Moments of the Hurwitz zeta function on the critical line

Estimating the moments of the Riemann zeta function $\zeta(s)$ on the critical line,

$$M_{k}(T)=\int_{T}^{2T}\left|\zeta\left(\tfrac{1}{2}+it\right)\right|^{2k}\,dt,$$

is a classical problem in analytic number theory (see [43, Chapter VII]). It is widely believed that $M_{k}(T)\sim c_{k}T(\log T)^{k^{2}}$ for all real $k\geqslant 0$, where $c_{k}$ is a fixed positive constant depending only on $k$. This conjecture is trivial for $k=0$, was proved by Hardy and Littlewood [20] for $k=1$, was proved by Ingham [30] for $k=2$, and is wide open in all other cases.

Despite the history and intractability of the problem, very precise conjectures for the exact value of $c_{k}$ exist. On the basis of number theoretic calculations, Conrey and Ghosh [13] conjectured the value of $c_{k}$ for $k=3$ and by a different, but still number theoretic, method Conrey and Gonek [14] conjectured the value of $c_{k}$ for $k=3,4$. Finally, using heuristics modeling $\zeta(s)$ by characteristic polynomials of random matrices from the Gaussian unitary ensemble, Keating and Snaith [32] conjectured the value of $c_{k}$ for all $k>0$, agreeing with the conjectures from [13] and [14].

The analogy with random matrix theory has led to many fruitful conjectures for moments of $L$-functions; see, for example, [12] and the references therein for details.

A weaker, and hence theoretically more tractable version of the above conjecture is the estimate $M_{k}(T)\asymp_{k}T(\log T)^{k^{2}}$. By work of Ramachandra [37, 38, 39], and Heath-Brown [28], the lower bound $M_{k}(T)\gg_{k}T(\log T)^{k^{2}}$ was known conditionally on the Riemann Hypothesis (RH) for $k>0$, and by work of Radziwiłł and Soundararajan [36], it was known unconditionally for all $k\geqslant 1$. Recent work of Heap and Soundararajan [27] establishes the lower bound unconditionally for all $k>0$.

For the upper bound, Soundararajan [41] had shown on RH that $M_{k}(T)\ll_{k,\epsilon}T(\log T)^{k^{2}+\epsilon}$ for every $\epsilon>0$ and $k>0$. Harper [21] removed the dependence on $\epsilon$, conditionally establishing the sharp upper bound for every $k>0$. The upper bound was known unconditionally for $k=1/n$, $n\in\mathbb{N}$ due to Heath-Brown [28], and for $k=1+1/n$, $n\in\mathbb{N}$ due to Bettin, Chandee and Radziwiłł [5]. Recently, Heap, Radziwiłł and Soundararajan [25] subsumed both of these results by proving the upper bound unconditionally for $0\leqslant k\leqslant 2$.

The object of this paper is to investigate analogous moments of the Hurwitz zeta function, $\zeta(s,\alpha)$. For $0<\alpha\leqslant 1$ and for $\Re s>1$, $\zeta(s,\alpha)$ is defined by the series

$$\zeta(s,\alpha)=\sum_{n=0}^{\infty}\frac{1}{(n+\alpha)^{s}}.$$

As with the Riemann zeta function, the Hurwitz zeta function can be continued to a meremorphic function on the entire complex plane with a simple pole at $s=1$ satisfying the Hurwitz relation (an analogue of the functional equation, see [33]). Clearly, $\zeta(s,1)=\zeta(s)$ and $\zeta(s,1/2)=(2^{s}-1)\zeta(s)$. For these values of $\alpha$, thus, $\zeta(s,\alpha)$ has an Euler product, derived from the usual Euler product for $\zeta(s)$. However, for $0<\alpha<1$, $\alpha\neq 1/2$, $\zeta(s,\alpha)$ does not have an Euler product, and, in fact, the behaviour of its zeroes is very different from that of $\zeta(s)$. Spira [42] showed that like $\zeta(s)$, $\zeta(s,\alpha)$ may have trivial zeros on the negative real line, and also that $\zeta(s,\alpha)$ is zero-free in the region $\Re s\geqslant 1+\alpha$. However, it is well-known, due to Davenport and Heilbronn [15] for the cases of rational or transcendental $\alpha$, and due to Cassels [11] for the case of algebraic irrational $\alpha$ that if $\alpha\neq 1/2,1$, then $\zeta(s,\alpha)$ always has a zero in the strip $1<\Re s<1+\delta$ for every $\delta>0$. Voronin showed [45] that for rational $\alpha\neq 1/2,1$, and fixed $\sigma_{1},\sigma_{2}$ with $1/2<\sigma_{1}<\sigma_{2}<1$, there are infinitely many zeros of $\zeta(s,\alpha)$ satisfying $\sigma_{1}<\Re s<\sigma_{2}$. This result was established also for transcendental $\alpha$ by Gonek [18]. Finally, Gonek [19] showed that if $\alpha=a/q$ with $(a,q)=1$, and $\varphi(q)=2$, where $\varphi$ is Euler’s totient function, then $\zeta(s,\alpha)$ has a positive proportion of its nontrivial zeros off the critical line $\Re s=1/2$. All of these are in contrast to the expected behaviour of $\zeta(s)$.

