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\centerline{\bf ON THE RIEMANN HYPOTHESIS}
\centerline{\bf FOR THE ZETA-FUNCTION}

\quad

\centerline{G. PUGLISI}

\quad

\quad

\centerline{\bf Abstract}

{\it\qquad\qquad The object of this paper is to give a proof of the conjecture made by Riemann}

{\it\qquad\qquad in 1859 about the complex zeros of the zeta-function} $\ \zeta(s)\ .\ $

\quad

\centerline{\bf Introduction}

\quad

We shall be concerned with the proof of the following

{\bf Theorem.}\ \ Let$\ \rho=\beta+i\gamma\ $ be any complex zero of the Riemann zeta function $\ \zeta(s)\ .\ $ Then
$$\beta={1\over2}\ \ .$$

\non

\centerline{\bf*****************}

\non

Before proceeding to prove this statement, I want to give here a detailed exposition of my method.

\quad

Put
$$M_V(s)\ =\ \sum_{n\le V}{\mu(n)\over n^s}\quad\ \big(s=\sigma+it\big)\ .$$

The first object is to define a class of integral functions$\ F_V(s)\ $such that

\item{a)} $F_V(\rho)=0\ $ for any complex zero $\ \rho=\beta+i\gamma\ .$

\item{b)} $F_V\ $converges to 1 as $\ V\rightarrow+\infty\ $ uniformly with respect to$\ s\ $in any bounded region on the right of the line $\ \sigma=\sup_{\rho}\beta$
 

This is performed by lemmas 2, 3, 4 and leads to definition (10), i.e.
$$F_V(s)\ \!=\ \!\zeta(s) M_V(s+s_V-1)$$

for a particular$\ s_V\ $near to 1.

More precisely, we distinguish two cases, according as $\ {1\over2}<\sup_{\rho}\beta < 1\ $ or $\ \sup_{\rho}\beta = 1\ .\ $ Using

Rouch\'e's theorem, it is not difficult to prove that in the former case
$$M_V(s_V)\ =\ 0\leqno(i)$$

for a suitable $\ s_V\ $ with $\ |s_V-1|\le V^{a-1}\ ,\ \ \!\sup_{\rho}\beta < a < 1\ $ and$\ V\ $sufficiently large.

In the latter case the same argument shows that $ (i) $ holds for a $ s_V $ which satisfies $\ |s_V-1|\le$

$\le\exp\big\{-(\log V)^{1/2}\big\}\ .$

Thus, in either case (see lemmas 2, 3) $\ F_V(s)\ $is regular in the whole plane. Furthermore

(see lemma 4), one may also prove that, for $\ |t|\ge 1$ 
$$F_V(s)\ \!=\ 1\ \!+\ \!\cases{O\Big(V^{a-\sigma- \delta}\ \!|t|^{\epsilon}\Big)\quad\ \ \big(a<\sigma+\delta<1\ \ ,\ \ 0<\delta<a-\sup_{\rho}\beta\big)\cr
\qquad\qquad\qquad\qquad\ \ in\ the\ former\ case\cr
O\Big(\exp\big\{-(\log V)^{1/4}\big\}\Big)\quad\ \big(\sigma\ge 1-\exp\{-(\log V)^{1/4}\}\big)\cr
\qquad\qquad\qquad\qquad\qquad\qquad \big(|t|\le\exp\{(\log V)^{1/2}\}\big)\cr
\qquad\qquad\qquad\qquad\ \ in\ the\ latter\ case\ \ .}\leqno(ii)$$

We shall next consider a zero $\ \rho_0=\beta_0+i\gamma_0\ $ with $\ \beta_0>\sup_{\rho}\beta - 1/200\ $ and

choose any $\ z_0>0\ $ such that

$\qquad\qquad\quad\ a-\beta_0<z_0\le\min\big({1-a\over5},\ \!{2a-1\over10}\big)\quad\ ,\quad \ \sup_{\rho}\beta < a < 1\ \ \!$

when $\ \ \sup_{\rho}\beta < 1\ ,\ $ while we choose any $\ \ \!z_0>0\ \ \!$ with $\ \ \!1-\beta_0<z_0\le 1/10$

if $\ \ \sup_{\rho}\beta = 1\ .\ $ Then, by $(ii)$
$$\bigg|\sum_{j=1}^{+\infty}\!{{(-z_0)}^j\over j!}\ \!D^j F_V(\rho_0+z_0)\bigg| = \big|F_V(\rho_0) - F_V(\rho_0+z_0)\big| = \big|F_V(\rho_0+z_0)\big|\ \!\!>\ \!\!{1\over2}\leqno(iii)$$

when V is great enough.

On the other hand, using classical tools it is not difficult to show that, when $\ \sigma<\sup_{\rho}\beta$
$$F_V(s)\ \ll\ \cases{(V|\gamma_0|)^{a-\sigma+\epsilon}\qquad\qquad\qquad\quad\ \!if\quad\ \sup_{\rho}\beta<1\cr
\big(V\sqrt{|\gamma_0|}\big)^{1-\sigma}\log^2(V|\gamma_0|)\qquad if\quad\ \sup_{\rho}\beta=1}$$

(the latter estimate being essentially an exercise).

This is done in lemmas 5, 8. As a consequence, by appealing to Cauchy's inequality, one

may see that
$$ {D^j F_V(\rho_0+z_0)\over j!}\ \!\ll\ \!\cases{{(V|\gamma_0|)^{2z_0}\over(2z_0)^j}\qquad\qquad\qquad\qquad if\quad\ \sup_{\rho}\beta < 1\cr\cr
{(V\sqrt{|\gamma_0|})^{2z_0}\over(2z_0)^j}\log^2(V|\gamma_0|)\qquad\ \!if\quad\ \sup_{\rho}\beta=1}\leqno(iv)$$

(see lemmas 6, 9). From$\ (iv)\ $it follows that
$$\bigg|\sum_{j>4z_0\log V}{(-z_0)^j\over j!} F_V(\rho_0+z_0)\bigg|\ \!<\ \!\epsilon\leqno(v)$$

if V is sufficiently large.\quad But when $\ j\le 4z_0\log V ,\ (iv)\ $ is of little use and we proceed

as follows. Define 
$$F_V(s)\ =\ \!\sum_{n=1}^{+\infty}{c_n\over n^s}\quad (\sigma>1)\quad , \quad p_j(u)\ \!=\ \!{1\over j!}\ e^{-u}\ \!u^j\quad \big(u\ge 0\big)\quad,\quad s_0=\rho_0+z_0\leqno(vi)$$

so that
$${1\over j!}\sum_{n\le Y}{c_n(\log n)^j\over n^{s_0}}\ \!=\ \!\sum_{n\le Y}{c_n\ \!p_j(\omega\log n)\over n^{s_0-\omega}}$$

where the sum
$$\sum_{n\le y}{c_n\over n^{s_0-\omega}}$$

is espressed in terms of $\ F_V(s)\ $ by means of Perron's formula \big(see (31), (49)\big).

Now the main problem is the treatment of the double sum
$$\eqalign{\sum_{1\le j\le 4z_0\log V}{{z_0}^j\over j!}&\sum_{U<n\le V^8}{c_n(\log n)^j\over n^{s_0}}\qquad\quad where\cr
&\qquad\qquad U = \cases{V^{1/10}\qquad\qquad\qquad\qquad\ if\quad\sup_{\rho}\beta< 1\cr
\exp\big\{(\log V)^{1/2}/4z_0\big\}\quad\ \ if\quad\sup_{\rho}\beta= 1\ \ .}}\leqno(vii)$$
\quad\ \ \ \big(see formulae (32), (51)\big). In both cases the crucial step is the following upper bound

provided by lemma 1 and (33)
$$\sum_{1\le j\le4z_0\log V}p_j(8z_0\log V)\ \!\ll\ \! V^{-{6z_0/5}}\ \!\longrightarrow\ 0$$

as $\ V\longrightarrow+\infty\ .\ $ If $\ 1/2<\sup_{\rho}\beta< 1\ $ the final result is the asymptotic

formula (37) which implies 
$$\bigg|\sum_{1\le j\le4z_0\log V}{{z_0}^j\over j!}\sum_{U<n\le V^8}{c_n(\log n)^j\over n^{\rho_0+z_0}}\bigg|\ \!<\ \!\epsilon\leqno(viii)$$ 

since, by$\ (vi),\ \ F_V(s_0-z_0) = F_V(\rho_0)=0\ .\ $

The proof of (54) is on the same lines, but one has to start from (51) instead of (32).

Finally, by means of straightforward arguments it may be shown that, in both cases

$$\bigg|\sum_{1\le j\le4z_0\log V}{{z_0}^j\over j!}\bigg(\sum_{n\le U}{c_n(\log n)^j\over n^{\rho_0+z_0}}+\sum_{n>V^8}{c_n(\log n)^j\over n^{\rho_0+z_0}}\bigg)\bigg| < \epsilon\ .\leqno(ix)$$

Collecting the estimates $\ (v),\ (viii),\ (ix)\ $ we finally obtain, if V is sufficiently large
$$\bigg|\sum_{j=1}^{+\infty}\!{{(-z_0)}^j\over j!}\ \!D^j F_V(\rho_0+z_0)\bigg|\ \!<\ \!3\epsilon\leqno(x)$$

either when $\ 1/2 < \sup_{\rho}\beta < 1\ $ or when $\ \sup_{\rho}\beta = 1\ .\ $

But$\ (x)\ $ is inconsistent with $\ (iii)\ .$

\non

\non

\non

\centerline{\bf Proof of the theorem}

\quad

We put
$$\ 0\ \!\le \sup_{\gamma> 10^3}\Big(\beta-{1\over2}\Big)\ \!=\  b\ \!\le\ \! {1\over2}\ \ .$$
and distinguish two main cases
$$\eqalign{&(i)\quad\ b = {1\over2}\quad\ \ then\quad\ \ b = b'=\limsup_{\gamma\rightarrow+\infty}\Big(\beta-{1\over2}\Big)\cr
&(ii)\quad 0<b<{1\over2}\quad\ \ then\ \ either\cr
&(ii)_1\ \ \!0\le b'< b = \max_{\gamma> 10^3} \Big(\beta-{1\over2}\Big)\quad or\cr
&(ii)_2\ \ \! b = b'\ \ .}\leqno(1)$$

Moreover
$$\eqalign{&if\ \ b\ \ is \ as \ in\ \ (1) ,\ \!(ii)_1\quad\ take\quad\rho_0 = b+{1\over2}+i\gamma_0\quad\ where\cr
&\qquad\qquad\qquad\qquad\quad\ \gamma_0 = \max\Big\{\gamma>10^3\ \!|\ \exists\ \!\rho = b+{1\over2}+i\gamma\Big\}\cr
&if\ \ b\ \ is\ \ as\ \ in\ \ (1) ,\ \!(ii)_2\quad then\quad\forall\ \!r>0\ \ \exists\ \!\rho_0 = \beta_0+i\gamma_0\ \ \!:\ \ \!\beta_0\ge b+(1-r)/2\cr
&if\ \ b\ \ is\ \ as\ \ in\ \ (1) ,\ \!(i)\quad then\quad\forall\ \!r>0\ \ \exists\ \!\rho_0 = \beta_0+i\gamma_0\ \ \!:\ \ \!\beta_0\ge 1-r/2\ \ .}\leqno(2)$$

We first prove

\non

{\bf Lemma 1.}\ \  When $\ x\ge 2$

$${1\over e^x}\!\sum_{1\le n\le x/2}{x^n\over n!}\ \le\ {x\over2e}\ \!\Big({2\over e}\Big)^{x/2}\ \ .$$

\non

{\bf Proof.}\ \ The following inequality 
$$\ n!\ \ge\ \!n^n e^{1-n}$$

is true for every $\ n\ge1 $ since it is trivial for$\ n=1\ $while, if it is true for some$\ n\ \!,\ $then
$$(n+1)!\ \ge\ \!(n+1)n^ne^{1-n}\ \!=\ \!e\Big({n\over n+1}\Big)^n(n+1)^{n+1}e^{-n}\ \!\ge\ (n+1)^{n+1}e^{1-(n+1)}\ .$$

