denote Euler’s function. For any Dirichlet character $\chi \pmod {q}$ (with $q > 1$) let $\tau (\chi ) = \sum \limits _{\ell \pmod {q}} \chi (\ell ) e^{\frac{2\pi i\ell }{q}}$ denote the associated Gauss sum and let $L(s, \chi )$ denote the Dirichlet L-function associated to $\chi .$

Theorem 1.1.

Let h, q be fixed coprime integers with $q > 2.$ Then for $x \rightarrow \infty$ we have the asymptotic formula

$$\sum _{n\le x} e^{2\pi i \frac{h A(n)}{q}} \; = \; {\left\{ \begin{array}{ll} C_{h,q} \cdot x \, (\log x)^{-1 + \frac{\mu (q)}{\phi (q)}} \Big (1 + \mathscr {O}\left( (\log x)^{-1}\right) \Big ) &{} \text {if} \; \mu (q) \ne 0,\\ &{}\\ \mathscr {O}\left( x\, e^{-c_0\,\sqrt{\log x}} \right) &{} \text {if} \; \mu (q) = 0, \end{array}\right. }$$

where $c_0>0$ is a constant depending at most on h, q,

$$C_{h,q} = \frac{V_{h,q}\cdot \sin \left( \frac{\mu (q) \,\pi }{\phi (q)} \right) }{\pi }\, \varGamma \left( 1 - \frac{\mu (q)}{\phi (q)} \right) \underset{\chi \ne \chi _0}{\prod _{\chi \!\!\!\!\!\pmod {q}}} \;L(1, \,\chi )^{\frac{\tau (\overline{\chi })\chi (h)}{\phi (q)}},$$

and

$$V_{h,q} := \exp \left[ - \frac{\mu (q)}{\phi (q)}\sum _{p\mid q}\sum _{k=1}^\infty \frac{1}{kp^{k}} \; + \; \sum _{p\,\mid \,q} \sum _{k=1}^\infty \frac{e^{\frac{2\pi i h p^k}{q}}}{k \, p^{k}} \; + \; \sum _p \sum _{k=2}^\infty \frac{e^{\frac{2\pi i p h k}{q}} - e^{\frac{2\pi i p^k h}{q}} }{k \, p^{k}}\right] .$$

Theorem 1.1 has the following easily proved corollary.

Corollary 1.2.

Let $q >1$ and let h be an arbitrary integer. Then

$$\sum _{n\le x} e^{2\pi i \frac{h A(n)}{q}} = \mathscr {O}\left( \frac{x}{\sqrt{\log x}} \right) .$$

The above corollary can then be used to obtain the desired uniform distribution theorem.

Theorem 1.3.

Let h, q be fixed integers with $q > 2.$ Then for $x\rightarrow \infty$, we have

$$\underset{A(n)\; \equiv \; h\!\!\!\!\pmod {q}}{\sum _{n \,\le \, x} 1} \; = \;\frac{x}{q} \; + \; \mathscr {O}\left( \frac{x}{\sqrt{\log x}} \right) .$$

We remark that the error term in theorem 1.3 can be replaced by a second order asymptotic term which is not uniformly distributed (mod q).

The proof of theorem (1.1) relies on explicitly constructing an L-function with coefficients of the form $e^{2\pi i \frac{h A(n)}{q}}$. It will turn out that this L-function will be a product of Dirichlet L-functions raised to complex powers. The techniques for obtaining asymptotic formulae and dealing with branch singularities arising from complex powers of ordinary L-series were first introduced by Selberg [7], and see also Tenenbaum [8] for a very nice exposition with different applications. In [4,5,6], one finds a larger class of additive functions where these methods can also be applied yielding similar results but with different constants.

