Estimates of sums related to the Nyman-Beurling criterion for the Riemann Hypothesis

TL;DR

This paper provides a sharp estimate for sums involving the Möbius function related to the Nyman-Beurling criterion for the Riemann Hypothesis, utilizing advanced tools from continued fractions and Fourier series.

Contribution

It introduces a novel, highly precise estimate for sums with the Möbius function in the context of the Nyman-Beurling criterion, advancing understanding of the Riemann Hypothesis.

Findings

Estimate is remarkably sharp compared to previous bounds

Utilizes innovative methods from continued fractions and Fourier series

Enhances analytical tools for studying the Möbius function in number theory

Abstract

We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis containing the M\"obius function. The estimate is remarkably sharp in comparison to estimates of other sums containing the M\"obius function. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.

Equations

$$\underset{N\to \infty }{\lim }{d}_N^2=0,$$

$$d_N^2=\underset{D_N}{\inf }\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\left|1-\zeta D_N\left(\frac{1}{2}+it\right)\right|}^2\frac{dt}{\frac{1}{4}+t^2}$$

$$D_N(s):=\sum _{n=1}^N\frac{a_n}{n^s},a_n\in \mathbb{C},$$

$$\sum _{|Im(\rho )|\le T}\frac{1}{|{\zeta }^{\prime }(\rho ){|}^2}\ll T^{\frac{3}{2}-\delta }$$

$$\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\left|1-\zeta V_N\left(\frac{1}{2}+it\right)\right|}^2\frac{dt}{\frac{1}{4}+t^2}\sim \frac{2+\gamma -\log 4\pi }{\log N}$$

$$V_N(s):=\sum _{n=1}^N\left(1-\frac{\log n}{\log N}\right)\frac{\mu (n)}{n^s}.$$

$$d_N^2$$

$$\displaystyle\ \ \ +\int_{-\infty}^{\infty}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}\left|D_{N}\left(\frac{1}{2}+it\right)\right|^{2}\frac{dt}{\frac{1}{4}+t^{2}}\Bigg{)}\>.$$

$$\sum _{1\le h,k\le N}{a}_h\overline{a_k}{h}^{-1/2}{k}^{-1/2}{\int }_{-\infty }^{\infty }{\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2{\left(\frac{h}{k}\right)}^{it}\frac{dt}{\frac{1}{4}+t^2}.$$

$$b_{h,k}$$

$$=\frac{\log 2\pi -\gamma }{2}\left(\frac{1}{h}+\frac{1}{k}\right)+\frac{k-h}{2hk}\log \frac{h}{k}-\frac{\pi }{2hk}\left(V\left(\frac{h}{k}\right)+V\left(\frac{k}{h}\right)\right),$$

$$V\left(\frac{h}{k}\right):=\sum _{m=1}^{k-1}\left\{\frac{mh}{k}\right\}\cot \left(\frac{\pi mh}{k}\right)$$

$$V\left(\frac{h}{k}\right)=-c_0\left(\frac{\bar{h}}{k}\right),$$

$$c_0\left(\frac{h}{k}\right):=\sum _{l=1}^{k-1}\frac{l}{k}\cot \left(\frac{\pi hl}{k}\right),$$

$$c_0\left(\frac{a}{q}\right)=\frac{1}{2}{D}_{sin}\left(0,\frac{a}{q}\right)=2q{\pi }^{-2}{D}_{\sin }\left(1,\frac{\bar{a}}{q}\right),$$

$$D_{\sin }(s,x):=\sum _{n=1}^{\infty }\frac{d(n)\sin (2\pi nx)}{n^s}$$

$$\sum _{n\ge 1}(-1{)}^n\log \frac{v_{n+1}}{v_n},$$

$$g(x):=\sum _{l\ge 1}\frac{1-2\{lx\}}{l},$$

$${\int }_{-\infty }^{\infty }{\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2{\left|D_N\left(\frac{1}{2}+it\right)\right|}^2\frac{dt}{\frac{1}{4}+t^2}$$

$$=\sum _{1\le h,k\le N}\mu (h)\mu (k)\left(1-\frac{\log h}{\log N}\right)\left(1-\frac{\log k}{\log N}\right)$$