Furthermore
$${d\over dt}\bigg(\Big({ex\over t}\Big)^t\bigg)\ \!=\ \!\Big({ex\over t}\Big)^t\log(x/t)\ \!> 0\quad\ \ \big(1\le t\le x/2\big)\ .$$

Hence
$$\sum_{1\le n\le x/2}{x^n\over n!}\ \le\ {1\over e}\!\sum_{1\le n\le x/2}\Big({e x\over n}\Big)^n\ \!\le\ {x\over2e}\ \!\max_{1\le n\le x/2}\ \!\Big({e x\over n}\Big)^n\ \!\le\ {x\over2e}\ \!\big(2e\big)^{x/2}\ .$$

\qed

According to (1), let now $\ a,\ r,\ v,\ w,\ T,\ \epsilon\ $ be real numbers such that

$$\eqalign{&\ \!0<\epsilon< 10^{-3}\quad ,\quad 0<120\epsilon\le 2r\le\min(1/2-b\ \!,\ \! 1/50)\quad,\quad 0<b<1/2\cr
&\qquad\ 1 > a\ge b+(1+2r)/2\quad ,\quad w = a+iv\quad,\quad2\le T\le v\le2T\ .}\leqno(3)$$

\non

{\bf Lemma 2.}\ \ Let $\ \epsilon\ ,\ r\ ,\ a\ ,\ b\ $ be as in (3) and let

$$\eqalign{&\ M_V(s)\ =\ \sum_{n\le V}{\mu(n)\over n^s}\quad ,\quad s\in\C \quad ,\quad V\ge V_0(\epsilon)\cr
&\qquad\ \ \!B(V)\ =\ \big\{s\in\C\ :\ |s-1|\le V^{a-1}\big\}\ .}$$

\quad

There exists a unique $\ s_V\in B(V)\ $ such that $\ M_V(s_V) = 0\ . $

\non

{\bf Proof.}\ \ Perron's formula gives

$$M_V(s)\ =\ {1\over2\pi i}\int_{c-iW}^{c+iW}\zeta^{-1}(s+z)\ {V^z\over z}\ dz\ +\ O\bigg({V^c\log V\over W}+{\log V\over V^{\sigma}}\bigg)\leqno(4)$$

where
$$c\ = \ \!\max(1-\sigma\ \!,\ 0) +(\log V)^{-1}\ \ ,\quad \sigma \ge a-r+\epsilon\quad,\quad W\ge2\ \ .$$

Also
$$\eqalign{&{1\over2\pi i}\int_{c-iW}^{c+iW}\zeta^{-1}(s+z)\ {V^z\over z}\ dz\ =\ \zeta^{-1}(s)\ +\ {1\over2\pi i}\Bigg(\int_{a-\sigma-r+\epsilon/2-iW}^{a-\sigma-r+\epsilon/2+iW}+\cr
&\quad\ \ +\int_{a-\sigma-r+\epsilon/2+iW}^{c+iW}-\int_{a-\sigma-r+\epsilon/2-iW}^{c-iW}\Bigg)\ \zeta^{-1}(s+z)\ {V^z\over z}\ dz  \ .}\leqno(5)$$

We have by (1), (3) $\quad \zeta^{-1}(s+z)\ \ll_{\epsilon}\ (W+|t|)^{\epsilon/2}\quad$for$\quad {\cal R}e(z)\ge a-\sigma-r+\epsilon/2\quad$ 


$\big($see  [T] \S 14.2\big). Hence
$$\eqalign{&\ \int_{a-\sigma-r+\epsilon/2-iW}^{a-\sigma-r+\epsilon/2+iW}\zeta^{-1}(s+z)\ {V^z\over z}\ dz\ \ll_{\epsilon}\ V^{a-\sigma-r+\epsilon/2}\ \!(W+|t|)^{\epsilon}\cr
&\ \ \int_{a-\sigma-r+\epsilon/2\pm iW}^{c\pm iW}\zeta^{-1}(s+z)\ {V^z\over z}\ dz\ \ll_{\epsilon}\ V^c W^{\epsilon-1}(1+|t|)^{\epsilon/2}\ .}\leqno (6)$$

From (4), (5), (6) it follows that
$$M_V(s)\ =\ \zeta^{-1}(s)\ +\ O_{\epsilon}\Big(V^c W^{\epsilon-1}(1+|t|)^{\epsilon/2}+\ V^{a-\sigma-r+\epsilon/2}\ \!(W+|t|)^{\epsilon}\Big)\leqno(7)$$

where
$$W\ge 2\ \ ,\ \ \sigma\ge a-r+\epsilon\ \ ,\ \ c\ = \ \!\max(1-\sigma\ \!,\ 0) +(\log V)^{-1}\ .$$

When $\ \ \!s\in \partial B_r(V)\ $ and $\ W = V\ge V_0(\epsilon)\ \!,\ (7)\ $gives at once
$$\eqalign{&|M_V(s)-\zeta^{-1}(s)|\ \le\ C(\epsilon)\ \! V^{a-1-r+3\epsilon/2}\ \!\le\ V^{a-1-r +2\epsilon}\cr
&\quad|\zeta^{-1}(s)|\ \ge\ |s-1|/2\ =\ V^{a-1}/2\ >\  V^{a-1-r+40\epsilon}\ \!.}$$

Therefore, by Rouch\'e's theorem 
$$\sharp\big\{s\in B_r(V)\ :\ M_V(s) = 0\big\}\ =\ \sharp\big\{s\in B_r(V)\ :\ \zeta^{-1}(s) = 0\big\}\ =\ 1\ \ .$$

\qed

Suppose now 
$$0<60\epsilon\le r\le 1/100\quad ,\quad b = 1/2\quad ,\quad a=1\quad ,\quad w\ \ as\ \ in\ \ (3)\ .\leqno(8)$$

\non

{\bf Lemma 3.}\ \ Let $\ b\ $ be as in (8) ,$\ M_V(s)\ $ as in lemma 2 , $\ V\ge V_0\ $ and let
$$B(V)\ =\ \Big\{s\in\C\ :\ |s-1|\le\exp\big\{-(\log V)^{1/2}\big\}\Big\}\ .$$

There exists a unique $\ s_V\in B(V)\ $ such that $\ M_V(s_V) = 0\ . $

\non

{\bf Proof.}\ \ When $\ \sigma\ge 1-c_0/\log(|t|+2)\ \ $\big(for a suitable $\ c_0\ ,\ 0<c_0\le 1\big)\ $ we have

$|\zeta(s)|\ll\log(|t|+2)\ \!.\ $ Then, if $\ \sigma\ge 1-c_0/2\log(|t|+2)\ $ and$\ c\ $ is as in (4)  
$$\eqalign{&\quad{1\over2\pi i}\int_{c-i(V+|t|)}^{c+i(V+|t|)}{1\over\zeta(s+z)}\ {V^z\over z}\ dz\ =\ {1\over\zeta(s)}\ +\ {1\over2\pi i}\Bigg(\int_{1-\sigma-{c_0\over\log(|t|+2)}-i(V+|t|)}^{1-\sigma-{c_0\over\log(|t|+2)}+i(V+|t|)}+\cr
&\quad\ \ \!+\int_{1-\sigma-{c_0\over\log(|t|+2)}+i(V+|t|)}^{c+i(V+|t|)}-\int_{1-\sigma-{c_0\over\log(|t|+2)}-i(V+|t|)}^{c-i(V+|t|)}\Bigg)\ \zeta^{-1}(s+z)\ {V^z\over z}\ dz\ =\cr
&\quad =\ {1\over\zeta(s)}\ +\ O\bigg(V^{1-\sigma-{c_0\over\log(|t|+2)}}\ \!\log^2(V+|t|) +\ \!{V^{c_0\over2\log(|t|+2)}\ \!\log(V +|t|)\over(V+|t|)}\bigg)\ \!=\cr
&\qquad\qquad\quad\ \ =\ {1\over\zeta(s)}\ \!+\ \!O\bigg(\exp\Big\{-{c_0\log V\over2\log(|t|+2)}\Big\}\ \!\log^2(V+|t|)\bigg)}$$

whence, according to (4)
$$|M_V(s)-\zeta^{-1}(s)|\ \le\ \ \!C_0\ \!\exp\Big\{-{c_0\log V\over6\log(|t|+2)}\Big\}\ \!\log^2(V+|t|)\ .\leqno(9)$$

On the other hand, when $\ s\in\partial B(V)\ $ and $\ V\ge V_0$ 
$$\eqalign{|\zeta^{-1}(s)|\ \ge\ |s-1|/&2\ =\ \exp\big\{-(\log V)^{1/2}\big\}/2\ \ge\cr
&\ \ \ge\ \ \! 2C_0\ \!\exp\Big\{-{c_0\log V\over6\log3}\Big\}\ \!\log^2(V+1)\ .}$$

and the lemma follows as before.

\qed

From now on we shall investigate the properties of the following integral function

$$\eqalign{&\qquad\ \!F_V(s)\ =\  \zeta(s)\ \!M_V(s+s_V-1)\cr
&V\ ,\ M_V(s)\ ,\ s_V\ \ as\ \ in\ \ lemmas\ \ 2\ \!,\ \!3\ \ \!.}\leqno(10)$$ 

We begin by proving

\non

{\bf Lemma 4.}\ \ If$\ F_V(s)\ $is defined by (10), then
$$F_V(s)\ =\ 1\ +\ \cases{\ O_{\epsilon}\Big(V^{a-\min(1,\ \!\sigma+r)+2\epsilon}\ \!(|t|+1)^{2\epsilon}\Big)\quad\big(\sigma\ge a+2\epsilon-r\big)\cr
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\ \!\big(|s-1|\ge \epsilon\big)\cr
\qquad\qquad\qquad\qquad\qquad\quad\ \!if\quad a\ \ is\ \ as\ \ in\ \ (3)\cr\cr
\ O\Big(\exp\Big\{-(\log V)^{1/4}\Big\}\Big)\quad\ \big(\sigma\ge 1-\exp\{-(\log V)^{1/4}\}\big)\cr
\qquad\qquad\quad\ \big(|s-1|\ge \log^{-1}V\ \ ,\ \ |t|\le\exp\{(\log V)^{1/2}\}\big)\cr
\qquad\qquad\qquad\qquad\qquad\quad if\quad a=1\ \ .}$$

\non

{\bf Proof.} \ \ Let $\ 0<b<1/2\ .\ \ \!$ By lemma 2 $\quad\!|1-s_V|\ \le\ V^{a-1}\le\ \epsilon\quad$if $\ \ V\ge \epsilon^{1/(a-1)}\ .$

Then, when $\ \sigma\ge a+2\epsilon-r$
$$\min\big(\sigma,\ {\cal R}e(s+s_V-1)\big)\ \ge\ a+2\epsilon-r - |s_V -1|\ \ge\ a-r +\epsilon\ .\leqno(11)$$ 

It follows from (7) with\ $ W=V\ $ and from (11) that
$$\eqalign{&M_V(s) - M_V(s+s_V-1)\ \!=\ \!\zeta^{-1}(s) - \zeta^{-1}(s+s_V-1)\ \!+\ \!O_{\epsilon}\Big(\big(V^{a-\sigma-r+3\epsilon/2}+V^{-1}\big) (|t|+1)^{\epsilon}\Big)\cr
&\qquad\qquad\qquad\big|\zeta^{-1}(s)-\zeta^{-1}(s+s_V-1)\big|\ =\ |s_V-1|\ \!\Big|{\zeta'\over\zeta^2}(s^*)\Big| \ll_{\epsilon}\ V^{a-1} (|t|+1)^{\epsilon}\cr
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\ \big({\cal R}e (s^*)\ge a-r+\epsilon\big)\ }$$

since $\quad\max\big(|\zeta(s)|\ \!,\ \!|\zeta(s)|^{-1} ,\ \!|(\zeta'/\zeta)(s)|\big) \ll_{\epsilon} (|t|+1)^{\epsilon\over2}\ \ \!$\ for $\ \ \sigma\ge a-r+\epsilon\ ,$


$|s-1|\ge\epsilon\ \ $\ (see [T] \S 14.2.5)\ .