2 On the function $L(s, \psi _{h/q})$

Let h, q be coprime integers with $q > 1$. In this paper we shall investigate the completely multiplicative function

$$\psi _{h/q}(n) := e^{\frac{2\pi i hA(n)}{q}}.$$

Then the L-function associated to $\psi _{h/q}$ is defined by the absolutely convergent series

$$\begin{aligned} L(s, \psi _{h/q}) := \; \sum _{n=1}^\infty \psi _{h/q}(n) n^{-s}, \end{aligned}$$

(2)

in the region $\mathfrak {R}(s) > 1,$ and has an Euler product representation (product over rational primes) of the form

$$\begin{aligned} L(s, \psi _{h/q}) := \prod _p \left( 1 - \frac{ e^{\frac{2\pi i h p}{q}} }{p^s} \right) ^{-1} . \end{aligned}$$

(3)

The Euler product (3) converges absolutely to a non-vanishing function for $\mathfrak {R}(s) > 1.$ We would like to show it has analytic continuation to a larger region.

Lemma 2.1.

Let $\mathfrak {R}(s) > 1.$ Then

$$\log \big (L(s, \psi _{h/q})\big ) = \sum _p \sum _{k=1}^\infty \frac{e^{\frac{2\pi i h p^k}{q}}}{k \, p^{sk}} \; + \; T_{h,q}(s)$$

where, for any $\varepsilon > 0$, the function

$$T_{h,q}(s) := \sum _p \sum _{k=2}^\infty \frac{e^{\frac{2\pi i p h k}{q}} - e^{\frac{2\pi i p^k h}{q}} }{k \, p^{sk}}$$

is holomorphic for $\mathfrak {R}(s) > \frac{1}{2} + \varepsilon$ and satisfies $|T_{h,q}(s)| = \mathscr {O}_\varepsilon \left( 1\right)$ where the $\mathscr {O}_\varepsilon$-constant is independent of q and depends at most on $\varepsilon$.

Proof.

Taking log’s, we obtain

$$\begin{aligned} \log \big ( L(s, \psi _{h/q}) \big )&= \sum _p \sum _{k=1}^\infty \frac{e^{\frac{2\pi i p h k}{q}}}{k \, p^{sk}}\\&= \sum _p \sum _{k=1}^\infty \frac{e^{\frac{2\pi i h p^k}{q}}}{k \, p^{sk}} \; + \;\sum _p \sum _{k=2}^\infty \frac{e^{\frac{2\pi i p h k}{q}} - e^{\frac{2\pi i p^k h}{q}} }{k \, p^{sk}}. \end{aligned}$$

Hence, we may take

$$T_{h,q}(s) = \sum _p \sum _{k=2}^\infty \frac{e^{\frac{2\pi i p h k}{q}} - e^{\frac{2\pi i p^k h}{q}} }{k \, p^{sk}},$$

which is easily seen to converge absolutely for $\mathfrak {R}(s) > \frac{1}{2}.$    $\square$

For $q > 2,$ let $\chi$ denote a Dirichlet character $\pmod {q}$ with associated Gauss sum $\tau (\chi ).$ We also let $\chi _0$ be the trivial character $\pmod {q}.$

We require the following lemma.

Lemma 2.2.

Let $h, q\in \mathbf Z$ with $q > 2$ and $(h,q) = 1.$ Then

$$ e^{\frac{2\pi i h}{q}} = \left( \frac{1}{\phi (q)} \underset{\chi \ne \chi _0}{\sum _{\chi \pmod {q}}} \tau (\chi )\cdot \overline{\chi (h)} \right) \; + \; \frac{\mu (q)}{\phi (q)}.$$

Proof.

Since $(h,q) = 1,$ it follows that for $\chi \pmod {q}$ with $\chi \ne \chi _0,$

$$\tau (\chi ) \, \overline{\chi (h)} = \sum _{\ell =1}^q \chi (\ell ) e^{\frac{2\pi i \ell h}{q}}.$$

This implies that

$$\begin{aligned} \underset{\chi \ne \chi _0}{\sum _{\chi \pmod {q}}} \tau (\chi )\, \overline{\chi (h)}&= (\phi (q)-1) \,e^{\frac{2\pi i h}{q}} \; + \; \underset{(\ell ,q)=1}{\sum _{\ell =2}^q }\left( \underset{\chi \ne \chi _0}{\sum _{\chi \pmod {q}}} \,\chi (\ell )\right) e^{\frac{2\pi i \ell h}{q}}\\&= (\phi (q)-1) \,e^{\frac{2\pi i h}{q}} \; - \; \underset{(\ell ,q)=1}{\sum _{\ell =1}^q } e^{\frac{2\pi i \ell h}{q}} \; + \; e^{\frac{2\pi i h}{q}}. \end{aligned}$$

