A logarithmic improvement in the Bombieri-Vinogradov theorem

TL;DR

This paper enhances the Bombieri-Vinogradov theorem by reducing the logarithmic factor from (log x)^(5/2) to (log x)^2 using advanced sieve techniques and Vaughan's identity.

Contribution

It introduces a weighted Vaughan's identity and leverages sieve estimates to achieve a significant logarithmic improvement in the theorem.

Findings

Reduced the logarithmic factor in the theorem from (log x)^(5/2) to (log x)^2

Provided both effective and non-effective versions of the improved result

Applied advanced sieve methods and weighted identities for the enhancement

Abstract

In this paper we improve the best known to date result of Dress-Iwaniec-Tenenbaum, getting (log x)^2 instead of (log x)^(5/2). We use a weighted form of Vaughan's identity, allowing a smooth truncation inside the procedure, and an estimate due to Barban-Vehov and Graham related to Selberg's sieve. We give effective and non-effective versions of the result.

Equations

$$\psi (x;q,a)=\sum _{\begin{array}{c}n\le x\\ n\equiv a(\mathrm{mod}q)\end{array}}\Lambda (n),$$

$$\sum _{q\le Q}\underset{2\le y\le x}{\max }\underset{\genfrac{}{}{0ex}{}{a}{(a,q)=1}}{\max }\left|\psi (y,q,a)-\frac{y}{\phi (q)}\right|{\ll }_A\frac{x}{(\log x{)}^A}.$$

$$\begin{array}{r}\sum _{q\le Q}\underset{2\le y\le x}{\max }\underset{\genfrac{}{}{0ex}{}{a}{(a,q)=1}}{\max }\left|\psi (y,q,a)-\frac{y}{\phi (q)}\right|{\ll }_A\frac{x}{(\log x{)}^{A-2}}.\end{array}$$

$$\sum _{\genfrac{}{}{0ex}{}{q\le Q}{l(q)>Q_1}}\underset{2\le y\le x}{\max }\underset{\genfrac{}{}{0ex}{}{a}{(a,q)=1}}{\max }\left|\psi (y;q,a)-\frac{\psi (y)}{\phi (q)}\right|\le C\left(\frac{x}{Q_1}+x^{\frac{1}{2}}Q+x^{\frac{2}{3}}{Q}^{\frac{1}{2}}+x^{\frac{5}{6}}\log \frac{Q}{Q_1}\right)(\log x{)}^{\frac{7}{2}},$$

$$\pi (x)=\sum _{p\le x}1\text{ and }\pi (x;q,a)=\sum _{\genfrac{}{}{0ex}{}{p\le x}{p\equiv a(\mathrm{mod}q)}}1.$$

$$\sum _{\genfrac{}{}{0ex}{}{q\le Q}{l(q)>Q_1}}\underset{2\le y\le x}{\max }\underset{\genfrac{}{}{0ex}{}{a}{(a,q)=1}}{\max }\left|\pi (y;q,a)-\frac{\pi (y)}{\phi (q)}\right|\ll x^{\frac{1}{2}}Q(\log x{)}^2+\frac{x}{Q_1}(\log x{)}^3+x^{\frac{13}{14}+\epsilon }(\log x{)}^4.$$

$$\psi (y,\chi )=\sum _{n\le y}\Lambda (n)\chi (n),$$

$$\sum _{q\le Q}\frac{q}{\phi (q)}\text{sideset}{}^{\ast }{\sum }_{\chi (q)}\underset{y\le x}{\max }|\psi (y,\chi )|\ll \left(x+Q^2{x}^{\frac{1}{2}}+Q{x}^{\frac{13}{14}+\epsilon }\right)(\log x{)}^2,$$

$$\psi (x,\chi )-\delta (\chi )x{\ll }_{A}x{e}^{-c\sqrt{\log x}}$$

$$\sum _{\genfrac{}{}{0ex}{}{q\le Q}{l(q)>Q_1}}\underset{2\le y\le x}{\max }\underset{\genfrac{}{}{0ex}{}{a}{(a,q)=1}}{\max }\left|\psi (y;q,a)-\frac{\psi (y)}{\phi (q)}\right|\ll x^{\frac{1}{2}}Q(\log x{)}^2+\frac{x}{Q_1}(\log x{)}^3+x^{\frac{13}{14}+\epsilon }(\log x{)}^4.$$