Hence
$$\eqalign{\big|F_V(s) - \zeta(s)\ \!M_V(s)\big|&\ =\ |\zeta(s)|\ \!|M_V(s) - M_V(s+s_V-1)|\ \ll_{\epsilon}\cr
&\ \!\ll_{\epsilon}\ (|t|+1)^{3\epsilon/2}\Big(V^{a-\sigma-r+3\epsilon/2} +\ \!V^{a-1}\Big)\ .}\leqno(12)$$

Also, again by (7)
$$(\zeta\ \!M_V)(s)\ \!=\ \!1 + O_{\epsilon}\Big(\big(V^{a-\sigma-r+3\epsilon/2}+V^{-1}\big) (|t|+1)^{3\epsilon/2}\Big)\leqno(13)$$

and the result follows from (12), (13).

Suppose now $\ b=1/2\ .\ \ \!$When $\ \sigma\ge 1-\exp\{-(\log V)^{1/4}\}\ $ and $\  |t|\le\exp\{(\log V)^{1/2}\}$

lemma 3 implies \big($ c_0\ $as in (9)\big)
$$\eqalign{\min\big(\sigma,\ {\cal R}e(s+s_V-1)\big)\ &\ge\ 1-\exp\{-(\log V)^{1/4}\} - \exp\{-(\log V)^{1/2}\}\ \ge\cr
&\ge\ 1-{c_0\over2}\big(\log(|t|+2)\big)^{-1}\ .}\leqno(14)$$

Since $\ \zeta'(s)\zeta^{-2}(s)\ll\log^2(|t|+2)\ $ for $\ \sigma\ge 1-c_0/\log(|t|+2)\ \!,\ \!$ we obtain from (9), (14)
$$\eqalign{&\ M_V(s) - M_V(s+s_V-1) = {1\over\zeta}(s) -{1\over\zeta}(s+s_V-1) + O\bigg(\exp\Big\{-{c_0\over6}{\log V\over\log(|t|+2)}\Big\}\ \!\cdot\cr
&\quad\cdot\big(\log(V+|t|)\big)^2\bigg) = (s_V-1)\ \!\zeta'(s_1)\ \!\zeta^{-2}(s_1) + O\Big(\exp\Big\{-{c_0\over8}\big(\log V\big)^{1/2}\big\}\Big) \ll\cr
&\qquad\qquad\qquad\qquad\qquad\qquad \ll\ \exp\Big\{-{c_0\over8}(\log V)^{1/2}\Big\}\cr
&\qquad\qquad\qquad\qquad\ \!(\zeta\ \!M_V)(s)\ -\ F_V(s)\ \ll\ \exp\big\{-(\log V)^{1/4}\big\}\cr
&\qquad\qquad\qquad\qquad\quad (\zeta\ \!M_V)(s) = 1 + O\Big(\exp\big\{-(\log V)^{1/4}\big\}\Big) \cr
&\qquad\! F_V(s) = (\zeta\ \!M_V)(s) + \big(F_V(s) - (\zeta\ \!M_V)(s)\big) =\ \!1 + O\big(\exp\big\{-(\log V)^{1/4}\big\}\big)\ \!.}$$

as required.

\qed

We now need a sharp estimate for $\ D^jF_V(s)\ $ when $\ \sigma>a\ .\ $ Put  
$$\eqalign{&\quad\ \ F_V(s)\ =\ \!\sum_{n=1}^{+\infty}{c_n\over n^s}\qquad\ (\sigma>1)\cr
&c_n\ =\ c_n(V)\ =\sum_{d|n,\ d\le V}\mu(d)\ \!\ \!d^{1-s_V}\ .}\leqno(15)$$

and suppose first $\ \ 0<b<1/2\ .$

\non

{\bf Lemma 5.}\ \ Let $\ \epsilon\ ,\ r\ ,\ a\ ,\ w\ ,\ T\ $ be as in (3) and let $\ F_V(s),\ c_n\ $ be defined by (10),

\qquad\qquad\quad\ (15) respectively. If 
$$s_0=w+r=a+r+iv\quad,\quad|\omega|\le1\quad,\quad r\le{\cal R}e\omega\le a -1/2$$

then, for $\ Y\ge2$
$$\eqalign{\sum_{n\le Y}{c_n\over n^{s_0-\omega}}\ &\ll_{\epsilon}\ \!(VT)^{{\cal R}e\omega}\bigg(1 + {\sqrt T\big(V(T+Y)\big)^{\epsilon}\over Y^r}\bigg)\ +\cr
&+\ \!Y^{{\cal R}e\omega}\bigg({(VT)^{{\cal R}e\omega}\over Y^{1+r}} + {1\over Y^a}\bigg)\big(V(T+Y)\big)^{\epsilon}\ .}$$

Furthermore
$$F_V(s_0-\omega+ z)\ \ll_{\epsilon}\ \!\big(V(T+|{\cal I}m z|)\big)^{\max(0,\ \!{\cal R}e (\omega-z)-r)+\epsilon}$$

when
$${\cal R}e z\ge{\cal R}e\omega + 1/2 -a-r\quad,\quad|s_0-\omega+z-1|\ge\log^{-1}\!T\ .$$

\non

{\bf Proof.}\ \ By (15) and Perron's formula
$$\eqalign{\sum_{n\le Y}{c_n\over n^{s_0-\omega}}\ &=\ \!{1\over2\pi i}\int_{1-a+{\cal R}e\omega-r+\log^{-1}\!V-iT_1}^{1-a+{\cal R}e\omega-r+\log^{-1}\!V+iT_1}F_V(s_0-\omega+z)\ Y^z\ {dz\over z}\ +\cr
&+\ O_{\epsilon}\bigg({Y^{1-a+{\cal R}e\omega-r+\epsilon} \log V\over T_1}\bigg)\qquad\big(2\le T_1\le Y\ ,\ |T_1-v|\ge v/2\big)\ .}\leqno(16)$$

Since $\ F_V(s)\ $ is regular, we have
$$\eqalign{&\qquad{1\over2\pi i}\int_{1-a+{\cal R}e\omega-r+\log^{-1}\!V-iT_1}^{1-a+{\cal R}e\omega-r+\log^{-1}\!V+iT_1}F_V(s_0-\omega+z)\ Y^z\ {dz\over z}\ =\cr
&=\ F_V(s_0-\omega)\ +\ {1\over2\pi i}\Bigg(\int_{-r-iT_1}^{-r+iT_1}+\int_{-r+iT_1}^{1-a+{\cal R}e\omega-r+\log^{-1}\!V+iT_1}+\cr
&\qquad\ \!-\int_{-r-iT_1}^{1-a+{\cal R}e\omega-r+\log^{-1}\!V-iT_1}\Bigg)F_V(s_0-\omega+z)\ Y^z\ {dz\over z}\ \ .}\leqno(17)$$

Suppose first that $\ -\epsilon\le {\cal R}e(z-\omega)+r\le 1-a+\log^{-1}\!V\ .\ $ Then, for $\ 2\le T_2\le V$

and $\ V\ge V_0(\epsilon)\ $ \big(see (6)\big) 

$$\eqalign{&\ M_V(s_0+s_V-1-\omega+z)\ =\ {1\over2\pi i}\int_{2-s_V-a+{\cal R}e(\omega-z)-r+2\log^{-1}\!V-iT_2}^{2-s_V-a+{\cal R}e(\omega-z)-r+2\log^{-1}\!V+iT_2}\ \!{1\over\zeta}\big(s_0\ \!+\cr
&\qquad\ +s_V-1-\omega+ z +\eta\big)\ V^{\eta}\ {d\eta\over\eta}\ \!+\ O\bigg({V^{1-a+{\cal R}e(\omega-z)-r}\log V\over T_2}\bigg)\ =\cr
&\ \ \!=\ {1\over2\pi i}\Bigg(\int_{1-s_V+{\cal R}e(\omega-z)-r-2\epsilon-iT_2}^{1-s_V+{\cal R}e(\omega-z)-r-2\epsilon+iT_2} +\int_{1-s_V+{\cal R}e(\omega-z)-r-2\epsilon+iT_2}^{2-s_V-a+{\cal R}e(\omega-z)-r+2\log^{-1}\!V+iT_2}+\cr
&-\int_{1-s_V+{\cal R}e(\omega-z)-r -2\epsilon-iT_2}^{2-s_V-a+{\cal R}e(\omega-z)-r+2\log^{-1}\!V-iT_2}\Bigg)\ \!{1\over\zeta}(s_0 +s_V-1-\omega+z+\eta)\ V^{\eta}\ {d\eta\over\eta}\ +\cr
&+\  {1\over\zeta}\big(s_0 +s_V-1-\omega+ z\big) + O\bigg({V^{1-a+{\cal R}e(\omega-z)-r}\log V\over T_2}\bigg) \ll_{\epsilon}V^{{\cal R}e(\omega-z)-r}\ \!\cdot\cr
&\cdot\big((T+|{\cal I}m z|+T_2)\ \!V^{-2}\big)^{\epsilon\over2} +\ \!\big(V^{1-a+{\cal R}e(\omega-z)-r}\log V\big)\ \!T_2^{-1} +\ \!\big(T+|{\cal I}m z|\big)^{\epsilon\over2} }$$

while, when $\ \ \!{1\over2}-a\le{\cal R}e(z-\omega)+r\le -\epsilon$
$$\eqalign{&\ \ \!M_V(s_0+s_V-1-\omega+z)\ =\ {1\over2\pi i}\Bigg(\int_{1-s_V+{\cal R}e(\omega-z)-r+iT_2}^{2-s_V-a+{\cal R}e(\omega-z)-r+2\log^{-1}\!V+iT_2}+\cr
&\ \ \!-\!\int_{1-s_V+{\cal R}e(\omega-z)-r-iT_2}^{2-s_V-a+{\cal R}e(\omega-z)-r+2\log^{-1}\!V-iT_2}\!+\int_{1-s_V+{\cal R}e(\omega-z)-r-iT_2}^{1-s_V+{\cal R}e(\omega-z)-r+iT_2}\Bigg)\ \!{1\over
\zeta}(s_0\ \!+\cr
&\quad\ \ \!+s_V -1-\omega+z+\eta)\ V^{\eta}\ {d\eta\over\eta}\ \!+\ \! O\bigg({V^{1-a+{\cal R}e(\omega-z)-r}\log V\over T_2}\bigg)\ \!\ll_{\epsilon}\cr
&\ll_{\epsilon}\ \! V^{{\cal R}e(\omega-z)-r}\big(T+|{\cal I}m z|+T_2\big)^{\epsilon\over2}\log V +\ \!\big(V^{1-a+{\cal R}e(\omega-z)-r}\log V\big)\ \!T_2^{-1}\ \!.}$$

Therefore, on choosing $\ T_2 = V^{1-a}\log V$ 
$$\eqalign{&M_V(s_0+s_V-1-\omega+z)\ \ll_{\epsilon}\ \!V^{\max(0,\ \!{\cal R}e(\omega-z)-r}\big(V(T+|{\cal I}m z|)\big)^{\epsilon\over2}\cr
&\qquad\qquad\ \Big(\ \!{1\over2}-a\le{\cal R}e(z-\omega)+r\le 1-a+\log^{-1}\!V\ \!\Big)\ .}\leqno(18)$$

Also, by a well known convexity argument
$$\eqalign{&\quad\zeta(s_0-\omega+ z)\ \ll_{\epsilon}\ \!\big(T+|{\cal I}m z|\big)^{\max(0,\ \!{\cal R}e(\omega-z)-r+\epsilon/2}\cr
\Big({1\over2}-a\le&\ \!{\cal R}e(z-\omega)+r\le 1-a+\log^{-1}\!V\ \ ,\ \ |s_0-\omega+z-1|\ge\log^{-1}\!T\Big)}\leqno(19)$$

When $\ {\cal R}e(z-\omega)-r\le 1-a+\log^{-1}\!V\ $ the latter bound in the lemma now follows

from (18), (19), while it is obvious if $\ {\cal R}e(z-\omega)-r\ge 1-a+\log^{-1}\!V\ .$
 
