The bound
$$
\max(I;1/I)\le
\exp\biggl(\frac{(\deg F+\deg G)(\beta/P)^JP}{(1-\beta/P)J}\biggr)
$$
should be
$$
\max(I;1/I)\le
\exp\biggl(\frac{2(\deg F+\deg G)(\beta/P)^JP}{(1-\beta/P)J}\biggr)
$$
P. 95, line 9.
In the expression
$
|b_F(j) - b_G(j)| \le (\deg F + \deg G) \beta^j / j,
$
the right-hand side should be
$
2 (\deg F + \deg G) \beta^j / j,
$
because there is a factor $2$ in equation (9.7).
P. 95, line after equation (9.15).
In the long expression,
$|t \beta|^J$ should be
$|z \beta|^J$.
P. 96, Exercise 9-8, part 2.
$k \ge 2$ should
be
$k \ge 3$.
P. 96, equation (21.3) and following.
The product given here is mis-identified as
$\mathscr{C}_1$ but should be $\mathscr{C}_2$.
The definition of $\mathscr{C}_1$ is in equation (0.3) on P. vii.
The definition of $\mathscr{C}_2$ is in equation (21.3) on P. 221.
Also, ``Recall that'' is incorrect here because
$\mathscr{C}_2$ has not been defined yet.
P. 105, Example 1.
$\displaystyle \prod_{p\ge 2}\biggl(1-\frac{1}{(p-1)p^{s+1}}-\frac{1}{(p-1)p^{s+2}}\biggr)$
should be
$\displaystyle \prod_{p\ge 2}\biggl(1+\frac{1}{(p-1)p^{s+1}}-\frac{1}{(p-1)p^{s+2}}\biggr)$
.
P. 107, Example 1.
$\displaystyle g_2(p)=g_3(p)=-\frac1{p(p-1)}$
should be
$\displaystyle g_2(p)=-g_3(p)=\frac1{p(p-1)}$
.
P. 111, line 4.
The reference should be
Lemma 10.1
and not to
Lemma 10.2
.
P. 111, Exercise 10-4, part 1.
``$E_n$ stands for the set of integers whose prime factors divide $n$''
should be
``$E_n$ stands for the set of multiples of $n$ whose prime factors
divide $n$''
.
P. 114, line 4.
The reference [4] should point to
volume 22
and not to
volume 21
.
P. 115, line 6 from bottom.
(7.4)
should be (7.5).
P. 117, last two lines.
At the end of each
line, $O(1)$
should be $O(\sqrt{x})$.
P. 119, Exercise 11-6.
The last line should end with period (.).
P. 122, line 7 from bottom.
Lemma 20.5
should be
Lemma 20.4.
P. 124, script.
The script is unbalanced in parentheses and should end with x)));}
P. 124, Exercise 12-2.
$0 < y < 3$
should be
$1 < y < 3$.
P. 126, Exercise 12-6.
The exercise ends with ``Conclude''. This is better
replaced by "Prove that $p_n\ge \tfrac43 n\log n$".
P. 128, line 5.
In the display,
$7/10$ should be
$7/5$.
P. 128, script.
This is the wrong script for this proof; it is the same script as on
P. 134. It should be the next one.
{Lambda(d) = my(dec = factor(d), P = dec[,1]);
if(#P != 1, return(0), return(log(P[1])));}
{check( upperlimit ) =
my(res = -2/3, mymax = 0, where = 1, aux);
for( n = 2, upperlimit,
res += Lambda(n)/n;
aux = max(abs(res - log(n)) , abs(res - log(n+1)));
if( aux > mymax, mymax = aux; where = n,));
print("When x <= ", upperlimit, " we have ");
print("|\sum_{n<=x}Lambda(n)/n - (log x-2/3)| <= ", mymax);
print("The maximum is reached around x = ", where);}
P. 129, Exercise 12-11, next to last line.
$$
M(f) \le \left( D(f) \sum_{p \le N} \frac{1}{p} \right)^{1/2}
\text{ should be }
|M(f)| \le \left( D(f) \sum_{p \le N} \frac{1}{p} \right)^{1/2}.
$$
(This stronger inequality is true, and is needed to assert the lower bound included in the
$o()$ on the last line.)