To study the moments of the Hurwitz zeta functions in the critical line, we define in analogy with $M_{k}(T)$,

(1)

$$M_{k}(T;\alpha)=\int_{T}^{2T}\left|\zeta\left(\tfrac{1}{2}+it,\alpha\right)\right|^{2k}\,dt,$$

so that $M_{k}(T;1)=M_{k}(T)$. One might expect the following:

When $k=1$, this is a theorem due to Rane [40, Theorem 2], with $c_{1}(\alpha)=1$. In fact, he proved for $0<\alpha\leqslant 1$ (not necessarily rational),

(2)

$$\begin{split}M_{1}(T;\alpha){}=T\log T+B(\alpha)T-\frac{1}{\alpha}+O\left(\frac{T^{1/2}\log T}{\alpha^{1/2}}\right)\end{split}$$

uniformly in $\alpha$ and $T$ with an effective constant $B(\alpha)$. This was improved further by several authors, with the current best error term due to Zhan [48, Theorem 2].

Winston Heap (private communication) has indicated that the above conjecture may not extend to irrational shifts $\alpha$ if $k\neq 1$, and the true behaviour in this case appears quite delicate. Heap and the author are currently exploring the higher moments and distributions of Hurwitz zeta functions with irrational shifts in an ongoing project. A Diophantine problem with a paucity of non-diagonal solutions connected to these moments was considered by Heap, the author, and Wooley in [26] (see also [6, Theorem 26]).

For $k=2$, the conjecture can be proved using methods for fourth moments of $L$-functions of degree $1$. This was done in an unpublished section of Andersson’s thesis [1, pp. 71-72]. We restate and reprove this result here for convenience:

We show later that this agrees with our conjecture for $c_{k}(\alpha)$. In principle, one could also work out the lower order terms in this asymptotic.

Our goal for the rest of the paper is to provide evidence for Conjecture 1.1 when $k\in\mathbb{N}$. In this case, $M_{k}(T;\alpha)$ can be related to the mean square of products of Dirichlet $L$-functions on the critical line.

To explain this connection, we fix some notation that will be used throughout the paper. We assume $\alpha=a/q$ with $(a,q)=1$ and $1\leqslant a\leqslant q$. Dirichlet characters will be denoted $\chi$ or $\nu$, and will be modulo $q$ unless noted otherwise. We will use bolded, lower case (Greek or Latin) letters such as $\boldsymbol{\ell}$ for tuples of natural numbers indexed by characters modulo $q$. Thus, if $\boldsymbol{\ell}$ is such a tuple, we think of it as a function $\boldsymbol{\ell}:\mathcal{D}(q)\to\mathbb{N}$ where $\mathcal{D}(q)$ is the set of Dirichlet characters modulo $q$. We denote $\boldsymbol{\ell}(\chi)$ as $\ell_{\chi}$. Further, we define,

$$|\boldsymbol{\ell}|=\sum_{\chi}\ell_{\chi},\,\lambda(\boldsymbol{\ell})=\sum_{\chi}\ell_{\chi}^{2},\,\mathcal{L}^{\boldsymbol{\ell}}(s)=\prod_{\chi}L(s,\chi)^{\ell_{\chi}}.$$

Here, and later, sums and products over $\chi$ or $\nu$ run over $\mathcal{D}(q)$. If $\boldsymbol{\ell}$ is clear from context, we suppress it and denote $\lambda(\boldsymbol{\ell})$ simply as $\lambda$. Finally, we denote by $d_{\boldsymbol{\ell}}(n)$ the coefficient of $n^{-s}$ in the Dirichlet series expansion of $\mathcal{L}^{\boldsymbol{\ell}}(s)$.