We also obtain from (16), (17), (18), (19)
$$\eqalign{&\sum_{n\le Y}{c_n\over n^{s_0-\omega}}\ \ll_{\epsilon}\ \!{V^{{\cal R}e\omega}\big(V(T+T_1)\big)^{\epsilon\over2}\over Y^r}\bigg(\!\log T_1\!\int_{-T_1}^{T_1}\big|\zeta\big(a-{\cal R}e\omega+i(v+y)\big)\big|^2\ \!{dy\over|y|+1}\bigg)^{1\over2} +\cr
&\qquad\quad\ \!+\ \!\bigg({\big(V(T+T_1)\big)^{{\cal R}e\omega}\over Y^r}\ \!+\ \!Y^{1-a+{\cal R}e\omega-r+\epsilon}\bigg)\ \!{\big(V(T+T_1)\big)^{\epsilon}\over T_1}\ \!+\ \!(VT)^{{\cal R}e\omega}\ .}\leqno(20)$$

Furthermore, when $\ T_1\ge2T\ $we put $\ L\ \!=\ \!\Big[{\log(T_1/T)\over\log2}\Big]\ .\ $ Then
$$\eqalign{&\qquad\qquad\qquad\quad\int_{-T_1}^{T_1}\big|\zeta\big(a-{\cal R}e\omega+i(v+y)\big)\big|^2\ \!{dy\over|y|+1}\ \ll\cr
&\ll \int_0^{2T}\big|\zeta(a-{\cal R}e\omega+it)\big|^2\ \!dt\ \!+\ \!{1\over T} \sum_{\ell=1}^L\ \!{1\over 2^{\ell}}\int_{2^{\ell}T}^{2^{\ell+1}T}\big|\zeta(a-{\cal R}e\omega+it)\big|^2\ \!dt\ \!\ll\cr
&\qquad\quad\ \ \ll\ T\log T\ \!+\ \!\sum_{\ell=1}^L\ \!\big(\ell+\log T\big)\ \ll\ T\log T\ +\ \log^2T_1\ .}\leqno(21)$$

Inserting the above bound in (20) and taking $\ T_1 =\theta Y\ \ \!(1/6\le\theta\le 1)\ \!,\ $ we finally have
$$\eqalign{\sum_{n\le Y}{c_n\over n^{s_0-\omega}}\ &\ll_{\epsilon}\ \!Y^{{\cal R}e\omega}\bigg({(VT)^{{\cal R}e\omega}\over Y^{1+r}} + {1\over Y^a}\bigg)\big(V(T+Y)\big)^{\epsilon}\ \!+\cr
&+\ (VT)^{{\cal R}e\omega}\bigg(1 + {\sqrt T\big(V(T+Y)\big)^{\epsilon}\over Y^r}\bigg)\ .}\leqno(22)$$

The proof of lemma 5 is now complete.

\qed

We shall deduce

\non

{\bf Lemma 6.}\ \ Let $\ \epsilon\ ,\ r\ ,\ a\ ,\ w\ ,\ T\ $ be as in (3) and let$\ F_V(s)\ $be defined by (10). If 
$$2r\ \!\le\ \!z_0\ \!\le\ \!\min\big(10\ \!r\ \!,\ \!(2a-1)/10\ \!,\ \!(1-a)/5\big)$$

then, for$\ j\ge 1$
$${1\over j!}\ \!D^j F_V(w+r)\ \ll\ {(VT)^{2z_0}\over(2z_0)^j}\quad .$$

\non

{\bf Proof.}\ \ Taking$\ \omega = 2z_0\ $in lemma 5, we have
$${\cal R}e\omega+{1\over2}-a-r\ \!\le\ \!{2a-1\over5} + {1-2a\over2} - r\ \! <\ \! 0\ .$$

Hence
$$F_V(s_0-2z_0+ z)\ \ll_{\epsilon}\ \!\big(VT)^{2z_0-r+\epsilon}\ \!\le\ \!(VT)^{2z_0}\leqno(23)$$

for $\ {\cal R}e z\ge 0\ $ and $\ |{\cal I}m z|\le 1\ .\ $ Now apply Cauchy's inequality to the circle $\ |s-s_0|\le 2z_0$

and obtain from (23)
$${1\over j!}\ \!D^j F_V(s_0)\ \ll\ {(VT)^{2z_0}\over(2z_0)^j}\ .$$

\qed

Following Bombieri \big(see[B], p. 46\big), define
$$p_j(u)\ =\ {1\over j!}\ e^{-u}\ \!u^j\quad \big(u\ge 0\big)$$

so that
$$\eqalign{&\qquad\ \sum_{j=0}^{+\infty}\ \!p_j(u) = 1\quad\ ,\quad\ p'_j(u)\ =\ p_{j-1}(u) - p_j(u)\quad\ ,\quad\ |p'_j(u)| \le 1\cr
&p'_j(u)\ \cases{>0\quad\ \ if\quad\ u<j\cr
<0\quad\ \ if\quad\ u>j}\quad\ ,\quad\ p_j(u)\le e^{-u/2}(360)^{-j}\quad when\quad u\ge20j\ \ .}\leqno(24)$$

Thus, using partial summation we may write
$$\eqalign{&{1\over j!}\sum_{n\le Y}{c_n(\log n)^j\over n^{s_0}}\ =\ {1\over\omega^j}\sum_{n\le Y}{c_n\ \!p_j(\omega\log n)\over n^{s_0-\omega}}\ =\ {1\over\omega^j}\Bigg(\sum_{n\le Y}{c_n\over n^{s_0-\omega}}\ \cdot\cr
&\qquad\quad\ \cdot p_j(\omega\log Y)\ -\ \omega\int_1^Y\sum_{n\le y}{c_n\over n^{s_0-\omega}}\ p'_j(\omega\log y)\ {dy\over y}\Bigg)\cr
&\qquad\qquad\qquad s_0=  w+r\quad\ ,\quad\ r\le\omega\le a-1/2\ \ .}\leqno(25)$$

Let further $\ Z_1\ge e^j/\omega\ $ and $\ Z_2\ge e^{20j}/\omega\ .\ $ Lemma 5, (24) and (25) then imply
$$\eqalign{&\qquad\qquad\ \ {1\over j!}\bigg|\sum_{Z_1<n\le Z_2}{c_n(\log n)^j\over n^{s_0}}\ \!\bigg|\ \le\ {1\over\omega^j}\ \!\bigg(\ \!\bigg|\sum_{Z_1<n\le Z_2}{c_n\over n^{s_0-\omega}}\ \!\bigg|\ {Z^{-\omega/2}\over(360)^j}\ +\cr
&\qquad\quad+\ \!\omega\int_{Z_1}^{Z_2}\bigg|\sum_{Z_1<n\le y}{c_n\over n^{s_0-\omega}}\ \!\bigg| - p_j'(\omega\log y)\ {dy\over y}\bigg)\ \ll_{\epsilon}\ \!{V^{\omega+\epsilon}\ \!T^{{1\over2}+\omega+\epsilon}\over Z_2^{\omega/2}(360\ \!\omega)^j}\ +\cr
&+ {V^{\omega+\epsilon}\ \!T^{{1\over2}+\omega+\epsilon}\over\omega^{j-1}}\!\int_{Z_1}^{Z_2}\!- p'_j(\omega\ \!\log y)\ {dy\over y}\ \ll\ V^{\omega+\epsilon}\ \!T^{{1\over2}+\omega+\epsilon}\bigg({Z_2^{-\omega/2}\over(360\ \!\omega)^j} + {(\log Z_1)^j\over\ j!\ \!Z_1^{\omega}}\bigg)\ .}\leqno(26)$$

We also define 
$$\eqalign{& J=[4z_0\log V]\cr
\big(2r\ \!\le\ \!z_0\ \!\le\ \!\min&\big(10\ \!r\ \!,\ \!(2a-1)/10\ \!,\ \!(1-a)/5\big)\big)\ .}\leqno(27)$$

The upper bound for $\ D^j F_V(w+r)\ $ provided by lemma 6 is a good result only if $j$ is great.

However the sum on the left of (22) can be dealt with in a different way.

By lemma 2 and (15)
$$\eqalign{&c_n\ \!=\sum_{d|n}\mu(d)\ \!\Big(\exp\big\{(1-s_V)\log d\big\}-1\Big)\ \!\ll\ \!V^{a-1}\sum_{d|n}\log d\ \ll\cr
&\qquad\quad\ \ll\ \!V^{a-1}\sum_{d|n}\log d\qquad\quad  when\qquad  1<n\le V\ .}\leqno(28)$$

Let now
$$\delta(Y)\ =\ \cases{ V^{a-1}\qquad\quad if\quad\ 2\le Y\le V\cr
 1\qquad\qquad\quad\! if\qquad\ Y> V\ \ \!.}\leqno(29)$$

Lemma 4 together with (28), (29) then give

$$\eqalign{&\quad\ \!\sum_{n\le Y}{c_n\over n^{s_0-\omega}}\ =\ 1\ \!+\ \!{1\over2\pi i}\bigg(\int_{\omega-r-iX}^{\omega-r+iX} +\int_{\omega-r+iX}^{1-a+\omega-r +\log^{-1}\!Y+iX}+\cr
&\quad\quad\!-\int_{\omega-r-iX}^{1-a+\omega-r+\log^{-1}\!Y-iX}\bigg)\Big(F_V(s_0-\omega+z)-1\Big)\ Y^z\ {dz\over z}\ \ \!+\cr
&\qquad\qquad\qquad\ \!+\ O_{\epsilon}\bigg(Y^{(\omega-r-a+\epsilon}\Big({Y\over X} + \delta(Y)\Big)\bigg)\ ,\cr
&\qquad\int_{\omega-r\pm iX}^{1-a+\omega-r+\log^{-1}\!Y\pm iX}\Big(F_V(s_0-\omega+z)-1\Big)\ Y^z\ {dz\over z}\ \ll_{\epsilon}\cr
&\quad\ \ll_{\epsilon} V^{-r+2\epsilon}\ \!T^{2\epsilon}\ \!Y^{1-a+\omega-r} X^{-1+2\epsilon}\ \!\ll_{\epsilon}\ \! V^{-r+2\epsilon}\ \!T^{2\epsilon}\ \!X^{-a+\omega}\ \!, \cr
&\qquad\! Y^{\omega-r-a+\epsilon}\Big({Y\over X} + \delta(Y)\Big)\ \!\ll\ \!V^{-a+\omega-r+\epsilon}+ V^{a-1}\ \!\ll\ V^{a-1}\cr
&\qquad\ \!\omega\ \ as \ \ in\ \ (25)\quad,\quad 2\ \!\le\ \!Y\ \!\le\ \! X\quad,\quad X\ \!\ge\ \!\max(V,3T)}\leqno(30)$$

and we may pass to the more difficult case when $\ j\le J\ .$

\non

{\bf Lemma 7.}\ \ Let $\ \epsilon\ ,\ r\ ,\ a\ ,\ w\ ,\ T\ $ be as in (3) and let$\ F_V(s)\ $be defined by (10), with 
$$V\ge T^{2/z_0}\ \ ,\quad 2r\ \!\le\ \!z_0\ \!\le\ \!\min\big(10\ \!r\ \!,\ \!(2a-1)/10\ \!,\ \!(1-a)/5\big)\ \ \!.$$

Then 
$$\eqalign{\sum_{1\le j\le 4z_0\log V}\!{{(-z_0)}^j\over j!}\ \!D^j F_V(w+r)\ &=\ F_V(w+r-z_0)\Big(1 + O\big(V^{-{z_0\over10}}\big)\Big)\ +\cr
&+\ O_{\epsilon}\Big(V^{{z_0-11r\over 10}+19\epsilon} +\ \!V^{-{z_0\over10}}\Big)\ .}$$

\non

{\bf Proof.}\ \ We first deduce from (30) (with $\ \omega=z_0$) 
$$\eqalign{&\sum_{n\le Y}{c_n\over n^{s_0-z_0}}\ \!=\ \!1 + {1\over2\pi i}\!\int_{z_0-r-iX}^{z_0-r+iX}\!\!\!\Big(F_V(s_0-z_0+z) -1\Big)\ \!Y^z\ \!{dz\over z} +\ \!O\Big( V^{a-1}\Big)\cr
&\qquad\qquad\qquad\qquad\ \big(2\le Y\le X\quad,\quad X\ge\max(V,3T)\big)\ .}\leqno(31)$$