$$\sum _{q\le x^{1/2}/(\log x{)}^B}\underset{y\le x}{\max }\underset{(a,q)=1}{\max }\left|\psi (y;q,a)-\frac{\mathrm{Li}y}{\phi (q)}\right|\ll x\frac{(\log \log x{)}^9}{\log x}.$$

$$\Lambda (n)={\lambda }_0(n)+{\lambda }_1(n)+{\lambda }_2(n)+{\lambda }_3(n),$$

$$\begin{array}{rl}& {\lambda }_1(n)=\sum _{d|n}\mu (d)\eta (d)\log \frac{n}{d},{\lambda }_2(n)=-\sum _{c\le U}\sum _{dc|n}\mu (d)\Lambda (c)\eta (d),\\ & {\lambda }_3(n)=\sum _{c>U}\sum _{dc|n}\mu (d)\Lambda (c)(1-\eta (d)).\end{array}$$

$$\begin{array}{rl}& {\Lambda }_1(n)=\sum _{\genfrac{}{}{0ex}{}{d|n}{d\le V}}\mu (d)\log \frac{n}{d}={\lambda }_1(n)-\sum _{\genfrac{}{}{0ex}{}{d|n}{d>V}}\mu (d)\eta (d)\log \frac{n}{d}={\lambda }_1(n)+{\lambda }_1^{\prime }(n),\\ & {\Lambda }_2(n)=-\sum _{c\le U}\sum _{\genfrac{}{}{0ex}{}{dc|n}{d\le V}}\mu (d)\Lambda (c)={\lambda }_2(n)+\sum _{c\le U}\sum _{\genfrac{}{}{0ex}{}{dc|n}{d>V}}\mu (d)\Lambda (c)\eta (d)={\lambda }_2(n)+{\lambda }_2^{\prime }(n),\\ & {\Lambda }_3(n)=\sum _{c>U}\sum _{\genfrac{}{}{0ex}{}{dc|n}{d>V}}\mu (d)\Lambda (c)={\lambda }_3(n)+\sum _{c>U}\sum _{\genfrac{}{}{0ex}{}{dc|n}{d>V}}\mu (d)\Lambda (c)\eta (d)={\lambda }_3(n)+{\lambda }_3^{\prime }(n).\end{array}$$

$$\Lambda (n)={\Lambda }_1(n)+{\Lambda }_2(n)+{\Lambda }_3(n),$$

$$\begin{array}{rl}\sum _{i=1}^3{\lambda }_i^{\prime }(n)& =\sum _{\genfrac{}{}{0ex}{}{d|n}{d>V}}\left(-\mu (d)\eta (d)\log \frac{n}{d}+\sum _{\genfrac{}{}{0ex}{}{c|\frac{n}{d}}{c\le U}}\mu (d)\Lambda (c)\eta (d)+\sum _{\genfrac{}{}{0ex}{}{c|\frac{n}{d}}{c>U}}\mu (d)\Lambda (c)\eta (d)\right)\\ & =\sum _{\genfrac{}{}{0ex}{}{d|n}{d>V}}\left(-\mu (d)\eta (d)\log \frac{n}{d}+\mu (d)\eta (d)\sum _{c|\frac{n}{d}}\Lambda (c)\right)=0,\end{array}$$

$$f_i(d)=\{\begin{array}{ll}\mu (d)\log \frac{N_i}{d},& d\le N_i,\\ 0,& d>N_i.\end{array}$$

$$\sum _{n=1}^N\left(\sum _{d_1|n}{f}_1(d_1)\right)\left(\sum _{d_2|n}{f}_2(d_2)\right)=N\log N_1+O(N).$$

$$\eta (t)=\frac{\log \frac{V_0}{t}}{\log \frac{V_0}{V}},V<t\le V_0.$$

$$\sum _{k\le Y}{\left|\sum _{d|k}\mu (d)\eta (d)\right|}^2\ll \frac{Y}{\log \frac{V_0}{V}}.$$

$$\sum _{q\le Q}\frac{q}{\phi (q)}\text{sideset}{}^{\ast }{\sum }_{\chi (q)}{\left|\sum _{m=m_0+1}^{m_0+M}{a}_m\chi (m)\right|}^2\le (M+Q^2)\sum _{m=m_0+1}^{m_0+M}|a_m{|}^2,$$