The proof is completed upon noting that the Ramanujan sum on the right side above can be evaluated as

$$\underset{(\ell ,q)=1}{\sum _{\ell =1}^q } e^{\frac{2\pi i \ell h}{q}} = \sum _{d\mid (q,h)} \mu \left( \frac{q}{d}\right) \,d \; = \; \mu (q).$$

   $\square$

Theorem 2.3.

Let $s\in \mathbf C$ with $\mathfrak {R}(s)>1.$ Then we have the representation

$$L(s, \psi _{h/q}) = \left( \underset{\chi \ne \chi _0}{\prod _{\chi \pmod {q}}} \;L(s,\overline{\chi })^{\frac{\tau (\chi )\overline{\chi (h)}}{\phi (q)}}\right) \cdot \zeta (s)^{\frac{\mu (q)}{\phi (q)}}\cdot e^{U_{h,q}(s)},$$

where

$$U_{h,q}(s) := - \frac{\mu (q)}{\phi (q)}\sum _{p\mid q}\sum _{k=1}^\infty \frac{1}{kp^{sk}} \; + \; \sum _{p\,\mid \,q} \sum _{k=1}^\infty \frac{e^{\frac{2\pi i h p^k}{q}}}{k \, p^{sk}} \; + \; \sum _p \sum _{k=2}^\infty \frac{e^{\frac{2\pi i p h k}{q}} - e^{\frac{2\pi i p^k h}{q}} }{k \, p^{sk}}.$$

Proof.

If we combine lemmas (2.1) and (2.2) it follows that for $\mathfrak {R}(s) > 1$,

$$\begin{aligned}&\log \big ( L(s, \psi _{h/q}) \big ) = \sum _p \sum _{k=1}^\infty \frac{e^{\frac{2\pi i h p^k}{q}}}{k \, p^{sk}} \; + \; T_{h,q}(s)\\&= \sum _{p\,\not \mid \,q} \sum _{k=1}^\infty \frac{e^{\frac{2\pi i h p^k}{q}}}{k \, p^{sk}} \; +\;\sum _{p\,\mid \,q} \sum _{k=1}^\infty \frac{e^{\frac{2\pi i h p^k}{q}}}{k \, p^{sk}} + \; T_{h,q}(s) \\&= \; \sum _{p\,\not \mid \,q} \sum _{k=1}^\infty \frac{ \left( \frac{1}{\phi (q)} \underset{\chi \ne \chi _0}{\sum \limits _{\chi \pmod {q}}} \tau (\chi )\cdot \overline{\chi (h\,p^k)} \; + \; \frac{\mu (q)}{\phi (q)}\right) }{k \, p^{sk}} \; +\;\sum _{p\,\mid \,q} \sum _{k=1}^\infty \frac{e^{\frac{2\pi i h p^k}{q}}}{k \, p^{sk}} + \; T_{h,q}(s). \end{aligned}$$

Hence

$$\begin{aligned} \log \big ( L(s, \psi _{h/q}) \big )&= \;\frac{1}{\phi (q)} \underset{\chi \ne \chi _0}{\sum _{\chi \!\!\!\!\pmod {q}}} \tau (\chi ) \overline{\chi (h)}\, \log (L(s, \overline{\chi }) \; + \; \frac{\mu (q)}{\phi (q)} \log \big ( \zeta (s) \big )\nonumber \\&- \frac{\mu (q)}{\phi (q)}\sum _{p\mid q}\sum _{k=1}^\infty \frac{1}{kp^{sk}} \; + \; \sum _{p\,\mid \,q} \sum _{k=1}^\infty \frac{e^{\frac{2\pi i h p^k}{q}}}{k \, p^{sk}} \; + \; T_{h,q}(s). \end{aligned}$$

The theorem immediately follows after taking exponentials.    $\square$

The representation of $L(s, \psi _{h/q})$ given in theorem 2.3 allows one to analytically continue the function $L(s, \psi _{h/q})$ to a larger region which lies to the left of the line $\mathfrak {R}(s) = 1 +\varepsilon$ ($\varepsilon > 0$). This is a region which does not include the branch points of $L(s, \psi _{h/q})$ at the zeros and poles of $L(s,\chi ), \zeta (s)$.