Hence, by (24), (25), (31)
$$\eqalign{&\qquad\qquad\quad\sum_{j=1}^J{{z_0}^j\over j!}\!\sum_{V^{1/10}<n\le X}{c_n(\log n)^j\over n^{s_0}}\  =\ \sum_{j=1}^J\sum_{V^{1/10}<n\le X}{c_n\ \!p_j(z_0\log n)\over n^{s_0-z_0}}\  =\cr
&\qquad\qquad\qquad\quad\ =\ \!\sum_{j=1}^J\bigg(p_j(z_0\log X)\ -\ \!z_0\int_{V^{1/10}}^Xp'_j(z_0\log Y)\ \!{dY\over Y}\bigg)\ +\cr
&\quad\ \ \!+\ \!{1\over2\pi i}\ \!\sum_{j=1}^J\ \!\int_{z_0-r-i X}^{z_0-r+iX}\Big(F_V(s_0-z_0 +z)-1\Big)\ \!\bigg(\big(X^z - V^{z/10}\big)\ \!p_j(z_0\log X)\ -\ \!z_0\ \!\cdot\cr
&\ \cdot\!\int_{V^{1/10}}^Xp'_j(z_0\log Y)\ \!\big(Y^z- V^{z/10}\big)\ \!{dY\over Y}\bigg)\ \!{dz\over z}\ +\ \Big(f(X) - f(V^{1/10})\Big)\sum_{j=1}^J\ \!p_j(z_0\log X)\ +\cr
&\qquad\ \!- z_0\int_{V^{1/10}}^X\Big(f(Y)- f(V^{1/10})\Big)\ \!\sum_{j=1}^J\ \!p'_j(z_0\log Y)\ {dY\over Y}\ =\ \ \!\sum_{j=1}^Jp_j\Big({z_0\log V\over10}\Big)\ +\cr
&+\ \!{1\over2\pi i}\ \!\sum_{j=1}^J\ \!\int_{z_0-r-i X}^{z_0-r+iX}\Big(F_V(s_0-z_0 +z)-1\Big)\ \!\bigg(p_j(z_0\log X)\ \!X^z- p_j\Big({z_0\log V\over10}\Big) V^{z/10}\ \!+\cr
&\quad\ \ - z_0\int_{V^{1/10}}^Xp'_j(z_0\log Y)\ \!Y^z\ \!{dY\over Y}\bigg)\ \!{dz\over z}\ \!+\ \!\Big(f(X) - f(V^{1/10})\Big)\sum_{j=1}^J\ \!p_j(z_0\log X)\ +\cr
&\qquad\qquad\qquad\quad - z_0\int_{V^{1/10}}^X\Big(f(Y)- f(V^{1/10})\Big)\ \!\sum_{j=1}^J\ \!p'_j(z_0\log Y)\ {dY\over Y}\cr
&\qquad\ \ \!where\qquad\ \ f(u)\ \!\ll\ \!V^{a-1}\quad,\quad J\ \ is\ as\ in\ \ (27)\quad\ ,\quad\ X \ge\max(V,3T)\ \ \!.}\leqno(32)$$

Taking $\ X=V^8,\ $ lemma 1 and (24) give the following estimates and relations
$$\eqalign{&\sum_{j=1}^Jp_j(z_0\log X)\ \!=\ \!{1\over V^{8z_0}}\!\!\sum_{j=1}^{4z_0\log V}{(8z_0\log V)^j\over j!}\ \!\le\ \!{4z_0\over e}\log V\Big({2\over e}\Big)^{\!4z_0\log V}\ll\ {1\over V^{6z_0/5}}\ ,\cr
&\qquad\ \ \bigg|\sum_{j=1}^Jp_j\Big({z_0\log V\over10}\Big) - 1\ \!\bigg|\ \!=\ \!{1\over V^{z_0/10}}+\ \!\sum_{j>J}p_j\Big({z_0\log V\over10}\Big)\ \!=\ \!{1\over V^{z_0/10}}\bigg(1 +\cr
&+\sum_{j>J}{1\over j!}\Big({z_0\log V\over10}\Big)^j\bigg)\ \!\le\ \!{1\over V^{z_0/10}}\bigg(1\ \!+\ \!{\big((z_0\log V)/10\big)^{J+1}\over(J+1)!}\sum_{h=0}^{+\infty}\Big({z_0\log V\over10(J+2)}\Big)^h\bigg)\le\cr
&\qquad\qquad\quad\ \ \le\ \!V^{-z_0/10}\bigg(1\ \!+\ \!\Big({e\over40}\Big)^{J+1}\ \!\sum_{h=0}^{+\infty}\Big({1\over40}\Big)^h\bigg)\ \!\ll\ \!V^{-z_0/10}\ ,\cr
&\ \!\sum_{j=1}^Jp'_j(z_0\log Y)\ \!=\ \!\sum_{j=1}^J\Big(p_{j-1}(z_0\log Y) - p_j(z_0\log Y)\Big)\ \!=\ \!Y^{-z_0} - p_J(z_0\log Y)\ ,\cr
&\qquad\qquad\qquad z_0\int_{V^{1/10}}^XY^{z-z_0}\ \!{dY\over Y}\ =\ {z_0\big(V^{(z-z_0)/10} - V^{8(z-z_0)}\big)\over z_0-z}\ ,\cr
&\quad\ z_0\int_{V^{1/10}}^Xp_J(z_0\log Y)\ Y^z\ \!{dY\over Y}\ =\ \!\Big({z_0\over z_0-z}\Big)^{J+1}\sum_{m=0}^J{(z_0-z)^m\over m!}\ \!\bigg(\Big({\log V\over10}\Big)^m\cdot\cr
&\qquad\qquad\qquad\qquad\quad\cdot V^{(z-z_0)/10} - \big(8\log V\big)^m\ \!V^{8(z-z_0)}\bigg)\ ,\cr
&\quad\ \!\sum_{m=0}^J{{z_0}^m\over m!}\bigg(\Big({\log V\over10}\Big)^m V^{-z_0/10} - \big(8\log V\big)^m\ \!V^{-8z_0}\bigg)\ =\ \!1\ \!+\ \!O\Big(V^{-z_0/10}\Big)\ .}\leqno(33)$$

Using (33), lemma 4 and the upper bounds (18), (19), (21), we then obtain
$${1\over2\pi i}\int_{z_0-r-iX}^{z_0-r+iX}\!\Big(F_V(s_0-z_0+z)-1\Big)\ \!V^{z\over10}\ {dz\over z}\ \ll_{\epsilon}\ \!V^{{z_0-r\over 10} -r+18\epsilon}\!\!\int_0^{V^8}\!{dy\over1+y}\ \!\ll_{\epsilon}\ \!V^{{z_0-11r\over 10}+19\epsilon}\ \!,$$
$$\eqalign{&{1\over2\pi i}\int_{z_0-r-iX}^{z_0-r+iX}\!\Big(F_V(s_0-z_0+z)-1\Big)\ V^{8z}\ {dz\over z}\ \!=\ \! F_V(s_0-z_0)-1\ \!+\ \!{1\over2\pi i}\Bigg(\int_{-z_0-iV^8}^{-z_0+iV^8}\!+\cr
&\ \ +\!\int_{-z_0+iV^8}^{z_0-r+iV^8}-\int_{-z_0-iV^8}^{z_0-r-iV^8}\Bigg)\ \!\Big(F_V(s_0-z_0+z)-1\Big)\ V^{8z}\ {dz\over z}\ \!=\ \! F_V(s_0-z_0)-1\ +\cr
&\quad\ \ \!+\ O_{\epsilon}\Bigg(V^{-(6z_0+r)+5\epsilon}\ \!T^{\epsilon}\bigg(\int_{-V^8}^{V^8}\Big(1+\big|\zeta\big(a-2z_0+r+i(v+y)\big)\big|^2\Big)\ \!{dy\over|y|+1}\bigg)^{1\over2} +\cr
&\quad\ \ +\ \! {V^{10z_0-9r+9\epsilon}\ \!T^{2z_0} +\ \!V^{8(z_0-r)+9\epsilon}\ \!T^{\epsilon}\over V^8}\Bigg)\ \!=\ \!F_V(s_0-z_0)\ \!-\ \!1\ \!+\ O_{\epsilon}\bigg({T^{1+3\epsilon\over2}\over V^{6z_0}}\bigg)}$$

so that, again by (33)
$$\eqalign{&{1\over2\pi i}\ \!\sum_{j=1}^J\ \!\int_{z_0-r-i X}^{z_0-r+iX}\Big(F_V(s_0-z_0 +z)-1\Big)\ \!\bigg(p_j(z_0\log X)\ \!X^z- p_j\Big({z_0\log V\over10}\Big) V^{z/10}\bigg)\ \!{dz\over z}\ \ll_{\epsilon}\cr
&\qquad\qquad\quad\ \!\ll_{\epsilon}\Big(F_V(s_0-z_0)\ \!-\ \!1\ \!+\ \!T^{1+3\epsilon\over2}\ \!V^{-6z_0}\Big)\ \!V^{-6z_0/5} +\ \!V^{{z_0-11r\over 10}+19\epsilon}\ .}\leqno(34)$$

Also, arguing as before
$$\ {z_0\over2\pi i}\int_{z_0-r-iX}^{z_0-r+iX}\!\Big(F_V(s_0-z_0+z)-1\Big)\ {V^{(z-z_0)/10} - V^{8(z-z_0)}\over z_0-z}\ {dz\over z}\ \ll_{\epsilon}\ \!V^{-{11r\over10}+18\epsilon}\ ,$$
$$\eqalign{&\qquad\quad\ \!{z_0\over2\pi i}\int_{z_0-r-iX}^{z_0-r+iX}\!\Big(F_V(s_0-z_0+z)-1\Big)\ {dz\over z}\!\int_{V^{1/10}}^Xp_J(z_0\log Y)\ Y^z\ \!{dY\over Y}\ =\cr
&\quad\ \ \ \!=\ \Big(F_V(s_0-z_0)-1\Big)\sum_{m=0}^J{{z_0}^m\over m!}\bigg(\Big({\log V\over10}\Big)^m V^{-z_0/10} - \big(8\log V\big)^m\ \!V^{-8z_0}\bigg)\ +\cr
&\qquad\ \ \!+\ \!\!O\Bigg(\bigg(\int_{-z_0-iV^8}^{-z_0+iV^8}+\!\int_{-z_0+iV^8}^{z_0-r+iV^8}+\int_{-z_0-iV^8}^{z_0-r-iV^8}\bigg)\ \!\Big(\big|F_V(s_0-z_0+z)\big|+1\Big)\ \!\cdot\cr
&\qquad\cdot\Big({z_0\over z_0-{\cal R}e z}\Big)^{J+1}\sum_{m=0}^J{(z_0-{\cal R}e z)^m\over m!}\ \!\bigg(\Big({\log V\over10}\Big)^mV^{({\cal R}e z-z_0)/10} +\ \!\big(8\log V\big)^m\ \!\cdot\cr
&\ \!\cdot V^{8({\cal R}e z-z_0)}\bigg)\ \!{|dz|\over|z|}\Bigg)\ \!=\ \!\Big(F_V(s_0-z_0)-1\Big)\bigg(1 + O\Big(V^{-{z_0/10}}\Big)\bigg) +\ \!O_{\epsilon}\bigg({V^{2z_0}\ \!T^{1+3\epsilon\over2}\over 2^J}\ \!+\cr
&+{{z_0}^{J+1}(V^9T)^{2z_0}\over r^{J+1} V^8}\bigg)\ \!=\ \!\Big(F_V(s_0-z_0)-1\Big)\bigg(1 + O\Big(V^{-{z_0/10}}\Big)\bigg) +\ \!O_{\epsilon}\Big(T^{1+3\epsilon\over2}\ \!V^{-{3z_0\over5}}\Big)\ .}$$

since $\ z_0\le 10r\ .\ $ Hence
$$\eqalign{&\quad\ \!{z_0\over2\pi i}\int_{z_0-r-i X}^{z_0-r+iX}\Big(F_V(s_0-z_0 +z)-1\Big)\ \!{dz\over z}\!\int_{V^{1/10}}^X\ \!\sum_{j=1}^Jp'_j(z_0\log Y)\ Y^z\ \!{dY\over Y}\ \!=\cr
&=\ \!\Big(1-\ \!F_V(s_0-z_0)\Big)\bigg(1 + O\Big(V^{-{z_0/10}}\Big)\bigg) +\ \!O_{\epsilon}\Big(T^{1+3\epsilon\over2}\ \!V^{-{3z_0\over5}}+\ \!V^{-{11r\over10}+18\epsilon}\Big)\ .}\leqno(35)$$