$$\begin{array}{rl}& \sum _{q\le Q}\frac{q}{\phi (q)}\text{sideset}{}^{\ast }{\sum }_{\chi (q)}\underset{y}{\max }\left|\sum _{m=m_0}^M\sum _{\genfrac{}{}{0ex}{}{n=n_0}{mn\le y}}^N{a}_m{b}_n\chi (mn)\right|\le \\ & c_3(M^{\prime }+Q^2{)}^{\frac{1}{2}}(N^{\prime }+Q^2{)}^{\frac{1}{2}}{\left(\sum _{m=m_0}^M|a_m{|}^2\right)}^{\frac{1}{2}}{\left(\sum _{n=n_0}^N|b_n{|}^2\right)}^{\frac{1}{2}}L(M,N),\end{array}$$

$$\Lambda (n)={\lambda }_0(n)+{\lambda }_1(n)+{\lambda }_2(n)+{\lambda }_3(n),$$

$$\psi (y,\chi )=s_0+s_1+s_2+s_3,$$

$$s_i=\sum _{n\le y}{\lambda }_i(n)\chi (n).$$

$$S_i=\sum _{q\le Q}\frac{q}{\phi (q)}\text{sideset}{}^{\ast }{\sum }_{\chi (q)}\underset{y\le x}{\max }|s_i|.$$

$$\sum _{q\le Q}\frac{q}{\phi (q)}\text{sideset}{}^{\ast }{\sum }_{\chi (q)}\underset{y\le x}{\max }|\psi (y,\chi )|\le S_0+S_1+S_2+S_3,$$

$$\begin{array}{rl}& S_0\ll U{Q}^2,\\ & s_1=\sum _{d\le y}\mu (d)\chi (d)\eta (d)\sum _{h\le \frac{y}{d}}\chi (h)\log h,\\ & s_2=-\sum _{\genfrac{}{}{0ex}{}{dcr\le y}{c\le U}}\chi (dcr)\mu (d)\Lambda (c)\eta (d),\\ & s_3=\sum _{n\le y}\chi (n)\sum _{c>U}\sum _{dc|n}\mu (d)\Lambda (c)(1-\eta (d)).\end{array}$$

$$|s_0|\le \sum _{n\le U}\Lambda (n)\ll U.$$

$$\eta (d)=\frac{\log \frac{V_0}{d}}{\log \frac{V_0}{V}},V\le d\le V_0.$$

Full text

1 Introduction

For integer number $a$ and $q\geq 1$, let

[TABLE]

where $\Lambda(n)$ is the von Mangoldt function. The Bombieri-Vinogradov theorem is an estimate for the error terms in the prime number theorem for arithmetic progressions averaged over all $q$ up to $x^{1/2}$, or, rather almost all $q$ up to $x^{\frac{1}{2}}$.

Theorem** (Bombieri-Vinogradov).**

Let $A$ be a given positive number and $Q\leq\frac{x^{1/2}}{(\log{x})^{B}},$ where $B=B(A)$. Then

[TABLE]

The implied constant in this theorem is not effective, since we have to take care of characters associated with those $q$ that have small prime factors. At the same time, effective versions - in which the effect of an exceptional character is avoided in one way or another - have been known since [9] and [15], and, very recently, [10]. We state the main result of this paper.

Theorem 1** (Bombieri-Vinogradov, ineffective).**

Let $A$ be a positive number and $Q\leq\frac{x^{1/2}}{(\log x)^{A}}$. Then we have the following bound

[TABLE]

The implied constant in Theorem 1 is ineffective. We give an effective version of the result above together with its applications in Section 1.1.

Previously, the best result of the type of Theorem 1 in the literature followed from [3]; it had $A-5/2$ instead of $A-2$. While [3] does not state the result in full – focusing on estimating a crucial sum – a complete form can be found in [14] (together with a fully explicit version). It is

[TABLE]

where $C$ is an explicit absolute constant (a similar fully explicit result was proven in [1] with $(\log x)^{\frac{9}{2}}$ instead of $(\log x)^{\frac{7}{2}}$). Another effective variant without explicit constants is given by Lenstra and Pomerance [9, Lemma 11.2] (with bigger power of $\log$) in their work on Gaussian periods.

Remark**.**

Define

[TABLE]

For $x\geq 4$, $1\leq Q_{1}\leq Q\leq x^{\frac{1}{2}}$ and any $\varepsilon>0$ we have

[TABLE]

The proof of the remark is exactly the same as in [1], we just have to change the power of $\log x$.

The key tool for the proof of Theorem 1 is Vaughan’s identity, which we have to get in an explicit version for our goal. Define

[TABLE]

the twisted summatory function for the von Mangoldt function $\Lambda$ and a Dirichlet character $\chi$ modulo $q$. The key tool in getting Theorem 1 is the following estimate.