Assume that $q > 1$ and $\chi \pmod {q}$. It is well known (see [2]) that the Dirichlet L-functions $L(\sigma +{ it},\chi )$) do not vanish in the region

$$\begin{aligned} \sigma \ge {\left\{ \begin{array}{ll} 1 - \frac{c_1}{\log q|t|} &{} \text {if} \; |t| \ge 1,\\ 1 - \frac{c_2}{\log q} &{} \text {if} \; |t| \le 1, \end{array}\right. } \qquad \quad (\text {for absolute constants} \; c_1,c_2 >0), \end{aligned}$$

(4)

unless $\chi$ is the exceptional real character which has a simple real zero (Siegel zero) near $s = 1.$

Similarly, $\zeta (\sigma +{ it})$ does not vanish for

$$\begin{aligned} \sigma \ge 1 - \frac{c_3}{\log (|t| + 2)}, \qquad \qquad (\text {for an absolute constant} \;c_3 > 0). \end{aligned}$$

(5)

Assume $q > 1$ and that there is no exceptional real character (mod q). It follows from (4) and (5) that $L(s, \psi _{h/q})$ is holomorphic in the region to the right of the contour $\mathscr {C}_q$ displayed in Figure 1.

Fig. 1

The contour $\mathscr {C}_q$

To construct the contour $\mathscr {C}_q$ first take a slit along the real axis from $1 - \frac{c_2}{\log q}$ to 1 and construct a line just above and just below the slit. Then take two asymptotes to the line $\mathfrak {R}(s) = 1$ with the property that if $\sigma + { it}$ is on the asymptote and $|t| \ge 1$, then $\sigma$ satisfies (4). If $q =1$, we do a similar construction using (5).

3 Proof of theorem 1.1

The proof of theorem 1.1 is based on the following theorem.

Theorem 3.1.

Let h, q be fixed coprime integers with $q > 2$ and $\mu (q) \ne 0.$ Then for $x\rightarrow \infty$ there exist absolute constants $c,c' > 0$ such that

$$\begin{aligned}&\sum _{n\le x} e^{2\pi i \frac{h A(n)}{q}}\\&= \frac{ \sin \left( \frac{\mu (q) \,\pi }{\phi (q)} \right) }{\pi } \int \limits _{1-\frac{c }{\sqrt{\log x}}}^1 \left( \underset{\chi \ne \chi _0}{\prod _{\chi \!\!\!\!\pmod {q}}} \;L(\sigma , \,\overline{\chi })^{\frac{\tau (\chi )\overline{\chi (h)}}{\phi (q)}}\right) \cdot |\zeta (\sigma )|^{\frac{\mu (q)}{\phi (q)}}\cdot e^{H_{h,q}(\sigma )} \; \frac{x^\sigma }{\sigma } \; d\sigma \\&+ \mathscr {O}\left( x e^{-c'\,\sqrt{\log x}} \right) . \end{aligned}$$

On the other hand if $\mu (q) = 0$, then $\sum \limits _{n\le x} e^{2\pi i \frac{h A(n)}{q}} = \mathscr {O}\left( x e^{-c'\,\sqrt{\log x}} \right) .$

Proof.

The proof of theorem 3.1 relies on the following lemma taken from [2].

Lemma 3.2.