P. 131, line 2 from bottom.
$\displaystyle \sum_{k\ge2}\sum_{p^k\le x}\frac{1}{kp^k}
=
\sum_{k\ge2}\sum_{p\le x}\frac{1}{kp^k}
-\sum_{k\ge2}\sum_{p^k> x}\frac{1}{kp^k}$
should be
$\displaystyle \sum_{k\ge2}\sum_{p^k\le x}\frac{1}{kp^k}
=
\sum_{k\ge2}\sum_{p\ge 2}\frac{1}{kp^k}
-\sum_{k\ge2}\sum_{p^k> x}\frac{1}{kp^k}$
(the second summation over $p$ should not be restricted to
$p\le x$).
P. 134, Further Reading.
treaty should be
treatise.
P. 141, line 6.
``non especially non-negative''
should be
``in particular, non-negative''
.
P. 142, line 3.
``for every primes''
should be
``for every prime''
.
Proof of Theorem 13.3, P. 143, line 5.
$\displaystyle G_p(X) = G(X) - \sum_{k \ge 1} G_p(D/p^k)$
should be
$\displaystyle G_p(X) = G(X) - \sum_{k \ge 1} g(p^k) G_p(D/p^k)$
.
P. 144, line 2.
The inequality
$\displaystyle 1/(1+x)\le 1+2x$ when $0\le x\le 1/2$
is not enough. What we need is
$\displaystyle 1/(1+x)= 1+\mathcal{O}^*(2x)$ when $0\le x\le 1/2$
.
P. 144, line 2 from bottom.
$\displaystyle \sqrt{2}(\log x + O^*(7/6))$ when $x \ge 2)$
should be
$\displaystyle \sqrt{2}(\log Q + O^*(7/6))$ when $Q \ge 2)$
(the variable name is $Q$ and not $x$).
P. 147, last line of Theorem 13.4 statement.
``where $C$ is as in Theorem 13.3.'' should be
``where $C$ is as in Theorem 13.3, taking $g(d) = f(d)/d$.''
P. 148, lines 6--7.
$Q$
should be
$D$ throughout.
P. 149.
Reference [6] indicates
volume 43
but it should be
volume 18.
P. 152, proof of Lemma 14.3, line 3 from bottom.
``second sum''
should be
``second product''.
P. 152, proof of Lemma 14.3, line 2 from bottom.
``recurrence hypotheses''
should be
``inductive hypothesis''.
P. 155, question 4 of Exercise 14-6.
``By combining Exer. 12-4''
should be
``By combining
Exer. 12-3'',
refering to the correct exercise.
P. 161, line 5 from bottom.
``Sect. 16''
should be
``Chap.
16''.
P. 162, line 7 from bottom.
``taking $\beta = \pi \alpha$''
should be
``taking $\beta = 2 \pi \alpha$''.
P. 167,
Exercise 15-14.
The notation for the sums and products is garbled in this
exercise. Here is the correct complete exercise.
Exercise 15-14.
For any real parameter $x$ larger than 2, we set
$\displaystyle
S(x) = \prod_{p\le x}\biggl(1-\frac{\chi_4(p)}{p}\biggr)^{-1},
\quad
T(x) = \sum_{n\le x}\frac{\chi_4(n)\Lambda(n)}{n\log n}
$
with the aim of showing that $\displaystyle L(1,\chi_4)=\lim_{x\rightarrow\infty}S(x)$.
1 ◇ By using the previous exercise and the remark following
it, show that $S(x)$ tends to some limit $\ell$ when $x$ goes to infinity.
2 ◇ Show that
$ T(x)-\log S(x) \ll x^{-1/2} $
and conclude that $S(x)$ also tends to $\ell$.
3 ◇ Show that, for any $\delta>0$, we have
$\displaystyle
\delta\int_1^\infty T(x)\frac{dx}{x^{1+\delta}}=
\log L(1+\delta,\chi_4).$
Conclude.
P. 168, line 8, Epilogue.
Theorem 12-2
should be
Theorem 15-2.
P. 168, line 11, Epilogue.
``who was barely twenty years old''
should be
``who was barely thirty years old''.
Furthermore
de la Vallée-Poussin
should be
de la Vallée Poussin, without hyphen.
P. 173, line 13.
In the displayed formula,
$\chi$
should be (twice)
$\chi_4$.
P. 176, Reference [3].
Sequence
should be
Sequences.
P. 176, Reference [8].
$n^2 a$
should be
$n^2+a$.
P. 176, Reference [9].
$n^2 1$
should be
$n^2 +1$.
P. 180, line 9.
``$\chi^k$ is multiplicative''
is not enough and should be
``$\chi^k$ is
completely multiplicative''.