$$\times \left[\frac{\log 2\pi -\gamma }{2}\left(\frac{1}{h}+\frac{1}{k}\right)+\frac{k-h}{2h\pi }\log \frac{h}{k}-\frac{\pi }{2hk}\left(V\left(\frac{h}{k}\right)+V\left(\frac{k}{h}\right)\right)\right].$$

$$\sum _{h\in I}\mu (h)\left(1-\frac{\log h}{\log N}\right)\frac{1}{h}V\left(\frac{h}{k}\right),$$

$$\sum _{h\in I}\mu (h)\left(1-\frac{\log h}{\log N}\right)g\left(\frac{h}{k}\right)$$

$$\sum _{Bk\le h<(1+\eta )Bk}\mu (h)g\left(\frac{h}{k}\right)=O\left((\eta Bk{)}^{1-\beta }\right).$$

$$\sum _{1\le h\le N}\frac{\mu (h)}{h}\left(1-\frac{\log h}{\log N}\right)V\left(\frac{h}{k}\right)=\sum _{1\le h\le k^{\prime }}\frac{\mu (h)}{h}\left(1-\frac{\log h}{\log N}\right)V\left(\frac{h}{k}\right)+O(k^{1-\kappa }),$$

$${\alpha }_0(x):=x,{\alpha }_l(x):=\alpha ({\alpha }_{l-1}(x)),\text{for l∈N}.$$

$$a_l(x):=\left[\frac{1}{{\alpha }_{l-1}(x)}\right].$$

$$x=[0;a_1(x),\dots ,a_l(x),\dots ]$$

$$x=[0;a_1(x),\dots ,a_L(x)],$$

$$\frac{p_l(x)}{q_l(x)}:=[0;a_1(x),\dots ,a_l(x)],where(p_l(x),q_l(x))=1.$$

$$p_{l+1}=a_{l+1}{p}_l+p_{l-1}$$

$$q_{l+1}=a_{l+1}{q}_l+q_{l-1}.$$

Full text

Key words: Riemann Hypothesis, Riemann zeta function, Nyman-Beurling-Báez-Duarte criterion.

2000 Mathematics Subject Classification: 30C15, 11M26, 42A16, 42A20

1. Introduction and statement of result

According to the approach of Nyman-Beurling-Báez-Duarte (see [1], [6]) to the Riemann Hypothesis, the Riemann Hypothesis is true if and only if

[TABLE]

where

[TABLE]

and the infimum is over all Dirichlet polynomials

[TABLE]

of length $N$ (see [7]).

Various authors (cf. [1], [2], [9]) have investigated the question, which asymptotics hold for $d_{N}$ if the Riemann Hypothesis is true.

In [7] the following was shown:

If the Riemann Hypothesis is true and if

[TABLE]

for some $\delta>0$, then

[TABLE]

for

[TABLE]

Moreover, it follows from work of Burnol [9] that among all Dirichlet polynomials $D_{N}(s)$ the infimum in (1.1) is assumed for $D_{N}(s)=V_{N}(s)$. It thus is of interest to obtain an unconditional estimate for the integral in (1.1).

If we expand the square in (1.1) we obtain

[TABLE]

The last integral evaluates as

[TABLE]

We have

[TABLE]

where

[TABLE]

is Vasyunin’s sum (see [21]).

It can be shown that

[TABLE]

where $h\bar{h}\equiv 1(\bmod k)$, with the cotangent sum

[TABLE]

where $h,k\in\mathbb{N}$, $k\geq 2$, $1\leq h\leq k$, $(h,k)=1$ (see [6], [13]).

Ishibashi [11] observed that $c_{0}$ is related to the value at $s=0$ or $s=1$ of the Estermann zeta function by the functional equation of the imaginary part. Namely

[TABLE]

where for $x\in\mathbb{R}$, $Re(s)>1$, we have:

[TABLE]

and $a\bar{a}\equiv 1(\bmod q)$.

If $x\in\mathbb{R}\setminus\mathbb{Q}$, then de la Bretèche and Tenenbaum [8] showed that the convergence of the above series at $s=1$ is equivalent to the convergence of

[TABLE]

where $u_{n}/v_{n}$ denotes the $n$-th partial quotient of $x$.

The irrational numbers for which this sum converges are called Wilton numbers.