Let

$$\delta (x) := {\left\{ \begin{array}{ll} 0, &{} \text {if} \; 0< x < 1,\\ \frac{1}{2}, &{} \text {if} \; x = 1,\\ 1, &{} \text {if} \; x > 1, \end{array}\right. }$$

then for $x, T > 0$, we have

$$ \left| \frac{1}{2\pi i} \int \limits _{c-iT}^{c+iT}\; \frac{x^s}{s}\; ds \; - \; \delta (x)\right| \; < \; {\left\{ \begin{array}{ll} x^c\cdot \min \left( 1, \frac{1}{T |\log x| } \right) , &{} \text {if}\; x \ne 1,\\ cT^{-1}, &{} \text {if} \; x = 1. \end{array}\right. } $$

It follows from lemma 3.2, for $x, T \gg 1$ and $c = 1 + \frac{1}{\log x}$, that

$$\begin{aligned} \frac{1}{2\pi i}\int \limits _{c-iT}^{c+iT} L\big (s, \psi _{h/q}\big ) \;\frac{x^s}{s} \; ds = \sum _{n\le x} \psi _{h/q}(n) \; + \; \mathscr {O}\left( \frac{x \log x}{T} \right) . \end{aligned}$$

(6)

Fix large constants $c_1, c_2 > 0.$ Next, shift the integral in (6) to the left and deform the line of integration to a contour

$$L^+ \; + \; \mathscr {C}_{T,x}\; + \; L^-$$

as in figure 2 below which contains two short horizontal lines:

$$L^{\pm } = \bigg \{\sigma \pm iT \;\, \bigg | \;\, 1 -\frac{c_1}{\log qT} \; \le \; \sigma \; \le \; 1+\frac{1}{\log x}\bigg \},$$

together with the contour $C_{T,x}$ which is similar to $C_q$ except that the two curves asymptotic to the line $\mathfrak {R}(s) = 1$ go from $1 -\frac{c_1}{\sqrt{\log qT}} + iT$ to $1 -\frac{c_2}{\sqrt{\log x}} +i\varepsilon$ and $1 -\frac{c_2}{\sqrt{\log x}} - i\varepsilon$ to $1 -\frac{c_1}{\sqrt{\log qT}} - iT$, respectively, for $0 <\varepsilon \rightarrow 0.$

Fig. 2

The contour $\mathscr {C}_{T,x}$

Now, by the zero-free regions (4), (5), the region to the right of the contour $L^+ + \mathscr {C}_{T,x} + L^-$ does not contain any branch points or poles of the L-functions $L(s, \chi )$ for any $\chi \pmod {q}$. It follows that

$$\begin{aligned} \frac{1}{2\pi i} \int \limits _{c-iT}^{c+iT} L\big (s, \psi _{h/q}\big ) \;\frac{x^s}{s} \; ds \; = \; \frac{1}{2\pi i}\left( \int _{L^+} + \int _{\mathscr {C}_\varepsilon } + \int _{L^-}\right) L\big (s, \psi _{h/q}\big ) \;\frac{x^s}{s} \; ds . \end{aligned}$$

(7)

The main contribution for the integral along $L^+ + \mathscr {C}_{T,x} + L^-$ in (7) comes from the integrals along the straight lines above and below the slit on the real axis$\Big [1-\frac{c_2}{\sqrt{\log x}}, \;1\Big ].$ These integrals cancel if the function $L\big (s, \psi _{h/q}\big )$ has no branch points or poles on the slit. It follows from theorem 2.3 that this will be the case if $\mu (q) = 0$. The remaining integrals in (7) can then be estimated as in the proof of the prime number theorem for arithmetic progressions (see [2]), yielding an error term of the form $\mathscr {O}\left( x e^{-c'\,\sqrt{\log x}} \right)$. This proves the second part of theorem 3.1.

Next, assume $\mu (q) \ne 0.$ In this case $L(s,\psi _{h/q})$ has a branch point at $s=1$ coming from the Riemann zeta function, it is necessary to keep track of the change in argument. Let $0^+i$ denote the upper part of the slit and let $0^- i$ denote the lower part of the slit. Then we have $\log [\zeta (\sigma +0^+ i) = \log |\zeta (\sigma )| - i\pi$ and $\log [\zeta (\sigma +0^- i) = \log |\zeta (\sigma )| + i\pi$.