P. 181, line 2.
Theorem 17.2
should be
Theorem 17.1.
P. 181, line 2.
``they vanish when $j > j_0k$''
should be
``they vanish when $j < j_0k$''.
P. 183, line 10.
In question 1 of Exercise 17-7,
$\displaystyle \ell(s)=\sum_{p\equiv 3[4]}\log\frac{1+p^{-s}}{1-p^{s}}$
should be
$\displaystyle \ell(s)=\sum_{p\equiv 3[4]}\log\frac{1+p^{-s}}{1-p^{-s}}$
.
P. 185, line 2.
``infinitely chains''
should be
``infinitely many chains''.
P. 185, line 5.
``lign''
should be
``row''.
P. 185, line 5.
``wave''
should be
``waive''.
P. 185, Reference [3].
``de composition''
should be
``décomposition''.
P. 185, Reference [9].
23 ()
should be
23 (1925).
P. 186, Reference [15].
``erratum 5 (1958)''
should be
``erratum 5 (1959)''.
P. 187, line 8.
``divisors of 3''
should be
``multiples of 3''.
P. 187, line 9.
``10 possibilities''
should be
``22 possibilities''.
P. 187, line 8.
should be
.
P. 190, line 4 from bottom.
In the displayed sum, the conditions on $d$ should be
$d \le \sqrt{n^2 + 1}, d
| (n^2+1)$
(add condition $d | (n^2+1)$)).
P. 191, line 10 from bottom.
developped
should be
developed.
P. 204, line 6 from bottom.
``no residue of the summand''
should be
``no residue of the integrand''.
P. 205, 207 and Index.
Cahen-Millen
should be Cahen-Mellin.
P. 210, line 13.
2 arctan$(T/\kappa)$
should
be 2i arctan$(T/\kappa)$,
so the sentence should be
``the first integral equals $2i\text{arctan}(T/\kappa)$, and the
absolute value of this is $\le \pi$, while ...''
P. 218, lines 3 and 5.
Lemma 19.1
should be Proposition 19.1.
P. 218, line 12.
Lemma 20.4
should be Lemma 20.5.
P. 218, Step 5.
The sentence
``In our region,
we have $\Gamma(\sigma + i t) \ll (1 + |t|)^{5/8} \exp - (\pi
|t|/2)$''
does not belong here: the region is $\sigma\ge 7/4$ which is not
introduced before Step 6. Please, simply ignore this
sentence.
P. 218, lineS 3 and 4 from bottom.
The integration bound
$2+iT$
should (twice)
be $3+iT$
and similarly the integration bound
$2-iT$
should (twice)
be $3-iT$.
P. 219, line 1.
For the first integral we should move the first absolute value
and write
$\displaystyle
\biggl| \int_{3 + iT}^{7/4 + i T} D(f_0,s) \Gamma(s) x^s \, ds \biggr|
$
P. 219, lines 1 and 4.
The integral sign
$\displaystyle\int^{3}_{3/4}$
should be
$\displaystyle\int^{2}_{3/4}$.
A similar confusion occurs
line 4 where the
integral sign
$\displaystyle\int^{2-iT}_{7/4-iT}$
should be
$\displaystyle\int^{3-iT}_{7/4-iT}$.
P. 219, line 3.
Lemma 19.1
should be Proposition 19.1.
P. 219, line 3.
Lemma 19.1
should be Proposition 19.1.
P. 219, line 8 from bottom.
The sentence
``on majorizing $(4
+ t^2)^{5/8}$ by $(2t)^{5/8}$ when $t \ge 2$'' is
incorrect. It should be
``on majorizing $(4
+ t^2)^{5/8}$ by $(2t^2)^{5/8}$ when $t \ge 2$''
.
This has consequences on the subsequent inequalities. This proof can
be improved in numerous way, but here is one that still leads to the
Claim of Step 7 without too many changes: simply improve the bound
$\int_0^2|\Gamma(7/4+it)|dt\le 2\Gamma(7/4)$ by using numerical
integration.
The Pari/GP script
$\texttt{intnum(t=0,2,abs(gamma(7/4+I*t)))}$
tells us that this integral is bounded above by $1.265$.