A basic ingredient in the papers [14], [18] have been the representations of Balazard, Martin in their papers [3], [4], of the function

[TABLE]

involving the Gauss transform from the theory of continued fractions as well as from the paper [19], by Marmi, Moussa and Yoccoz.

These concepts and results also play an important role in the present paper. We shall represent them in the next section.

We now describe our result and its relation to the Nyman-Beurling criterion:

From (1.2) and (1.3) we obtain

[TABLE]

We investigate a partial sum, in which $k$ is kept fixed, namely (up to a constant depending only on $h$):

[TABLE]

which coincides with

[TABLE]

($I$ is a suitable interval).

We prove the following result:

Theorem 1.1**.**

Let $0<\delta<D/2$, $k^{2\delta}\leq B\leq k^{D}$, where $k^{-\delta}\leq\eta\leq 1$. Then there is a positive constant $\beta$ depending only on $\delta$ and $D$, such that

[TABLE]

Remark. This is one of the few cases of a remarkably sharp estimate of a sum containing the Möbius function that is better than the trivial estimate by a factor, which is a positive power of the number of the terms.

Corollary 1.2**.**

Let $\gamma>0$, $\delta>0$, $\epsilon>0$ fixed, $k\geq N^{\delta}$. Then there is $\kappa>0$ depending only on $\gamma,\delta,\epsilon$, such that

[TABLE]

where $k^{\prime}=k^{1+\epsilon}$.

Proof.

This follows from Theorem 1.1 by integration by parts. ∎

2. Continued Fractions

In this section we provide some basic facts related to the theory of continued fractions. We also recall facts and results from the paper [4].

Definition 2.1**.**

Let $X:=[0,1]\setminus\mathbb{Q}$. For $x\in(0,1)$ we set $\alpha(x):=\{1/x\}$. For $x\in X$ we define recursively

[TABLE]

This definition is also valid for $x\in\mathbb{Q}$ and $l\in\mathbb{N}$, whenever $\alpha_{l-1}(x)\neq 0$. We set

[TABLE]

We have:

Lemma 2.2**.**

For $x\in X$,

[TABLE]

*the continued fraction expansion of $x$.

For $x\in(0,1)\cap\mathbb{Q}$ we have*

[TABLE]

*where $L$ is the last $l$, for which $\alpha_{l-1}\neq 0$.

We define the partial quotient of $p_{l}(x),q_{l}(x)$ by*

[TABLE]

We have

[TABLE]

[TABLE]

Proof.

(cf. [10], p. 7.) ∎

Definition 2.3**.**

For $r\in(0,1)\cap\mathbb{Q}$ let

[TABLE]

Then we call $L$ the depth of $r$.

Definition 2.4**.**

Let $x\in X$. Then for $l\in\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$, we set:

[TABLE]

(by convention $\beta_{-1}=1$) and

[TABLE]

*so that $\gamma_{0}(x)=\log(1/x).$

Let $r\in(0,1)\cap\mathbb{Q}$ be a rational number of depth $L$. Then we set:*

[TABLE]

[TABLE]

For $x\in X$ we define Wilton’s function $\mathcal{W}(x)$ by

[TABLE]

for which the series is convergent.

Definition 2.5**.**

*(definitions from Sec. 4.1 of [4])

For $\lambda\geq 0$ let*

[TABLE]

For $x>0$ let

[TABLE]

For $x\in X$ let

[TABLE]

For a rational number $r$ of depth $L$ let

[TABLE]

Lemma 2.6**.**

We have

[TABLE]

Proof.

This is Proposition 33 of [4]. ∎

Definition 2.7**.**

Let

[TABLE]

Lemma 2.8**.**

The series $g(x)$ and $\mathcal{W}(x)$ converge for the same values $x\in[0,1]$ and we have

[TABLE]

in each point of convergence.

Proof.

This is Proposition 28 of [4]. ∎

Definition 2.9**.**

For $n\in\mathbb{N}$, $x\in X$ or $x$ a rational number with depth $\geq n$, we define

[TABLE]

where $l(x):=\log(1/x)$ and the operator $T\>:\>L^{p}\rightarrow L^{p}$ is defined by

[TABLE]

Lemma 2.10**.**

Let $x\in X$ be a Wilton number or a rational number. Then we have:

[TABLE]

Proof.