Proposition 1** (Vaughan’s inequality, improved).**

For $x\geq 4$ and any $\varepsilon>0$ we have

[TABLE]

where $Q$ is any positive real number and $\sideset{}{{}^{*}}{\sum}_{\chi(q)}$ means a sum over all primitive characters $\chi(\operatorname{mod}q)$.

The improvement here consists in having a factor of $(\log x)^{2}$, rather than $(\log x)^{\frac{5}{2}}$ or $(\log x)^{3}$. In order to prove Proposition 1 we use the weighted version of Vaughan’s identity (see Lemma 1) and an estimate due to Barban-Vehov [2] and Graham [6]. While Graham uses the Siegel-Walfisz theorem, there is an effective (and explicit) version of it in [7]. We follow methods developed in [7] in the proof.

Proposition 1 allows us to prove the Bombieri-Vinogradov theorem in the form of Theorem 1 and, hence, Corollary 1. In addition to Theorem 1, the proof uses the Siegel-Walfisz theorem, which states that

[TABLE]

uniformly for $q\leq(\log x)^{A}$. Here $A>0$ is a fixed real number, $c$ is an absolute positive constant, and $\delta(\chi)=1$ if $\chi$ is principal and is zero otherwise. The implied constant in the Bombieri-Vinogradov theorem is ineffective since the implied constant in the Siegel-Walfisz theorem is ineffective. To prove Corollary 1 we use the Siegel-Walfisz theorem to deal with moduli $q\leq Q$ having small prime divisors and Theorem 1 to deal with the sum over the remaining moduli.

1.1 Effectivity

We formulate the corollary of the main result.

Corollary 1** (Bombieri-Vinogradov, with exceptional character taken out).**

Let $x\geq 4$, $1\leq Q_{1}\leq Q\leq x^{\frac{1}{2}}$. Denote by $l(q)$ the smallest prime divisor of $q$. Then for any positive $\varepsilon>0$ we have

[TABLE]

The implied constant is effective and can be made explicit using [7] together with the best available constant in Pólya-Vinogradov inequality given in [5]. The effectivity is attained by getting rid of those moduli that have small prime divisors, thus of a possible exceptional character.

The recent work of Liu [10] gives us a genunely effective Bombiei-Vinogradov theorem. This is ultimately due to the fact that we can use an effective Landau-Page result (see [12], [8] and also [17, Chapter 10]), which is non-trivial up to $(\log x)^{2}$ instead of making a standard ineffective step on applying Siegel-Walfisz theorem.

Theorem** (Liu, 2017).**

There exists an effective positive constant $B$ such that

[TABLE]

In [10] various applications of the statement above are considered, such as an asymptotic formula for the representation of a large integer as the sum of two squares and a prime and Titchmarsh divisor problem (both results obviously become effective).

Acknowledgements

The author is grateful to Henryk Iwaniec for his crucial advice. The author also thanks her former supervisor Harald Helfgott for his help.

2 Proof of Theorem 1

Auxiliary lemmas

We start with a so-called weighted Vaughan identity. It allows us to get cancellation in type II sums.

Lemma 1** (Weighted Vaughan identity).**

Let $U,V\geq 1$, $n>U$. Define a function $\eta(t):\mathbb{Z}^{+}\to\mathbb{R}$ with $\eta(t)=1$ for $t\leq V$. We have

[TABLE]

where $\lambda_{0}(n)=\Lambda(n)$ for $n\leq U$ and equals to [math] for $n\leq U$, and

[TABLE]

Proof.

Let $n>U$, since otherwise the statement is trivial. Define the following quantities

[TABLE]

Vaughan’s identity in its classical form is

[TABLE]

so it remains to show that $\lambda_{1}^{\prime}(n)+\lambda_{2}^{\prime}(n)+\lambda_{3}^{\prime}(n)=0$ for every $n$. Let us rewrite this sum

[TABLE]

where in the last equality we used the fact that $\sum_{x|y}\Lambda(x)=\log y$. ∎

Lemma 2** (Graham [6]).**

Let $1\leq N_{1}\leq N_{2}\leq N$ and define

[TABLE]

We have

[TABLE]

From the lemma above one can deduce

Corollary 2**.**

Define a function $\eta(t)$, that is equal to $1$ for $t\leq V$, to [math] for $t>V_{0}$ and

[TABLE]

Then

[TABLE]

The constant here can be made explicit using [7].