Finally, according to (32)
$$\eqalign{&\big(f(X) - f(V^{1/10})\big)\sum_{j=1}^J p_j(z_0\log X)\ \!\ll\ \! V^{a-1}\quad ,\quad z_0\!\int_{V^{1/10}}^X\sum_{j=1}^Jp'_j(z_0\log Y)\ \!\Big(f(Y)\ +\cr
&- f(V^{1/10})\Big)\ {dY\over Y}\ \!=\ \!\int_{{z_0\over10}\log V}^{8z_0\log V}\!\Big(f(e^{u/z_0})- f(V^{1/10})\Big)\ e^{-u}\Big(1-{u^J\over J!}\Big)\ du\ \ll\ \!V^{a-1}\ .}\leqno(36)$$

Inserting (34), (35), (36) in the identity (32) and using (33), we now obtain
$$\sum_{j=1}^J{{z_0}^j\over j!}\!\sum_{V^{1/10}<n\le V^8}{c_n(\log n)^j\over n^{s_0}}\ =\ F_V(s_0-z_0)\Big(1 + O\big(V^{-{z_0\over10}}\big)\Big)\ \!+\ \!O_{\epsilon}\Big(V^{{z_0-11r\over 10}+19\epsilon} +\ \!V^{-{z_0\over10}}\Big)\leqno(37)$$

since $\ \ \!V\ge T^{2/z_0}\ .$

On the other hand (28) yields, when $\ |z|\le 2z_0$
$$\eqalign{&\sum_{n\le V^{1/10}}{c_n\over n^{s_0+z}}\ \!-1\ \ll\   V^{a-1}\sum_{d\le V^{1/10}}\ \!{\log d\over d^{a+r-2z_0}}\!\sum_{\ell\le V^{1/10}/d}\ell^{2z_0-r -a}\ \!\ll\cr
&\qquad\qquad\quad\ll\ V^{9(a-1)+2z_0-r\over10}\sum_{d\le V^{1/10}}{\log d\over d}\ \ll\ V^{{9(a-1)\over10}+{z_0\over5}}}$$

whence, by Cauchy's inequality
$$\sum_{j=1}^J{{z_0}^j\over j!}\!\sum_{n\le V^{1/10}}{c_n(\log n)^j\over n^{s_0}}\ \ll\ V^{{9(a-1)\over10}+{z_0\over5}}\sum_{j=1}^J{1\over\ \!2^j}\ \!\le\ V^{{9(a-1)\over10}+{z_0\over5}}\ .\leqno(38)$$

Furthermore, by (26)\quad \big(with $\ \omega=2z_0\ \!,\ \ \!Z_1=V^8\ \!,\ \ \!Z_2\ge e^{10j/z_0}\big)$ 
$${1\over j!}\bigg|\sum_{V^8<n\le Z}{c_n(\log n)^j\over n^{s_0}}\ \!\bigg|\ \ll\ \!V^{2z_0+\epsilon}\ \!T^{{1\over2}+2z_0+\epsilon}\bigg({Z_2^{-z_0}\over(720z_0)^j}\ \!+\ \!{(8\log V)^j\over V^{16z_0} j!}\bigg)$$

while
$${T^{{1\over2}+2z_0+\epsilon}\over V^{14z_0-\epsilon}}\sum_{j=1}^J{z_0^j(8\log V)^j\over j!}\ \!\ll\ \!V^{-6z_0+\epsilon}\ T^{{1\over2}+2z_0+\epsilon}\ .$$

Thus
$$\sum_{j=1}^J{{z_0}^j\over j!}\sum_{n> V^8}{c_n(\log n)^j\over n^{s_0}}\ \ll_{\epsilon}\ \!V^{-6z_0+\epsilon}\ \!T^{{1\over2}+2z_0+\epsilon}\ .\leqno(39)$$

Fitting together the results in (37), (38), (39), we finally deduce
$$\sum_{j=1}^J{{z_0}^j\over j!}\sum_{n=1}^{\infty}{c_n(\log n)^j\over n^{s_0}}\ \! \!=\ \!F_V(s_0-z_0)\Big(1\ \!+\ \!O\big(V^{-{z_0\over10}}\big)\Big)\ \!+\ \!O_{\epsilon}\Big(V^{{z_0-11r\over 10}+19\epsilon} +\ \!V^{-{z_0\over10}}\Big)$$

since $\ \ V\ge T^{2/z_0}\ .\ $ This establishes lemma 7.

\qed

When $\ b=1/2\ $ the corresponding results are as follows.

\non

{\bf Lemma 8.}\ \ Let $\ \epsilon\ ,\ r\ ,\ a\ ,\ w\ ,\ T\ $ be as in (8) and let $\ F_V(s),\ c_n\ $ be defined by (10),

\qquad\qquad\quad\ (15) respectively. If 
$$s_0=w+r=1+r+iv\quad,\quad|\omega|\le1\quad,\quad r\le{\cal R}e\omega\le 1/2$$

then, for $\ Y\ge2$

$$\eqalign{\sum_{n\le Y}{c_n\over n^{s_0-\omega}}\ &\ll\ \!(V\sqrt T)^{{\cal R}e\omega}\bigg(\log^2(VT) + {\sqrt T\log^2\big(V(T+Y)\big)\over rY^r}\bigg)\ +\cr
&+\ \!Y^{{\cal R}e\omega}\bigg({(V\sqrt T)^{{\cal R}e\omega}\over Y^{1+r}} + {1\over Y}\bigg)\log^2\big(V(T+Y)\big)\ .}$$

Furthermore, if
$${\cal R}e z\ge{\cal R}e\omega - 1/2 -r\quad,\quad|s_0-\omega+z-1|\ge\log^{-1}\!T\ .$$

then
$$F_V(s_0-\omega+ z)\ \ll\ \!\big(V(T+|{\cal I}m z|)^{1\over2}\big)^{\max(0,\ \!{\cal R}e (\omega-z)-r)}\log^2\big(V(T+|{\cal I}m z|)\big)\ .$$

\non

{\bf Proof.}\ \ We start with the relations analogous to (16), (17), namely
$$\sum_{n\le Y}{c_n\over n^{s_0-\omega}}\ \!=\ \!{1\over2\pi i}\int_{{\cal R}e\omega-r +\log^{-1}\!V-iT_1}^{{\cal R}e\omega-r  +\log^{-1}\!V+iT_1}F_V(s_0-\omega+z)\ Y^z\ \!{dz\over z}\ \!+\ \!O_{\epsilon}\bigg({Y^{{\cal R}e\omega-r+\epsilon}\log V\over T_1}\bigg)\leqno(40)$$

where
$$2\le T_1\le Y\ \ ,\ \ |T_1-v|\ge v/2$$

and
$$\eqalign{&\qquad\ \!{1\over2\pi i}\int_{{\cal R}e\omega-r+\log^{-1}\!V-iT_1}^{{\cal R}e\omega-r+\log^{-1}\!V+iT_1}F_V(s_0-\omega+z)\ Y^z\ {dz\over z}\ =\cr
&=\ F_V(s_0-\omega)\ +\ {1\over2\pi i}\Bigg(\int_{-r-iT_1}^{-r+iT_1}+\int_{-r+iT_1}^{{\cal R}e\omega-r+\log^{-1}\!V+iT_1}+\cr
&\qquad\! -\int_{-r-iT_1}^{{\cal R}e\omega-r+\log^{-1}\!V-iT_1}\Bigg)\ \!F_V(s_0-\omega+z)\ Y^z\ {dz\over z}\ \ .}\leqno(41)$$

But now, when $\ -{1\over2}\le{\cal R}e(z-\omega)+r\le\log^{-1}\!V\ $ we have the crude upper bound 
$$M_V(s_0+s_V-1-\omega+z)\ =\ \!\sum_{n\le V}{\mu(n)\over n^{s_0+s_v-1-\omega+z}}\ \ll\ V^{\max(0,\ \!{\cal R}e(\omega-z)-r)}\ \!\log V\leqno(42)$$

while  \big(see (19) and [T]\ $ \S\ 5.14\big)$

$$\zeta\big(s_0-\omega+z\big)\ \ll\ (T+|{\cal I}m z|)^{\max(0,\ \!{{\cal R}e(\omega-z)-r)\over2})}\log(T+|{\cal I}m z|)\leqno(43)$$

for
$$-{1\over2}\le{\cal R}e(z-\omega)+r\le \log^{-1}\!V\quad,\quad|s_0-\omega+z-1|\ge\log^{-1}\!T\ \ \! .$$

The latter bound of the lemma now follows from (42), (43) if $\ {\cal R}e(z-\omega)+r\le \log^{-1}\!V\ $

and it is trivial for $\ {\cal R}e(z-\omega)+r\ge \log^{-1}\!V\ .\ $

Moreover \big(see (21)\big) 
$$\int_{-T_1}^{T_1}\big|\zeta\big(1-{\cal R}e\omega+i(v+y)\big)\big|^2\ \!{dy\over|y|+1}\ \ll\ T\log T\ \!+\ \!\log^2T_1\leqno(44)$$

and from (40), (41), (42), (43) it follows that \big(see (20), (22)\big)

$$\eqalign{&\sum_{n\le Y}{c_n\over n^{s_0-\omega}}\ \!\ll\ \!{V^{{\cal R}e\omega}\log\big(V(T+T_1)\big)\over Y^r}\bigg(\log T_1\!\!\int_{-T_1}^{T_1}\big|\zeta\big(1-{\cal R}e\omega+i(v+y)\big)\big|^2\ \!{dy\over|y|+1}\bigg)^{1\over2} +\cr
&\quad\!+\ \!\bigg({\big(V(T+T_1)^{1\over2}\big)^{{\cal R}e\omega}\over Y^r}\ \!+\ \!Y^{{\cal R}e\omega-r+\epsilon}\bigg)\ \!{\log^2\big(V(T+T_1)\big)\over T_1}\ \!+\ \!(VT)^{{\cal R}e\omega}\log^2(VT)\ \ll\cr
&\qquad\qquad\qquad\ \ \!\ll\ \!(V\sqrt T)^{{\cal R}e\omega}\bigg(\log^2(VT)\ \!+\ \!{\sqrt T\ \! \!\log^2\big(V(T+Y)\big)\over r\ \!\!Y^r}\bigg)\   +\cr
&\qquad\qquad\qquad\qquad\ +\ \!Y^{{\cal R}e\omega}\bigg({(V\sqrt T)^{{\cal R}e\omega}\over Y^{1+r}} + {1\over Y}\bigg)\log^2\big(V(T+Y)\big)}\leqno(45)$$

for a suitable $\ T_1\asymp Y\ .$ 

\qed

As we did for lemma 6, we now deduce

\non

{\bf Lemma 9.}\ \ Let $\ \epsilon\ ,\ r\ ,\ a\ ,\ w\ ,\ T\ $ be as in (8) and let$\ F_V(s)\ $be defined by (10). If 
$$r\ \!\le\ \!z_0\ \!\le\ \!\min\big(10\ \!r\ \!,\ \!1/10\big)$$

then, for$\ j\ge 1$
$${1\over j!}\ \!D^j F_V(w+r)\ \ll\ {(V\sqrt T)^{2z_0}\log^2(VT)\over(2z_0)^j}\quad .$$

\non

{\bf Proof.}\ \ Take $\ \omega=2z_0\ $ in lemma 8. Since $\ {\cal R}e\omega - 1/2 - r\le r< 0\ \!,\ $ we have

for $\ {\cal R}e z\ge 0\ ,\ |{\cal I}m z|\le 1$
$$F_V(s_0- 2z_0+z)\ \!\ll\ \! (V\sqrt T)^{2z_0 -z -r}\log^2(VT)\ \!\le\ \!(V\sqrt T)^{2z_0}\log^2(VT)$$

and Cauchy's inequality gives at once
$${1\over j!}\ \!D^j F_V(s_0)\ \ll\ {(V\sqrt T)^{2z_0}\log^2(VT)\over(2z_0)^j}\ \ .$$