This follows directly from the definition of Wilton’s function $\mathcal{W}(x)$. ∎

Lemma 2.11**.**

For $n\in\mathbb{N}$, $x\in X$ or $x$ a rational number with depth $\geq n+1$ we have

[TABLE]

Proof.

This follows from Lemma 2.10 by the same computation as in the proof of Lemma 2.10 in [18], which is valid also for $x$ a rational number with depth $\geq n+1$. ∎

Lemma 2.12**.**

For $m\in\mathbb{N}_{0}$, $x\in X$ or $x$ a rational number of depth $\geq m+1$ we have

[TABLE]

Proof.

This is Lemma 2.11 of [18], whose proof is also valid for rational $x$ of depth $\geq m+1$. ∎

Definition 2.13**.**

Let $\mathcal{E}$ be a measurable subset of $(0,1)$. The measure $m$ is defined by

[TABLE]

Lemma 2.14**.**

The measure $m$ is invariant with respect to the map $\alpha$, i.e.

[TABLE]

for all measurable subsets $\mathcal{E}\subseteq(0,1)$.

Proof.

This result is well-known. ∎

Lemma 2.15**.**

Let $n\in\mathbb{N}$. For $f\in L^{p}$, we have

[TABLE]

where

[TABLE]

Proof.

For the proof of this result, due to Marmi, Moussa and Yoccoz [19], see [18] Lemma 2.8, (ii). ∎

Definition 2.16**.**

Let $s\in\mathbb{N}$, $b_{0}=0$ and $b_{1},\ldots,b_{s}\in\mathbb{N}$. The cell of depth $s$, $\mathcal{C}(b_{1},\ldots,b_{s})$ is the interval with the endpoints $[0;b_{1},\ldots,b_{s}]$ and $[0;b_{1},\ldots,b_{s-1},b_{s}+1]$.

Lemma 2.17**.**

In the interior of the cell $\mathcal{C}(b_{1},\ldots,b_{s})$ of depth $s$, the functions $a_{j},p_{j},q_{j}$ are constants for $j\leq s$,

[TABLE]

The endpoints of $\mathcal{C}(b_{1},\ldots,b_{s})$ are

[TABLE]

*For $x\in X$ and $s\in\mathbb{N}$ there is a unique cell of depth $s$ that contains $x$.

Within a cell of depth $s$ we have the derivatives*

[TABLE]

[TABLE]

Proof.

See [4], sections 2.3 and 2.4. ∎

Lemma 2.18**.**

Let $\mathcal{C}(b_{1},\ldots,b_{s})$ be as in Definition 2.16. Then we have

[TABLE]

Proof.

We compare the measure of cells of depths $j$ and $j+1$. By (2.1) of Lemma 2.16, we know that $\mathcal{C}(b_{1},\ldots,b_{j})$ has the endpoints

[TABLE]

and thus the length

[TABLE]

The cell $\mathcal{C}(b_{1},\ldots,b_{j+1})$ has the length

[TABLE]

Thus

[TABLE]

From (2.2) we obtain:

[TABLE]

which concludes the proof of Lemma 2.18. ∎

Lemma 2.19**.**

We have

[TABLE]

Proof.

This follows from

[TABLE]

∎

Lemma 2.20**.**

Let $C_{1}>0$ be sufficiently large. Then there exists $C_{2}>0$, such that for $s\geq s_{0}$,

[TABLE]

Proof.

By partition into cells of depth $s$ we obtain from Lemmas 2.18 and 2.19 the following:

[TABLE]

The integral in (2.3) is

[TABLE]

where $X_{1},\ldots,X_{s}$ are i.i.d. exponentially distributed random variables with rate 2. The distribution of the sum $X_{1}+\cdots+X_{s}$ is the Gamma distribution with density

[TABLE]

Lemma 2.20 now follows from (2.3) and (2.4). ∎

3. Vaughan’s identity

We now express the Möbius function by Vaughan’s identity.

Lemma 3.1**.**

Let $w>1$. For $h\in\mathbb{N}$ we have:

[TABLE]

where

[TABLE]

for

[TABLE]

[TABLE]

and

[TABLE]

Proof.