By the standard proof of the prime number theorem for arithmetic progressions it follows that (with an error $\mathscr {O}\big (e^{-c'\sqrt{\log x}} \big )$) the right hand side of (7) is asymptotic to

$$\begin{aligned} \mathscr {I}_{\text {slit}}&:=\frac{-1}{2\pi i} \int \limits _{1-\frac{c}{\sqrt{\log x}}}^1 \left[ \exp \Big (\log \left( L\left( \sigma + 0^+i, \; \psi _{h/q} \right) \right) \Big ) \right. \nonumber \\&\left. - \exp \Big (\log \left( L\left( \sigma -0^-i, \; \psi _{h/q} \right) \right) \Big )\right] \frac{x^\sigma }{\sigma } \; d\sigma . \end{aligned}$$

(8)

We may evaluate $\mathscr {I}_{\text {slit}}$ using theorem 2.3. This gives

$$\begin{aligned} \mathscr {I}_{\text {slit}}&= \;\frac{-1}{2\pi i} \int \limits _{1-\frac{c}{\sqrt{\log x}}}^1 \left( \underset{\chi \ne \chi _0}{\prod _{\chi \!\!\!\!\pmod {q}}} \;L(\sigma ,\,\overline{\chi })^{\frac{\tau (\chi )\overline{\chi (h)}}{\phi (q)}}\right) \cdot e^{U_{h,q}(\sigma )}\\&\cdot \left[ \exp \left( \frac{\mu (q)}{\phi (q)}\Big ( \log |\zeta (\sigma )| -i\pi \Big )\right) \; - \; \exp \left( \frac{\mu (q)}{\phi (q)}\Big ( \log |\zeta (\sigma )| +i\pi \Big )^{} \right) \right] \frac{x^\sigma }{\sigma } \; d\sigma \\&\\&= \; \frac{\sin \left( \frac{\mu (q) \,\pi }{\phi (q)} \right) }{\pi } \int \limits _{1-\frac{c}{\sqrt{\log x}}}^1 \left( \underset{\chi \ne \chi _0}{\prod _{\chi \!\!\!\!\pmod {q}}} \;L(\sigma , \,\overline{\chi })^{\frac{\tau (\chi )\overline{\chi (h)}}{\phi (q)}}\right) \cdot |\zeta (\sigma )|^{\frac{\mu (q)}{\phi (q)}}\cdot e^{U_{h,q}(\sigma )} \; \frac{x^\sigma }{\sigma } \; d\sigma . \end{aligned}$$

As in the previous case when $\mu (q) = 0,$ the remaining integrals in (7) can then be estimated as in the proof of the prime number theorem for arithmetic progressions, yielding an error term of the form $\mathscr {O}\left( x e^{-c'\,\sqrt{\log x}} \right)$. This completes the proof of theorem 3.1.    $\square$

The proof of theorem 1.1 follows from theorem 3.1 if we can obtain an asymptotic formula for the integral

$$\begin{aligned} \mathscr {I}_{\text {slit}}\; = \; \frac{\sin \left( \frac{\mu (q) \,\pi }{\phi (q)} \right) }{\pi } \int \limits _{1-\frac{c}{\sqrt{\log x}}}^1 \left( \underset{\chi \ne \chi _0}{\prod _{\chi \!\!\!\!\pmod {q}}} \;L(\sigma , \,\overline{\chi })^{\frac{\tau (\chi )\overline{\chi (h)}}{\phi (q)}}\right) \cdot |\zeta (\sigma )|^{\frac{\mu (q)}{\phi (q)}}\cdot e^{U_{h,q}(\sigma )} \; \frac{x^\sigma }{\sigma } \; d\sigma . \end{aligned}$$

(9)