\qed

To complete the treatment of $\ D^jF_V(s)\ \!,\ $ we suppose $\ b=1/2\ $ and $\ j\le J\ .$

\non

{\bf Lemma 10.}\ \ Let $\ \epsilon\ ,\ r\ ,\ a\ ,\ w\ ,\ T\ $ be as in (8) and let$\ F_V(s)\ $be defined by (10), with
$$V\ge\max\big(T^{1/z_0},\ \!\exp\big\{r^{-2}\big\}\big)\quad,\quad r\le z_0\le \min\big(10\ \!r\ \!,\ \!1/10\big)\ .$$

Then
$$\eqalign{&\sum_{1\le j\le 4z_0\log V}{(-z_0)^j\over j!}\ \!D^j F_V(w+r)\ =\ F_V(w+r-z_0)\bigg(1 + O\bigg(\exp\Big\{\!-{(\log V)^{1/2}\over4}\Big\}\bigg)\bigg)\ +\cr
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\!+\ O\bigg(\exp\Big\{\!-{r(\log V)^{1/2}\over5z_0}\Big\}\bigg)\ .}$$

\non

{\bf Proof.}\ \ Using (45) and proceeding as in (26) we first obtain
$$\eqalign{&{1\over j!}\sum_{Z_1<n\le Z_2}{c_n(\log n)^j\over n^{s_0}}\ \!\ll\ \!{V^{\omega}\over r}\ \!T^{1+\omega\over2}\bigg({Z_2^{-\omega/2}\over(360\ \!\omega)^j} + {(\log Z_1)^j\over\ j!\ \!Z_1^{\omega}}\bigg)\log^2(VT)\cr
&\quad s_0=  w+r\quad ,\quad r\le\omega\le 1/2\quad,\quad Z_1\ge e^j/\omega\quad,\quad Z_2\ge e^{20j}/\omega\ \ .}\leqno(46)$$

Secondly, the inequality analogous to (28) is
$$c_n\ \!\ll\ \!\exp\{-(\log V)^{1/2}\}\ \!\sum_{d|n}\log d\qquad\  when\qquad  1<n\le V\ .\leqno(47)$$ 

by lemma 3 and (15). Perron's formula and (47) then imply \big(see (30)\big) 
$$\eqalign{&\quad\!\sum_{n\le Y}{c_n\over n^{s_0-\omega}}\ \!=\ \!1\ \!+\ \!{1\over2\pi i}\int_{\omega-r +\log^{-1}\!Y-iX}^{\omega-r+\log^{-1}\!Y+iX}\Big(F_V(s_0-\omega+z) -1\Big)\ Y^z\ {dz\over z}\ +\cr
&\qquad\qquad\qquad+\ \!O_{\epsilon}\bigg(Y^{\omega-r+\epsilon}\Big({1\over X}+{\delta(Y)\over Y}\Big)\bigg)}$$

where
$$\eqalign{&\qquad\qquad\qquad\quad\ \ \omega\ \ as\ \ in\ \ (46)\cr
&\delta(Y)\ =\ \cases{\exp\big\{-(\log V)^{1/2}\big\}\quad\ if\quad\ \!2\le Y\le V\cr
1\qquad\qquad\qquad\qquad\quad\ \!if\qquad Y> V}\cr
&\quad\! 2\ \!\le\ \!Y\ \!\le\ \! X\quad,\quad X\ \!\ge\ \!\max\big(V,3T\!+\!|{\cal I}m\eta|\big)\cr
&\quad\!Y^{\omega-r+\epsilon}\Big({1\over X}+{\delta(Y)\over Y}\Big)\ \!\ll\ \!\exp\{-(\log V)^{1/2}\}}$$ 

and
$${1\over2\pi i}\int_{\omega-r +\log^{-1}\!X\pm iX}^{\omega-r+\log^{-1}\!Y\pm iX}\Big(F_V(s_0-\omega+z) - 1\Big)\ Y^z\ {dz\over z}\ \!\ll\ \!{\log^2X\over X}\ .$$

Hence, on taking $\ \omega=z_0\ $ \big(see (31)\big)
$$\eqalign{&\sum_{n\le Y}{c_n\over n^{s_0-z_0}}\ \!=\ \!{1\over2\pi i}\!\int_{z_0-r+\log^{-1}\!X-iX}^{z_0-r+\log^{-1}\!X+iX}\!\!\!(F_V(s_0-z_0+z)\ \!Y^z\ \!{dz\over z}\ +\cr
&\qquad\qquad\qquad\quad\ \ \!+\ O\Big(\exp\big\{-(\log V)^{1/2}\big\}\Big)\cr
&\qquad\qquad\qquad\quad\big(2\le Y\ \ ,\ \ \max(T,V,Y)\le X\big)\ .}\leqno(48)$$

Moreover
$$\eqalign{&\sum_{1<n\le V}{|c_n|\over n^{1+r}}\ \!\ll\ \!{1\over e^{\sqrt{\log V}}}\sum_{d\le V}{\log d\over d^{1+r}}\sum_{\ell\le V/d}{1\over \ell^{1+r}}\ \!\ll\ \!{V^{-r}\over r\ \!e^{\sqrt{\log V}}}\sum_{d\le V}{\log d\over d}\ \!\ll\ \!{\log^{5/2}V\over V^r\ \!e^{\sqrt{\log V}}}\ \!\ll\ \!{1\over V^r}\ ,\cr
&\quad\ \sum_{n> V}{|c_n|\over n^{1+r}}\ \!\le\ \!\sum_{n> V}{d(n)\over n^{1+r}}\ \!\le\ \!(1+r)\!\int_V^{+\infty}{\log u+O\big(1/u\big)\over u^{1+r}}\ \!du\ \!\ll\ \!{V^{-r}\over r}\ \!\log V\ \!\le\ \!{\log^{3/2}V\over V^r}\ }$$

since $\ V\ge\exp\big\{r^{-2}\}\ .\ $ Thus
$$\quad\ |F_V(1+r+iy)-1|\ \le\ \sum_{n>1}{|c_n|\over n^{1+r}}\ \ll\ V^{-r}\log^{3/2}V\qquad\quad\big(y\in\r\big)\ \ .\leqno (49)$$

We also recall that 
$$\eqalign{&\qquad\qquad\qquad\ \ J=[4z_0\log V]\cr
&where,\ now\qquad r\ \!\le\ \!z_0\ \!\le\ \!\min\big(10\ \!r\ \!,\ \!1/10\big)\ \ .\qquad\qquad}\leqno(50)$$

Putting $\ X = V^8\ ,\ \ \!U = \exp\big\{(\log V)^{1/2}/4z_0\big\}\ \!,\ $ we obtain from (48)\quad\big(see (32)\big)
$$\eqalign{&\sum_{j=1}^J{{z_0}^j\over j!}\!\sum_{U<n\le V^8}{c_n(\log n)^j\over n^{s_0}}\ \!=\ \!\sum_{j=1}^Jp_j(z_0\log U)\ \!+\ \! {1\over2\pi i}\int_{z_0-r+(8\log V)^{-1}-i V^8}^{z_0-r+(8\log V)^{-1}+iV^8}\!\!\Big(F_V(s_0-z_0 +z)\ +\cr
&\ - 1\Big)\bigg(\sum_{j=1}^Jp_j(8z_0\log V)\ \!V^{8z}-\sum_{j=1}^Jp_j(z_0\log U)\ \!U^z -\ \!z_0\!\int_U^{V^8}\sum_{j=1}^Jp'_j(z_0\log Y)\ Y^z\ \!{dY\over Y}\bigg)\ \!{dz\over z}\ +\cr
&\qquad\!+ \Big(f(V^8)- f(U)\Big)\sum_{j=1}^Jp_j(8z_0\log V)\ \!-\ \!z_0\!\int_U^{V^8}\Big(f(Y)- f(U)\Big)\sum_{j=1}^Jp'_j(z_0\log Y)\ {dY\over Y}\cr
&\qquad\ where\qquad\qquad f(u)\ \ll\ \exp\big\{-(\log V)^{1/2}\big\}\qquad\  and\qquad\quad J\ \ is\ \ as\ \ in\ \ (50)\ \ .}\leqno(51)$$

Furthermore \big(see (33)\big)
$$\eqalign{&\qquad\qquad\quad\ \Big|\sum_{j=1}^Jp_j\big(z_0\log U\big) - 1\ \!\Big|\ \le\ \exp\Big\{\!-{(\log V)^{1/2}\over4}\Big\}\bigg(1\ \!+\cr
&\ \!+ \Big({e\over16z_0(\log V)^{1/2}}\Big)^{J+1}\sum_{h=0}^{+\infty}\big(16z_0(\log V)^{1/2}\big)^{-h}\bigg)\ \!\ll\ \! \exp\Big\{\!-{(\log V)^{1/2}\over4}\Big\}\ ,\cr
&\quad\ \ z_0\!\int_U^{V^8}Y^{z-z_0}\ \!{dY\over Y}\ =\ {z_0\over z_0-z}\bigg(\!\exp\Big\{{z-z_0\over4z_0}(\log V)^{1/2}\Big\}\ \!-\ \!V^{8(z-z_0)}\bigg)\ ,\cr
&z_0\!\int_U^{V^8}p_J(z_o\log U)\ Y^z\ \!{dY\over Y}\ \!=\ \!\Big({z_0\over z_0-z}\Big)^{J+1}\sum_{m=0}^J{(z_0-z)^m\over m!}\bigg(\Big({(\log V)^{1/2}\over4z_0}\Big)^m\cdot\cr
&\qquad\qquad\quad\ \ \!\cdot\exp\Big\{{z-z_0\over4z_0}(\log V)^{1/2}\Big\}\ \!-\ \!\big(8\log V\big)^m V^{8(z-z_0)}\bigg)\ ,\cr
&\quad\sum_{m=0}^J{{z_0}^m\over m!}\bigg({\big(\log U\big)^m\over U^{z_0}} - {\big(8\log V\big)^m\over V^{8z_0}}\bigg)\ \!=\ 1 +\ \!O\bigg(\!\exp\Big\{\!-{(\log V)^{1/2}\over4}\Big\}\bigg)\ .}\leqno(52)$$