To see how products of the form $\mathcal{L}^{\boldsymbol{\ell}}(s)$ arise naturally in considering Conjecture 1.1 for $\alpha=a/q$, we observe that for $\Re{s}>1$, the orthogonality of Dirichlet characters gives

$$\begin{split}\zeta(s,\alpha)&{}=\frac{q^{s}}{\varphi(q)}\sum_{\chi}\overline{\chi}(a)L(s,\chi).\end{split}$$

By analytic continuation, this equality holds everywhere. Thus, by the multinomial theorem

(3)

$$\begin{split}|\zeta(s,\alpha)|^{2k}&{}=\bigg{|}\frac{q^{ks}}{\varphi(q)^{k}}\sum_{|\boldsymbol{\ell}|=k}\binom{k}{\boldsymbol{\ell}}\prod_{\chi}\bigg{\{}\overline{\chi}(a)L(s,\chi)\bigg{\}}^{\ell_{\chi}}\bigg{|}^{2}\\ &{}=\frac{q^{2k\sigma}}{\varphi(q)^{2k}}\sum_{\begin{subarray}{c}|\boldsymbol{\ell}^{(1)}|=k\\ |\boldsymbol{\ell}^{(2)}|=k\end{subarray}}\binom{k}{\boldsymbol{\ell}^{(1)}}\binom{k}{\boldsymbol{\ell}^{(2)}}s(a;\boldsymbol{\ell}^{(1)},\boldsymbol{\ell}^{(2)})\mathcal{L}^{\boldsymbol{\ell}^{(1)}}(s)\overline{\mathcal{L}^{\boldsymbol{\ell}^{(2)}}(s)},\end{split}$$

where $\binom{k}{\boldsymbol{\ell}}=k!/\prod_{\chi}\ell_{\chi}!$ are multinomial coefficients, the sums runs over $\boldsymbol{\ell}$ such that $|\boldsymbol{\ell}|=\sum_{\chi}\ell_{\chi}=k$, and $s(a;\boldsymbol{\ell}^{(1)},\boldsymbol{\ell}^{(2)})=\prod_{\chi}\chi(a)^{\ell^{(2)}_{\chi}-\ell^{(1)}_{\chi}}$. In particular, when we integrate both sides from $1/2+iT$ to $1/2+i2T$, the terms of this sum whose phase oscillates will probably not contribute to the main term. The terms that do not have oscillations correspond to the diagonal terms $\boldsymbol{\ell}^{(1)}=\boldsymbol{\ell}^{(2)}$ where the phases of each term in the product cancel out, yielding a positive real number. Thus, heuristically,

(4)

$$\begin{split}M_{k}(T;\alpha)&{}=\int_{T}^{2T}\left|\zeta\left(\tfrac{1}{2}+it,\alpha\right)\right|^{2k}\,dt\\ &{}\approx\frac{q^{k}}{\varphi(q)^{2k}}\sum_{|\boldsymbol{\ell}|=k}\binom{k}{\boldsymbol{\ell}}^{2}\int_{T}^{2T}\left|\mathcal{L}^{\boldsymbol{\ell}}\left(\tfrac{1}{2}+it\right)\right|^{2}\,dt.\end{split}$$

whence, the problem of estimating $M_{k}(T;\alpha)$ naturally reduces to studying the mean square of $\mathcal{L}^{\boldsymbol{\ell}}(s)$ along the critical line.

To study such moments, we will use a hybrid Euler-Hadamard product, a tool introduced originally by Gonek, Hughes and Keating [17] in the context of the Riemann zeta function. Specifically, we will need the following version for Dirichlet $L$-functions in the $t$-aspect:

Such a hybrid Euler-Hadamard product was proved by Bui and Keating [9] in their study of moments in the $q$-aspect of Dirichlet $L$-functions at the central point $s=1/2$ (see [9, Remark 1]). Similar hybrid Euler-Hadamard products have been used in the literature for studying moments in many other contexts such as for for orthogonal and symplectic families of $L$-functions [10]; for $\zeta^{\prime}(s)$ [8]; for the Dedekind zeta function $\zeta_{\mathbb{K}}(s)$ of a Galois extension $\mathbb{K}$ of $\mathbb{Q}$ [23]; for quadratic Dirichlet $L$-functions over function fields [7], [3]; for normalized symmetric square $L$-functions associated with $SL_{2}(\mathbb{Z})$ eigenforms [16]; and for quadratic Dirichlet $L$-functions over function fields associated to irreducible polynomials [2].