Since we have assumed q is fixed, it immediately follows that for arbitrarily large $c \gg 1$ and $x \rightarrow \infty ,$ we have

$$\begin{aligned}&I_{\text {slit}}\; = \; \frac{\sin \left( \frac{\mu (q) \,\pi }{\phi (q)} \right) }{\pi } \int \limits _{1-\frac{c \log \log x}{\log x}}^1 \left( \underset{\chi \ne \chi _0}{\prod _{\chi \!\!\!\!\pmod {q}}} \;L(\sigma , \,\overline{\chi })^{\frac{\tau (\chi )\overline{\chi (h)}}{\phi (q)}}\right) \cdot |\zeta (\sigma )|^{\frac{\mu (q)}{\phi (q)}} \,\cdot \,e^{U_{h,q}(\sigma )} \; \frac{x^\sigma }{\sigma } \; d\sigma \;\, \\&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + \;\, \mathscr {O}\left( \frac{x}{(\log x)^c} \right) . \end{aligned}$$

Now, in the region $1-\frac{c \log \log x}{\log x} \le \sigma \le 1$,

$$\underset{\chi \ne \chi _0}{\prod _{\chi \!\!\!\!\pmod {q}}} \;L(\sigma , \,\overline{\chi })^{\frac{\tau (\chi )\overline{\chi (h)}}{\phi (q)}}\cdot \frac{e^{H_{h,q}(\sigma )}}{\sigma } = \underset{\chi \ne \chi _0}{\prod _{\chi \!\!\!\!\pmod {q}}} \;L(1, \,\chi )^{\frac{\tau (\overline{\chi })\chi (h)}{\phi (q)}}\cdot e^{U_{h,q}(1)} \; + \; \mathscr {O}\left( \frac{\log \log x}{\log x} \right) .$$

Consequently,

$$\begin{aligned} I_{\text {slit}}\;&= \; \frac{ \sin \left( \frac{\mu (q) \,\pi }{\phi (q)} \right) }{\pi } \underset{\chi \ne \chi _0}{\prod _{\chi \!\!\!\!\pmod {q}}} \;L(1, \,\chi )^{\frac{\tau (\overline{\chi })\chi (h)}{\phi (q)}}\cdot e^{U_{h,q}(1)} \int \limits _{1-\frac{c \log \log x}{\log x}}^1 \zeta (\sigma )^{\frac{\mu (q)}{\phi (q)}}\; x^\sigma \; d\sigma \nonumber \\&+ \mathscr {O}\left( \frac{\log \log x}{\log x} \left| \;\; \int \limits _{1-\frac{c \log \log x}{\log x}}^1 \zeta (\sigma )^{\frac{\mu (q)}{\phi (q)}}\; x^\sigma \; d\sigma \;\;\right| \right) . \end{aligned}$$

(10)

It remains to compute the integral of $|\zeta (\sigma )|^{\frac{\mu (q)}{\phi (q)}}$ occurring in (10). For $\sigma$ very close to 1, we have

$$|\zeta (\sigma )|^{\frac{\mu (q)}{\phi (q)}} = \left( \frac{1}{|\sigma -1|} \;+\; \mathscr {O}(1) \right) ^{\frac{\mu (q)}{\phi (q)}} = \left( \frac{1}{|\sigma -1|} \right) ^{\frac{\mu (q)}{\phi (q)}} \; + \; \mathscr {O}\left( \left( \frac{1}{|\sigma -1|} \right) ^{\frac{\mu (q)}{\phi (q)}-1} \right) .$$

It follows that

$$\begin{aligned} \int \limits _{1-\frac{c \log \log x}{\log x}}^1 |\zeta (\sigma )|^{\frac{\mu (q)}{\phi (q)}}\; x^\sigma \; d\sigma \; = \; \varGamma \left( 1 - \frac{\mu (q)}{\phi (q)} \right) \; \frac{x}{(\log x)^{1-\frac{\mu (q)}{\phi (q)}}} \; + \; \mathscr {O}\left( \frac{x}{(\log x)^{2-\frac{\mu (q)}{\phi (q)}}} \right) . \end{aligned}$$

(11)