Cf. [12]. ∎

We now obtain

[TABLE]

where

[TABLE]

From the definitions for $c_{1}$ and $c_{4}$ in Lemma 3.1 we obtain:

[TABLE]

where

[TABLE]

We now choose

[TABLE]

depending only on $\delta,\eta$ and $D$. The subsequent steps are all valid, if $\delta_{0}$ is sufficiently small.

We partition the sum ${\textstyle\sum}_{1}$ as follows:

[TABLE]

with

[TABLE]

and

[TABLE]

A trivial estimate of the sum ${\textstyle\sum}_{2}$ gives:

[TABLE]

The sums ${\textstyle\sum}_{1,1}$ and ${\textstyle\sum}_{3}$ are estimated by the same method.

In the next section we shall give the details of the estimate of ${\textstyle\sum}_{1,1}$, whereas the estimate of ${\textstyle\sum}_{1,2}$ will be given in the last section.

4. The sums ${\textstyle\sum}_{1,1}$ and ${\textstyle\sum}_{3}$

We now choose $s_{0}:=[\delta_{1}\log k]$, where $\delta_{1}>0$ depends only on $\delta_{0}$ and $D$. In the sequel we shall always assume that $\delta_{1}$ is sufficiently small.

For the tuplets $(d_{1},d_{2},e_{1},e_{2},\gamma)$ appearing in the sum ${\textstyle\sum}_{1,1}$ at least one of the two cases must hold:

[TABLE]

[TABLE]

since otherwise we would have the inequalities

[TABLE]

which in turn would lead to the contradiction

[TABLE]

Since the cases 1) and 2) are symmetric, it suffices to treat case 1 only.

We write

[TABLE]

where

[TABLE]

We estimate ${\textstyle\sum}_{1,1}^{(in)}$ and ${\textstyle\sum}_{1,1}^{(end)}$ trivially by

[TABLE]

[TABLE]

and now describe the estimate of the sums ${\textstyle\sum}_{1,1}(C)$.

We set

[TABLE]

and define $e_{2}^{(0)}$ and $e_{2}^{(1)}$ as follows:

Let

[TABLE]

[TABLE]

[TABLE]

We define

[TABLE]

and obtain

[TABLE]

We now make use of the antisymmetry of the function $g$:

[TABLE]

We replace the points

[TABLE]

by the mirror images of the points

[TABLE]

namely by

[TABLE]

and due to (4.1) we obtain

[TABLE]

We have

[TABLE]

We now partition the range of summation over the points $\Pi(\vec{d\>})(e_{2}^{(1)}-v)k^{-1}$ in cells of depth $s_{0}$ and assume that $\delta_{1}$ is chosen so small that $q_{s_{0}}(x)\leq k^{\delta_{0}/10}$ for all cells $\mathcal{C}$ of depth $s_{0}$ with the possible exceptions of cells $\mathcal{C}$ of depth $s_{0}$ with total measure $k^{-2\delta_{2}}$ for some $\delta_{2}>0$.

This can be achieved because of Lemma 2.20. Denote the union of these exceptional cells by $\mathscr{C}_{ex}$.

For the points

[TABLE]

we estimate the difference in (4.5) by the trivial bound

[TABLE]

We obtain

[TABLE]

where ${\textstyle\sum}_{ex}$ is extended over all tuplets $(d_{1},d_{2},e_{1},e_{2},\gamma)$ with

[TABLE]

and

[TABLE]

For the other cells we use that

[TABLE]

and substitute

[TABLE]

This leads to the problem to estimate the differences

[TABLE]

and

[TABLE]

This can be done by appeal to Lemmas 2.6, 2.17 and (4.6). The sum containing the terms $T^{s_{0}+1}W(x)$ can be estimated using Lemma 2.12. We obtain

[TABLE]

By (4.1), (4.2), (4.3) we get:

[TABLE]

By the same method we derive

[TABLE]

[TABLE]

5. The sum ${\textstyle\sum}_{1,2}$

We have

[TABLE]

where

[TABLE]

Therefore

[TABLE]

with

[TABLE]

We observe that

[TABLE]

where $d_{4}$ denotes the divisor function sum of order $4$.

Let

[TABLE]

We recursively define the sequence $(\mathcal{R}_{l})$ of regions that exhaust $\mathcal{R}$.