Hence, by (42), (43), (44), (49), (52)\quad \big(see (34),\ (35),\ (36)\big)

$$\eqalign{&\qquad\ \ \!{1\over2\pi i}\int_{z_0-r+(8\log V)^{-1}-iV^8}^{z_0-r+(8\log V)^{-1}+iV^8}\!\Big(F_V(s_0-z_0+z)-1\Big)\ \!U^z\ {dz\over z}\ \!\ll\ \!\bigg(\int_{z_0-r+(8\log V)^{-1}\pm iV^8}^{z_0\pm iV^8} +\cr
&\qquad+\int_{z_0-iV^8}^{z_0+iV^8}\bigg)\Big(F_V(s_0-z_0+z)-1\Big)\ \!U^z\ {dz\over z}\ \!\ll\ \!U^{z_0}\bigg({\log^2V\over V^8} + {\log^{3/2}V\over V^r}\!\!\int_0^{V^8}\!\!{dy\over y+r}\bigg)\ \!\ll\cr
&\qquad\qquad\qquad\quad\ll\ \!V^{-r}\exp\Big\{{(\log V)^{1/2}\over4}\Big\}\log^{5/2}V\ \!\ll\ \!V^{-r}\exp\Big\{{(\log V)^{1/2}\over3}\Big\}\ ,\cr
&{1\over2\pi i}\int_{z_0-r+(8\log V)^{-1}-iV^8}^{z_0-r+(8\log V)^{-1}+iV^8}\!\!\!\Big(F_V(s_0-z_0+z)-1\Big)\ \!V^{8z}\ \!{dz\over z}\ \!=\  \!F_V(s_0-z_0) -1 + O\bigg({\sqrt T\log^2V\over V^{6z_0}}\bigg)\ \!,\cr
&{1\over2\pi i}\sum_{j=1}^J\int_{z_0-r+(8\log V)^{-1}-iV^8}^{z_0-r+(8\log V)^{-1}+iV^8}\Big(F_V(s_0-z_0 +z) - 1\Big)\bigg(\sum_{j=1}^Jp_j(8z_0\log V)\ \!V^{8z}-\sum_{j=1}^Jp_j(z_0\log U)\ \!\cdot\cr
&\ \ \cdot U^z\bigg)\ \!{dz\over z}\ \!\ll\ \!\Big(\big|(F_V(s_0-z_0)-1\big|\ \!+\ \!T^{1\over2}\ \!V^{-6z_0}\log^2V\Big)\ \!V^{-6z_0/5} +\ \!V^{-r}\exp\Big\{{(\log V)^{1/2}\over3}\Big\}\ ,\cr
&\qquad\qquad{z_0\over2\pi i}\int_{z_0-r+(8\log V)^{-1}-iV^8}^{z_0-r+(8\log V)^{-1}+iV^8}\!\Big(F_V(s_0-z_0+z)-1\Big)\ {U^{z-z_0} - V^{8(z-z_0)}\over z_0-z}\ {dz\over z}\ \ll\cr
&\qquad\qquad\qquad\qquad\qquad\qquad\ \ll\ {\log^3V\over r}\exp\Big\{\!-{r(\log V)^{1/2}\over4z_0}\Big\}\ ,\cr
&\qquad\ \ {z_0\over2\pi i}\int_{z_0-r+(8\log V)^{-1}-iV^8}^{z_0-r+(8\log V)^{-1}+iV^8}\!\Big(F_V(s_0-z_0+z)-1\Big)\ {dz\over z}\!\int_{U}^{V^8}p_J(z_0\log Y)\ Y^z\ \!{dY\over Y}\ =\cr
&\qquad\quad\ =\ \Big(F_V(s_0-z_0)-1\Big)\ \!\bigg(1 +\ \!O\bigg(\!\exp\Big\{\!-{(\log V)^{1/2}\over4}\Big\}\bigg)\bigg)\ \!+\ \!O\bigg({T^{1\over2}\log^2V\over V^{3z_0/5}}\bigg)\ ,\cr
&\qquad\! {z_0\over2\pi i}\int_{z_0-r+(8\log V)^{-1}-iV^8}^{z_0-r+(8\log V)^{-1}+iV^8}\Big(F_V(s_0-z_0 +z)-1\Big)\ \!{dz\over z}\int_U^{V^8}\ \!\sum_{j=1}^Jp'_j(z_0\log Y)\ Y^z\ \!{dY\over Y}\ \!=\cr
&\qquad\quad\ \ =\ \Big(1- F_V(s_0-z_0)\Big)\bigg(1 +\ \!O\Big(\!\exp\Big\{\!-{(\log V)^{1/2}\over4}\Big\}\Big)\bigg)\ \!+\ \!O\bigg({T^{1\over2}\log^2V\over V^{3z_0/5}}\ +\cr
&\qquad\qquad\qquad\qquad\qquad\qquad+ {\log^3V\over r}\exp\Big\{\!-{r(\log V)^{1/2}\over4z_0}\Big\}\bigg)\ ,\cr
&\qquad\qquad\qquad\qquad\quad\sum_{j=1}^Jp_j(z_0\log U)\ \!=\ \!1 + O\Big(\exp\Big\{\!-{(\log V)^{1/2}\over4}\Big\}\Big)\ ,\cr
&\qquad\qquad\qquad\ \ \Big(f(V^8) - f(U)\Big)\sum_{j=1}^Jp_j(8z_0\log V)\ \ll\ \exp\big\{-(\log V)^{1/2}\big\}\ ,\cr
&\qquad\qquad\ z_0\!\int_U^{V^8}\Big(f(Y)- f(U)\Big)\sum_{j=1}^Jp'_j(z_0\log Y)\ {dY\over Y}\ \ll\ \exp\big\{-(\log V)^{1/2}\big\}\ .}\leqno(53)$$

When the estimates and the relations in (53) are substituted in the identity (51),

we get the asymptotic formula analogous to (37), namely 
$$\eqalign{\sum_{j=1}^J{{z_0}^j\over j!}\sum_{U<n\le V^8}{c_n(\log n)^j\over n^{s_0}}\ &=\ F_V(s_0-z_0)\ \!\bigg(1 +\ \!O\bigg(\!\exp\Big\{\!-{(\log V)^{1/2}\over4}\Big\}\bigg)\bigg)\ \!+\cr
&+\ O\bigg(\exp\Big\{\!-{r(\log V)^{1/2}\over5z_0}\Big\}\bigg)}\leqno(54)$$

where $\ \ U = \exp\big\{(\log V)^{1/2}/4z_0\big\}\ $ and
$$V\ge\max\big(T^{1/z_0},\ \exp\big\{r^{-2}\big\}\big)\ .\leqno(55)$$

Next, by (47) $\ \big(|z|\le 2z_0\big)$
$$\eqalign{&\sum_{n\le U}{c_n\over n^{s_0+z}} - 1\ \!\ll\ \!\exp\big\{\!-(\log V)^{1/2}\big\}\!\sum_{d\le U}\ \!{\log d\over d^{1+r-2z_0}}\!\sum_{\ell\le\ U/d}\ell^{2z_0-r-1}\ \!\ll\cr
&\quad\!\ll\ \!\exp\Big\{\!-{2z_0+r\over4z_0}(\log V)^{1/2}\Big\}\!\sum_{d\le U}\ \!{\log d\over d}\ \ll\ \!{1\over r^2}\exp\Big\{\!-{(\log V)^{1/2}\over2}\Big\}\ .}$$

Hence, by (55) and Cauchy's inequality\ \ \big(see (38)\big) 
$$\sum_{j=1}^J{{z_0}^j\over j!}\sum_{n\le U}{c_n(\log n)^j\over n^{s_0}}\ \ll\ \exp\Big\{\!-{(\log V)^{1/2}\over4}\Big\}\ \ .\leqno(56)$$

Moreover, the upperbound 
$$\sum_{j=1}^J{{z_0}^j\over j!}\sum_{n> V^8}{c_n(\log n)^j\over n^{s_0}}\ \ll\ {1\over r}\ \!V^{-6z_0}\ \!T^{{1\over2}+z_0}\log^2(VT)\leqno(57)$$

which corresponds to (39) can be similarly deduced on using (46), instead of (26), with

the same choice of $\ \omega,\ \!Z_1,\ \!Z_2\ .$ 

Collecting formulae (54), (56), (57), we finally have 
$$\eqalign{\sum_{j=1}^J{{z_0}^j\over j!}\sum_{n=1}^{+\infty}{c_n(\log n)^j\over n^{s_0}}\ &=\ F_V(s_0-z_0)\ \!\bigg(1 +\ \!O\bigg(\!\exp\Big\{\!-{(\log V)^{1/2}\over4}\Big\}\bigg)\bigg)\ \!+\cr
&+\ O\bigg(\exp\Big\{\!-{r(\log V)^{1/2}\over 5z_0}\Big\}\bigg)}$$ 

subject to (55). This proves lemma 10.

\qed

\centerline{\bf ****************}

\non

Let $\ b\ $ and $\ \rho_0 =\beta_0+i\gamma_0\ $ be as in $\ (1)\ \!(ii)\ $and (2), respectively. 

According to (3) put
$$\eqalign{&\qquad\quad\ T=2\gamma_0/3\quad,\quad r = \min\Big({1\over100}\ \!,\ { 1-2b\over27}\ \!,\ {2b\over23}\Big)\cr
&\epsilon\le r/60\quad,\quad a = b + (1+2r)/2\quad,\quad w = a + iv = a + i\gamma_0\cr
&\qquad\qquad\qquad\ z_0 = w+r-\rho_0 = a +r - \beta_0\ .}\leqno(58)$$

Then, by (2)
$$2r\le z_0\le 5r/2\le\min\big((2a-1)/10\ \!,\ \!(1-a)/5\big)\quad,\quad F_V(w+r-z_0) = F_V(\rho_0) = 0\ \ \!.\leqno(59)$$

Lemmas 4, 6, 7 and (59) imply, if $\ V\ge T^{1/r}$

$$\eqalign{&F_V(s_0) = 1 + O_{\epsilon}\Big(V^{2(\epsilon-r)}\ \!T^{2\epsilon}\Big) = 1 +  O_{\epsilon}\Big(V^{-2r+3\epsilon}\Big)\ \!,\cr
&\quad\ \!\sum_{1\le j\le4z_0\log V}{(\rho_0-s_0)^j\over j!}\ D^jF_V(s_0)\ \ll\ V^{-r/5}}$$

and
$$\eqalign{&\quad\sum_{j>4z_0\log V}{(\rho_0-s_0)^j\over j!}\ D^jF_V(s_0)\ \!\ll\ \!(VT)^{2z_0}\sum_{j>4z_0\log V}2^{-j}\ll\cr
&\qquad\qquad\ \!\ll\ V^{z_0(3+2r-4\log2)}\ \!\ll\ \!V^{-7z_0/10}\ \!\ll\ \!V^{-7r/5}\ .}$$

Hence, by (59)

$$\eqalign{0=\big|F_V(\rho_0)\big|\ \!&=\ \!\bigg|\sum_{j=0}^{+\infty}{(\rho_0-s_0)^j\over j!}\ D^jF_V(s_0)\bigg|\ \!=\ \!\Big|F_V(s_0)+ O\Big(V^{-r/5}\Big)\Big|\ =\cr
&=\ \!\Big|1+O\Big(V^{-r/5}\Big)\Big|\ \!>\ \! {1\over2}}$$

if $\ V\ $ is great.

By (1) we then have either $\ b=0\ $ or $\  b= 1/2\ .$
 
Suppose now $\ b= 1/2\ $ and let $\ \rho_0 =\beta_0+i\gamma_0\ $ be as in (2). 

According to (8), we have $\ a = 1\ .\ $ Put further $\ r = 1/100\ $ and take
$$ T\ ,\ \epsilon\ ,\ v\ ,\ w\ ,\ z_0$$

as in (58). By (2)
$$ r< z_0\le 3r/2<1/10\quad,\quad F_V(w+r-z_0) = F_V(\rho_0) = 0$$

and lemmas 4, 9, 10 imply, when $\ V\ge\exp\big\{\max\big(r^{-2},\ \!\log^2(2T)\big)\big\}$ 
$$\eqalign{&\qquad\qquad\qquad\quad\! F_V(s_0) = 1 + O\Big(\exp\Big\{-(\log V)^{1/4}\Big\}\Big)\ ,\cr
&\qquad\ \sum_{1\le j\le4z_0\log V}{(\rho_0-s_0)^j\over j!}\ D^jF_V(s_0)\ \ll\ \exp\Big\{\!-{2(\log V)^{1/2}\over15}\Big\}\ ,\cr
&\sum_{j>4z_0\log V}{(\rho_0-s_0)^j\over j!}\ D^jF_V(s_0)\ \!\ll\ \!(V\sqrt T)^{2z_0}\log^2(VT)\sum_{j>4z_0\log V}2^{-j}\ll\cr
&\qquad\qquad\ll\ \!V^{z_0\left(2-4\log2+1/\sqrt{\log V}\right)}\log^2V\ \!\ll\ \!V^{-7/10^4}\log^2V\ .}$$

Then, as before
$$0\ \!=\ \!\big|F_V(\rho_0)\big|\ =\ \Big|1 + O\Big(\exp\Big\{\!-(\log V)^{1/4}\Big\}\Big)\Big|\ \!>\ \!{1\over2}$$

if $\ V\ $ is sufficiently large.

Hence $\ b = 0\ $ and the proof of the theorem is now complete.

\qed

\non

\centerline{\bf References}

\non

\item{[B]} E. Bombieri, {\it Le Grand Crible dans la Th\'eorie Analytique des Nombres} (seconde edition revue et augment\'ee), Ast\'erisque 18, 1987/1974

\item{[T]} E.C. Titchmarsh, {\it The theory of the Riemann zeta--function} (second edition revised by D.R. Heath--Brown), Clarendon Press, Oxford, 1986











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