Combining equations (10) and (11) we obtain

$$\begin{aligned} I_{\text {slit}}=\frac{\sin \left( \frac{\mu (q) \,\pi }{\phi (q)} \right) }{\pi }\, \varGamma \left( 1 - \frac{\mu (q)}{\phi (q)} \right) \underset{\chi \ne \chi _0}{\prod _{\chi \!\!\!\!\pmod {q}}} L(1, \,\chi )^{\frac{\tau (\overline{\chi })\chi (h)}{\phi (q)}}\,&e^{U_{h,q} (1)} \; \frac{x}{(\log x)^{1-\frac{\mu (q)}{\phi (q)}}} \;\\ +&\mathscr {O}\left( \frac{x}{(\log x)^{2-\frac{\mu (q)}{\phi (q)}}} \right) . \end{aligned}$$

Remark: As pointed out to me by Gérald Tenenbaum, it is also possible to deduce Corollary 1.2 directly from theorem 2.3 by using theorem II.5.2 of [8]. In this manner one can obtain an explicit asymptotic expansion which, furthermore, is valid for values of q tending to infinity with x.

4 Examples of equidistribution (mod 3) and (mod 9)

Equidistribution (mod 3): Theorem (1.1) says that for $h = 1, \; q = 3:$

$$\begin{aligned} \sum _{n \, \le \, x} e^{\frac{2\pi i A(n)}{3}} \;&= \;{\frac{-V_{1,3}}{ \pi }\;\varGamma \left( \frac{3}{2}\right) \underset{\chi \ne \chi _0}{\prod \limits _{\chi \pmod {3}}} \;L(1, \,\chi )^{ \frac{G(\overline{\chi })}{2}} } \frac{x}{(\log x)^{\frac{3}{2}}}\; { \Big (1 + \mathscr {O}\left( \frac{1}{\log x}\right) \Big )}\\&\approx \; (-0.503073 + 0.24042\, i) \, \frac{x}{(\log x)^{\frac{3}{2}}}. \end{aligned}$$

We computed the above sum for $x = 10^7$ and obtained

$$\sum _{n \, \le \, 10^7} e^{\frac{2\pi i A(n)}{3}} \approx -98,423.00 + 55,650.79\, i.$$

Our theorem predicts that

$$\sum _{n \, \le \, 10^7} e^{\frac{2\pi i A(n)}{3}} \; \approx \; -88,870.8 + 42,471.7 \;i.$$

Since $\log \left( 10^7\right) \approx 16.1$ is small, this explains the discrepancy between the actual and predicted results.

As $x \rightarrow \infty ,$ we have

$$\begin{aligned} \underset{A(n)\; \equiv \; a \pmod {3}}{\sum _{n\, \le \, x} } \;\;&= \;\; \frac{1}{3}\sum _{h = 0}^2 \sum _{n\, \le \, x} e^{ \frac{2\pi i A(n)h}{3} } e^{-\frac{2\pi i h\,a}{3}} \;\; = \;\; \frac{x}{3} \; + c_a \; \frac{x}{(\log x)^{\frac{3}{2}}} \; + \; \mathscr {O}\left( \frac{x}{(\log x)^{\frac{5}{2}}} \right) \end{aligned}$$

where

$$c_0 = -0.335382, \qquad c_1 \; \approx \;0.306498, \qquad c_2 \; \approx \; 0.0288842.$$

Equidistribution (mod 9):

Our theorem says that for $h \ne 3,6$ ($1\le h<9)$ and $q = 9$:

$$\sum _{n \, \le \, x} e^{\frac{2\pi i h A(n)}{9}}\; = \; \mathscr {O}\left( x\, e^{-c_0\sqrt{\log x}} \right) .$$

Surprisingly!! there is a huge amount of cancellation when $x = 10^7:$

$$ \sum _{n \, \le \, 10^7} e^{\frac{2\pi i h A(n)}{9}}\; \approx \; {\left\{ \begin{array}{ll} -315.2 -140.4\, i &{} \text {if} \; h = 1,\\ \;282.2 - 543.4\, i &{} \text {if} \; h = 2,\\ \; 94.5 + 321.9 \, i &{} \text {if} \; h = 4,\\ \; 94.5 - 321.9 \, i &{} \text {if} \; h = 5,\\ \;282.2 + 543.4\, i &{} \text {if} \; h = 7,\\ -315.2 +140.4\, i &{} \text {if} \; h = 8.\\ \end{array}\right. }$$