With $P(s,\chi)$ and $Z(s,\chi)$ as in Theorem 1.3, we define

$$\mathcal{P}_{X}^{\boldsymbol{\ell}}(s)=\prod_{\chi}P_{X}(s,\chi)^{\ell_{\chi}},\,\mathcal{Z}_{X}^{\boldsymbol{\ell}}(s)=\prod_{\chi}Z_{X}(s,\chi)^{\ell_{\chi}}.$$

We can view $\mathcal{L}^{\boldsymbol{\ell}}(s)$ as an $L$-function of degree $|\boldsymbol{\ell}|$, $\mathcal{P}_{X}^{\boldsymbol{\ell}}(s)$ as an approximation to its Euler product, and $\mathcal{Z}_{X}^{\boldsymbol{\ell}}(s)$ as an approximation to its Hadamard product. Roughly, Theorem 1.3 implies that $\mathcal{L}^{\boldsymbol{\ell}}(s)\approx\mathcal{P}_{X}^{\boldsymbol{\ell}}(s)\mathcal{Z}_{X}^{\boldsymbol{\ell}}(s)$.

As is usually the case with hybrid Euler-Hadamard products, $X$ mediates between the primes and zeroes; if we want to take fewer primes in the Euler product we must take more zeroes in the Hadamard product and vice-versa.

For $X$ growing relatively slowly with $T$, we expect the two terms in the decomposition $\mathcal{L}^{\boldsymbol{\ell}}(s)\approx\mathcal{P}_{X}^{\boldsymbol{\ell}}(s)\mathcal{Z}_{X}^{\boldsymbol{\ell}}(s)$ to behave like independent random variables due to a separation of scales. This is analogous to the splitting conjecture of Gonek, Hughes and Keating [17, Conjecture 2]. Concretely, we have:

On [17, p. 511], it is suggested that this splitting conjecture holds for a much wider range of $X$ and $T$ with $X=o(T)$. Recently, Heap [24] has justified this suggestion. He proved on RH that the splitting conjecture for $\zeta(s)$ holds for every $k>0$ and a much wider range of $X$ provided one requires only an order of magnitude result, instead of an asymptotic. He also established the splitting conjecture for $k=1$ and $k=2$ for wider ranges of $X$ both with and without RH.

The mean square of $\mathcal{P}_{X}^{\boldsymbol{\ell}}(s)$ can be computed exactly.

One could prove a similar result uniformly in $c$ on any vertical line $\Re{s}=\sigma$ with $1>\sigma\geqslant c\geqslant 1/2$ given $X\ll_{\epsilon}(\log T)^{1/(1-c+\epsilon)}$ , but we choose not to do so for conciseness. Note that the product over $p$ in (6) is conditionally convergent but not absolutely convergent.

For the mean square of $\mathcal{Z}_{X}^{\boldsymbol{\ell}}(s)$, we use random matrix theory to model each $L$-function appearing in the product by random unitary matrices. One expects that the matrices representing distinct $L$-functions behave independently as in [23, Conjecture 2]. This leads to:

It is clear that one can use Conjectures 1.4 and 1.6 together with Theorem 1.5 to get a conjectural asymptotic for $\int_{T}^{2T}|\mathcal{L}^{\boldsymbol{\ell}}(1/2+it)|^{2}\,dt$. Precisely, we get,

Note that for a fixed $q$, the above says that the mean square of a product of Dirichlet $L$-functions grows as $\asymp_{k,q}T(\log T)^{\lambda}$. This is known for $|\boldsymbol{\ell}|\leqslant 2$, and we shall show that in these cases our predicted constant matches up.

Due to the conditional hypotheses, the above theorem is really a conjecture. We note here that Heap made a similar conjecture about moments of products of $L$-functions from the Selberg class (see [23, Section 6]) using the recipe of Conrey, Farmer, Keating, Rubinstein and Snaith [12]. Specializing to Dirichlet $L$-functions, one can recover the above conjecture.

He also discussed how such conjectures could be reproduced by using hybrid Euler-Hadamard products under appropriate hypotheses. However, since he has not worked out the details of this approach in this specific context, we do so here for completeness.