We also need the large sieve inequality as stated in a classical form in, for example [11, p.561],

[TABLE]

from which it follows that

Lemma 3** (Large sieve inequality).**

Let $a_{m}$, $b_{n}$ be arbitrary complex numbers. Then

[TABLE]

where $c_{3}=2.64...$, $L(M,N)=\log(2MN)$ and $M^{\prime}=M-m_{0}+1$, $N^{\prime}=N-n_{0}+1$ are the number of terms in the sums over $m$ and $n$ respectively.

For the proof see [1, Lemma 6.1].

Proof of Proposition 1

We proceed now with the proof of Proposition 1. Fix arbitrary real numbers $Q>0$ and $x\geq 4$. Without loss of generality we can assume that $2\leq Q\leq x^{1/2}$ and decompose the von Mangoldt function using a weighted form of Vaughan’s identity, namely Lemma 1.

[TABLE]

where $\lambda_{i}(n)$, $i=0,1,2,3$ are as in the statement of the lemma and $U,V,V_{0}$ are parameters. Notice also that we are free to choose $\eta(t)$ as we wish, we only need to fulfill the conditions stated in Lemma 1.

Assume $y\leq x$, $q\leq Q$, and $\chi$ is a character mod $q$. We use the above decomposition to write

[TABLE]

where

[TABLE]

Denote the contributions to our main sum by

[TABLE]

Easily we obtain

[TABLE]

where

[TABLE]

Here in bounding $S_{0}$ we used Chebychev’s estimate

[TABLE]

In what follows we choose $\eta(\cdot)$ from the paper by Graham, see [6]:

[TABLE]

We remind that $\eta(d)=1$ for $d\leq V$ and $\eta(d)=0$ for $d>V_{0}$. This choice allows us to win $\log^{\frac{1}{2}}$ in the last sum, that is of type II.

Type I sums

We start with linear sums among $s_{i}$ and work with $s_{1}$ first. Write

[TABLE]

and exchange the sum and the integral

[TABLE]

Denote the summands $\sigma_{1}$ and $\sigma_{2}$. Then

[TABLE]

If $q=1$, then we have only trivial $\chi\operatorname{mod}q$ and

[TABLE]

If $q>1$ and $\chi$ is a primitive character $\operatorname{mod}q$, we use the Pólya-Vinogradov inequality(see [5] for explicit results): for all $x,y$ we have

[TABLE]

Then

[TABLE]

Further

[TABLE]

and

[TABLE]

Type II sums

Now we work with $s_{2}$ and want to use dyadic decomposition. Write

[TABLE]

where we introduced a new parameter $w$, that should be smaller than $U$ and will be chosen later. We deal first with the linear part of $s_{2}$, namely $s_{2}^{\prime}$. Write

[TABLE]

Since we have the bound

[TABLE]

then proceeding as for $s_{1}$ via Pólya-Vinogradov inequality and using the fact that $cd=t\leq wV_{0}$ we get

[TABLE]

where the $x$ term comes from the contribution of $q=1$ and $Q^{\frac{5}{2}}wV_{0}$ from the remaining $q\neq 1$.

Next consider $s_{2}^{\prime\prime}$. Writing $s_{2}^{\prime\prime}$ as a dyadic sum we have

[TABLE]

Using the triangle inequality

[TABLE]

By the large sieve inequality we get

[TABLE]

where $M^{\prime}$ and $K^{\prime}$ are the number of terms in sums over $c$ and $t$ respectively and

[TABLE]

and

[TABLE]

By Chebyshev’s estimate

[TABLE]

then using the estimates $M^{\prime}\leq M$, $K^{\prime}\leq\frac{x}{M}$ we have

[TABLE]

To bound $\sigma_{2}(M)$ we use a result of Corollary 2 and get

[TABLE]

Putting it together we obtain

[TABLE]

where we applied the bound

[TABLE]

We continue with an estimate for $S_{3}$ and use of the large sieve inequality (3) and properties of $\eta(\cdot)$ from Lemma 2. Writing $s_{3}$ as a dyadic sum we have

[TABLE]

Using the triangle inequality

[TABLE]

where $a_{m}=\Lambda(m)$ and $c_{k}=\sum_{{d|k},\;{d\leq V}}\mu(d)(1-\eta(d))$. Now apply the large sieve inequality (3) to get