Let $\mathcal{R}_{0}$ be the union of squares with sides parallel to the coordinate axes, the coordinates of whose vertices are integer multiples of $2^{-0}=1$ and side lengths $2^{-0}=1$, that are completely contained in $\mathcal{R}$.

The recursion $l\rightarrow l+1$ : Assume that $\mathcal{R}_{l}$ has been defined. We let $\mathcal{S}_{l}$ be the union of squares of sides parallel to the coordinate axes of side lengths $2^{-(l+1)}$, the coordinates of their vertices are integer multiples of $2^{-(l+1)}$, whose interiors are contained in the set $\mathcal{R}\setminus\mathcal{R}_{l}$. We set

[TABLE]

We choose $l_{0}$ as the minimal $l$, such that

[TABLE]

for all squares $Q$ of side lengths $2^{-l_{0}}$ contained in $\mathcal{R}$.

Since the boundaries of $\mathcal{R}$ are straight lines, the trivial bound (4.7) gives

[TABLE]

Let $Q$ be a square of $\mathcal{R}_{l_{0}}$ and

[TABLE]

We set

[TABLE]

Because of (5.4) we have

[TABLE]

[TABLE]

for $\delta_{0}$ sufficiently small.

We now estimate

[TABLE]

By the Cauchy-Schwarz inequality we have:

[TABLE]

We have

[TABLE]

where

[TABLE]

We now make use of the representation

[TABLE]

which follows from Lemmas 2.8 and 2.11.

We set

[TABLE]

and replace ${\textstyle\sum}_{g}(Q,u_{1},u_{2})$ by

[TABLE]

by removing the terms $T^{s_{0}+1}\mathcal{W}(\gamma)$, $\delta(\gamma)$ in (5.10).

By Lemmas 2.11 and 2.12 it follows that

[TABLE]

We now partition the sum ${\textstyle\sum}_{g}^{{}^{\prime}}(Q,u_{1},u_{2})$ as follows

[TABLE]

where in ${\textstyle\sum}_{g,1}^{{}^{\prime}}$ the summation is extended over all $\gamma$, for which

[TABLE]

and in ${\textstyle\sum}_{g,2}^{{}^{\prime}}$ the summation is extended over the other values of $\gamma$.

By the definition of $\mathscr{C}_{ex}$, for the $\gamma$ appearing in ${\textstyle\sum}_{g,1}^{{}^{\prime}}$ the intervals

[TABLE]

belong to cells $\mathcal{C}_{1}$ resp. $\mathcal{C}_{2}$ of depth $s_{0}$ as described in Lemma 2.20.

Let $\mathcal{I}$ be an interval of length 1. By Lemma 2.20 we have

[TABLE]

with the exception of at most

[TABLE]

In ${\textstyle\sum}_{g,1}^{{}^{\prime}}$ we replace the terms $p(\gamma,u_{1},u_{2})$ by their averages

[TABLE]

The intervals

[TABLE]

are contained in some cells $\mathcal{C}_{i}$ $(i=1,2)$ of depth $s_{0}$.

We set

[TABLE]

where

[TABLE]

By Lemma 2.20 we have

[TABLE]

We now write

[TABLE]

[TABLE]

and show that $R_{2}(Q,u_{1},u_{2})$ is small in average. We have

[TABLE]

We apply Lemma 2.15 for the estimate of

[TABLE]

and obtain by the Cauchy-Schwarz inequality

[TABLE]

In $\sum_{Q}{}^{{}^{\prime}}$ the sum is extended over all squares $Q$ whose union is $\mathcal{R}_{l_{0}}$.

We replace the integrands $g(u_{i}x/k)$ in (5.17) by finite partial sums of their Fourier series

[TABLE]

where

[TABLE]

By Parseval’s equation we obtain

[TABLE]

We finally also replace ${\textstyle\sum}_{g,2}^{{}^{\prime}}(Q,u_{1},u_{2})$ by

[TABLE]

where $\gamma$ is extended over the same values as in formula (5.13).

From (5.9), (5.11), (5. 12), (5.13), (5.14), (5.15), (5.16), (5.19) and (5.20) we obtain:

[TABLE]

where

[TABLE]

From the identity

[TABLE]

we obtain

[TABLE]

[TABLE]

6. Conclusion

Theorem 1.1 now follows from (3.1), (3.2), (4.11), (4.12) and (5.22).