It is evident that our previous discussion about (4) and Theorem 1.7 can together be used to compute the correct value of $c_{k}(\alpha)$ in Conjecture 1.1.

Since the current levels of technology can handle second moments and fourth moments of $\zeta(s)$ really well, it is natural to hope that we can prove Conjectures 1.4 and 1.6 for $|\boldsymbol{\ell}|\leqslant 2$. We define the Kronecker delta $\boldsymbol{\delta}^{\chi}$ by

$$\delta_{\nu}^{\chi}=\begin{cases}1&\text{ if }\chi=\nu\\ 0&\text{ if }\chi\neq\nu.\end{cases}$$

Then, we can prove:

The above theorem can almost certainly be extended to the case $|\boldsymbol{\ell}|=2$. This corresponds to $\boldsymbol{\ell}=\boldsymbol{\delta}^{\chi}+\boldsymbol{\delta}^{\nu}$, and $\mathcal{L}^{\boldsymbol{\ell}}(s)=L(s,\chi)L(s,\nu)$ with $\chi$ and $\nu$ not necessarily distinct characters modulo $q$.

We note first that some of these have already been proved. The case $\boldsymbol{\ell}=2\boldsymbol{\delta}^{\chi_{0}}$ where $\chi_{0}$ is the principal character modulo $q$ was essentially proved by Gonek, Hughes, and Keating [17, Theorem 3]. More generally, the case $\boldsymbol{\ell}=\boldsymbol{\delta}^{\chi_{0}}+\boldsymbol{\delta}^{\chi}$ where $\chi$ is a (not necessarily primitive) quadratic Dirichlet character modulo $q$ was essentially proved by Heap [23, Theorem 3]. To see this, note from (12) that $Z_{X}(s,\chi)$ depends only on the primitive character $\chi^{*}$ modulo $q^{*}(\chi)$ that induces $\chi$. In particular, one can replace $L(s,\chi_{0})^{2}$ with $\zeta(s)^{2}$ and $L(s,\chi_{0})L(s,\chi)$ with $\zeta(s)L(s,\chi^{*})=\zeta_{\mathbb{K}}(s)$ where $\mathbb{K}$ is a quadratic extension of $\mathbb{Q}$ and $\zeta_{\mathbb{K}}(s)$ is its Dedekind zeta function. Analogues of splitting for these products is precisely what was proven in these papers.

By following both these arguments, one should be able to extend to the general case $\boldsymbol{\ell}=\boldsymbol{\delta}^{\chi}+\boldsymbol{\delta}^{\nu}$. To do so, one would need a moment result for the product of two primitive Dirichlet $L$-functions and a short Dirichlet polynomial, generalizing that of [22]. That is, we would need an asymptotic for

(9)

$$\int_{T}^{2T}\left|L(s,\chi)L(s,\nu)\sum_{n\leqslant T^{\theta}}\frac{a_{n}}{n^{s}}\right|^{2}\,dt,$$

where $\chi$ and $\nu$ are any primitive characters with conductor dividing $q$, and some $0<\theta<1$ sufficiently large. Such asymptotics exist in the special cases of $\zeta(s)^{2}$ [29, 4] and $\zeta(s)L(s,\chi)$ [22], for any character $\chi$. Proving (9) and the splitting conjecture for $\boldsymbol{\ell}=\boldsymbol{\delta}^{\chi}+\boldsymbol{\delta}^{\nu}$ for more general $\chi,\nu$ by using the methods of [23], [22] and [17] as outlined above should be possible but long and technical. Thus, we do not pursue this here.

Note that Theorem 1.8 and Theorem 1.9 together establish Conjecture 1.1 with $k=1$ and $\alpha$ rational, giving an alternate proof of the leading term of Rane’s asymptotic (2) in this case.

Lastly, as a final piece of evidence for Conjecture 1.1, we prove the following results about upper and lower bounds:

In principle, it should be possible to remove the $\epsilon$ in the upper bound by using the methods of Harper [21].

The rest of the paper is structured as follows. In Section 2, we sketch a proof of Theorem 1.3; in Section 3, we prove Theorem 1.5; in Section 4, we provide some evidence for Conjecture 1.6; in Section 5, we prove Theorem 1.9; and in Section 6, we prove Theorems 1.2, 1.7, 1.8, and 1.10 and verify our conjectured constants are correct in the known cases (viz. $|\boldsymbol{\ell}|\leqslant 2$ or $k\leqslant 2$).