[TABLE]

where

[TABLE]

and

[TABLE]

where $M^{\prime}$ and $K^{\prime}$ denote the number of terms in the sums over $m$ and $k$, respectively. From the definition of $M^{\prime}$ and $N^{\prime}$ we conclude

[TABLE]

By Chebyshev’s estimate we have an upper bound

[TABLE]

Thus by Cauchy inequality

[TABLE]

Further

[TABLE]

and

[TABLE]

Thus we have

[TABLE]

We take $\eta(\cdot)$ from the paper by Graham, see Corollary 2 and [6]:

[TABLE]

so that

[TABLE]

On applying Lemma 2 we obtain

[TABLE]

that implies

[TABLE]

Since

[TABLE]

then

[TABLE]

Finally we have to adjust the parameters $U,V,V_{0},w$. We repeat our previous estimates

[TABLE]

Combining the results above and taking $U=V$ we get

[TABLE]

where

[TABLE]

Now let’s specify $V$ and $V_{0}$. We introduce a parameter $0<\alpha<\frac{1}{2}$ to be chosen later. We subdivide into two cases

$x^{\alpha}\leq Q\leq x^{\frac{1}{2}}$, 2. 2.

$Q\leq x^{\alpha}$

and denote $\operatorname{R}(x,Q,w,V,V_{0})$, $\operatorname{G}(x,w,V,V_{0})$ as $\operatorname{R}_{1},\operatorname{R}_{2}$ and, respectively $\operatorname{G}_{1}$ and $\operatorname{G}_{2}$. If $x^{\alpha}\leq Q\leq x^{\frac{1}{2}}$, then $V=\frac{x^{\beta_{1}}}{Q}$. We choose $V_{0}=\frac{x^{\delta_{1}}}{Q}$ and $w=\frac{x^{\gamma_{1}}}{Q}$. Then putting that into previous expression $\operatorname{R}(x,Q,w,V,V_{0})$ we get for the factor

[TABLE]

If $Q\leq x^{\alpha}$, we let $V=x^{\beta_{2}}$, $V_{0}=x^{\delta_{2}}$, $w=x^{\gamma_{2}}$ and get

[TABLE]

Let $0<\varepsilon<\frac{1}{14}$. We keep in mind conditions $\alpha<\frac{1}{2}$, $\gamma_{1}<\beta_{1}$, $\delta_{1}>\beta_{1}$ and put

[TABLE]

Then

[TABLE]

where we used

[TABLE]

Similarly to satisfy $\gamma_{2}<\beta_{2}$, $\delta_{2}>\beta_{2}$ we put

[TABLE]

we obtain

[TABLE]

where we used

[TABLE]

Now we bound $\operatorname{G}(x,Q,w,V,V_{0})$. We notice that with our choice of parameters above $\log\frac{V_{0}}{V}\gg\log x$, where the implied constant depends on $\beta_{i},\delta_{i}$. Thus $\operatorname{G}_{1}\ll(\log x)^{2}$ and similarly $\operatorname{G}_{2}\ll(\log x)^{2}$. Finally, we have

[TABLE]

The power $\frac{13}{14}+\frac{\varepsilon}{2}$ is optimal here. Indeed, let us show first that $\alpha>\frac{3}{7}$. The system

[TABLE]

brings us to

[TABLE]

Solving this we obtain $\delta_{1}\leq\frac{5\alpha}{2}-\frac{1}{2}$. Further since $Q^{\frac{3}{2}}x^{1-\frac{\beta_{1}}{2}}\leq Q^{2}x^{\frac{1}{2}}$, we get

[TABLE]

Thus $\alpha>\frac{3}{7}$. We use that to obtain the fact that the term $Qx^{A}$ has $A>\frac{13}{14}$. Since $Q^{\frac{5}{2}}x^{\gamma_{2}+\delta_{2}}\leq Q^{2}x^{\frac{1}{2}}$, we get $\gamma_{2}+\delta_{2}\leq\frac{1}{2}-\frac{\alpha}{2}<\frac{2}{7}$. The inequality $Qx^{1-\frac{\beta_{2}}{2}}\leq Qx^{A}$ gives us $\delta_{2}>\beta_{2}\geq 2(1-A)$. Similarly for $\gamma_{2}$ we obtain $\gamma_{2}\geq 2(1-A)$ because of the term $Qx^{1-\frac{\gamma_{2}}{2}}\leq Qx^{A}$. Combining all of this we get

[TABLE]

and thus $A>\frac{13}{14}$.