The proof of Theorem 1.3 is very similar to [17, Theorem 1] and [9, Theorem 1]. The main difference lies in the fact that we are not assuming that the character is primitive.

Recall that if $\chi$ and $\chi^{*}$ are as in the theorem, then

(10)

$$L(s,\chi)=L(s,\chi^{*})\prod_{p\mid q}\left(1-\frac{\chi^{*}(p)}{p^{s}}\right).$$

Further, by inspection we see that if $P(s,\cdot)$ and $Z(s,\cdot)$ are as in the theorem, then

(11)

$$P_{X}(s,\chi)=P_{X}(s,\chi^{*})\prod_{p\mid q}\left(1-\frac{\chi^{*}(p)}{p^{s}}\right),$$

(12)

$$Z_{X}(s,\chi)=Z_{X}(s,\chi^{*}).$$

Clearly, (10),(11) and (12) show that we can assume without loss of generality that $\chi$ is a primitive character modulo $q$.

At this point, one can follow [17, Theorem 1] and [9, Theorem 1] mutatis mutandis to complete the proof.

We briefly discuss some notation for this section. Recall that $d_{\boldsymbol{\ell}}(n)$ is the coefficient of $n^{-s}$ in the Dirichlet series of $\mathcal{L}^{\boldsymbol{\ell}}(s)$. $d_{\boldsymbol{\ell}}(n)$ is essentially a divisor function ‘twisted’ by the Dirichlet characters modulo $q$. We also use $d_{k}(n)$ for the true divisor function, i.e., the coefficient of $n^{-s}$ in $\zeta(s)^{k}$. In particular, it is immediate from writing $d_{\boldsymbol{\ell}}(n)$ out as a convolution that $|d_{\boldsymbol{\ell}}(n)|\leqslant d_{k}(n)$ for every $n\in\mathbb{N}$. We will use the notation $\mathscr{S}_{q}(X)$ to denote the set of $X$-smooth (also known as $X$-friable) numbers which are coprime to $q$. That is,

$$\mathscr{S}_{q}(X)=\{n\in\mathbb{N}:p\mid n\implies p\leqslant X\text{ and }p\nmid q\}.$$

We will need as a lemma, Mertens’ theorem for arithmetic progressions:

We also have the following, which is immediate from [9, Lemma 3]:

For the rest of this section, we will fix $s=1/2+it$. Now, we want to estimate $\int_{T}^{2T}\left|\mathcal{P}_{X}^{\boldsymbol{\ell}}(s)\right|^{2}dt$ assuming that $q^{2}<X\ll_{\epsilon}(\log T)^{2-\epsilon}$. Clearly, by Lemma 3.2,

$$\frac{1}{T}\int_{T}^{2T}|\mathcal{P}_{X}^{\boldsymbol{\ell}}(s)|^{2}\,dt=\left(\frac{1}{T}\int_{T}^{2T}|\mathcal{P}_{X}^{*\boldsymbol{\ell}}(s)|^{2}\,dt\right)\left(1+O_{k}\left(\frac{1}{\log X}\right)\right),$$

and so it suffices to compute $\int_{T}^{2T}|\mathcal{P}_{X}^{*\boldsymbol{\ell}}(s)|^{2}\,dt$.

From the definition of $\mathcal{P}_{X}^{*\boldsymbol{\ell}}(s)$ in Lemma 3.2, it follows that if

(13)

$$\mathcal{P}_{X}^{*\boldsymbol{\ell}}(s)=\sum_{n=1}^{\infty}\frac{\beta_{\boldsymbol{\ell}}(n)}{n^{s}},$$

then $\beta_{\boldsymbol{\ell}}(n)$ is multiplicative and supported on $\mathscr{S}_{q}(X)$, $|\beta_{\boldsymbol{\ell}}(n)|\leqslant d_{2k}(n)$ for all $n$, and finally for $n\in\mathscr{S}_{q}(\sqrt{X})$ and $p\in\mathscr{S}_{q}(X)$, we have $\beta_{\boldsymbol{\ell}}(n)=d_{\boldsymbol{\ell}}(n)$ and $\beta_{\boldsymbol{\ell}}(p)=d_{\boldsymbol{\ell}}(p)$.