``We thus find that, on majorizing $(4+t^2)^{5/8}$ by $(2t^2)^{5/8}$ when $t\ge 2$,
$$
\left|\int_{7/4-i\infty}^{7/4+i\infty}
D(f_0,s)\Gamma(s) x^sds\right|
\le 2x^{7/4}\biggl(
203\cdot 1.265
+160\cdot\sqrt{2\pi}\cdot e^{1/12}\cdot 2^{5/8}
\int_2^\infty t^{3/2}\log t \, e^{-\pi t/2}dt
\biggr)\;.
$$
We have now gone far enough in the analysis of the problem to
finish by numerical integration:
$\texttt{aux = intnum( t = 2, [[+1], Pi/2],
t^(3/2)*log(t)*exp(-Pi/2*t));}$
followed by
$\texttt{2*(257 + 673 * aux)}$.
''
This would enable us to improve the inequality of Step 7 in
$$
\frac1{2\pi}\left|\int_{7/4-i\infty}^{7/4+i\infty}
D(f_0,s)\Gamma(s) x^sds\right|
\le 680\cdot x^{7/4}\;.
$$
P. 220, Exercise 21-1.
$\pi^2x^2/12$
should be
$6x^2/\pi^2$.
P. 223, Eq. (22.1).
$\mathscr{C}_1$
should be
$\mathscr{C}_2$.
P. 223, line 2 from bottom.
The second factor
$\displaystyle\left(1-\frac{n}{x+L}\right)$
should be
$\displaystyle\left(1-\frac{n}{x}\right)$.
P. 224, line 1.
$\mathscr{C}_1$
should be
$\mathscr{C}_2$.
P. 225, line 1.
$\displaystyle\limsup_{n\rightarrow\infty}$
should be
$\displaystyle\limsup_{N\rightarrow\infty}$
.
P. 226, line 8 from bottom.
The second integral,
$\displaystyle\int_0^1|\log t|dt$
should be
$\displaystyle\int_0^1(\log t)^2dt$
.
P. 228, line 2 from bottom, P. 229, line 9
and P. 230, line 12.
$\mathscr{C}_1$
should be
$\mathscr{C}_2$.
P. 234, line 14 from bottom.
$ |\zeta(\sigma)|^3 |\zeta(\sigma + i t)|^4 |\zeta(\sigma + 2 i t)|
\ge 0 $
should be
$ |\zeta(\sigma)|^3 |\zeta(\sigma + i t)|^4 |\zeta(\sigma + 2 i t)|
\ge 1 $
(lower bound is $1$, not $0$).
P. 235, Exercise 23-1.
The initial sentence
``Let $D(t) = \sum_{n \ge 1} a_n n^{i t}$ and $D^*(t) = \sum_{n \ge 1} a^*_n n^{i t}$
be two Dirichlet series, both absolutely convergent for $\Re s \ge
0$''
is confusing and, strictly speaking,
wrong. It can be ``Let $D(s) = \sum_{n \ge 1} a_n n^{-s}$ and $D^*(s) = \sum_{n \ge 1} a^*_n n^{-s}$
be two Dirichlet series, both absolutely convergent for $\Re s \ge
0$'', but later, one should replace $D(t)$ by $D(it)$ and
$D^*(2t)$ by $D^*(2it)$.
Another way to fix this is to simply say ``Let $D(t) = \sum_{n \ge 1} a_n n^{i t}\) and \(D^*(t) = \sum_{n \ge 1} a^*_n n^{i t}$
be two series, both absolutely convergent for all real $t$.''
P. 235, line 10.
$[s-1|$
should be
$|s-1|$.
P. 235, line 14.
The proof is confusing at this point. It is better to add:
``We have proved that
$|1/\zeta(\sigma+it)|\le 20(\sigma-1)$ when $|s-1|\le 3/5$ and
$\sigma\ge 3/4$. This implies the claimed inequality when $|s-1|\le 1/2.$''
P. 236, line 3 from bottom.
``We now take $x=y=1/5^9$''
should be
``We now take $x=y=1/6^9$''
.
P. 238, line 14.
$\displaystyle\int_0^T\frac{2dt}{1+t}\le 2\log(1+T)$
should be
$\displaystyle\int_0^T\frac{2dt}{1+t}= 2\log(1+T)$
.
P. 238, line 9.
The statement
``As for the first part, we use Lemma 7.1''
is incorrect and comes from a confusion of the true content of Lemma 7.1.
It is better to write:
On invoking Cor. 23.1 and Lemma 7.1, we see that the second sum is bounded is absolute value by
$$
\mathcal{O}\biggl(\frac{x}{(\log x)^A}\biggr)
\sum_{m\le \sqrt{x}}\frac{\log m}m
\ll_A \frac{x}{(\log x)^{A-2}}\;.
$$
As for the first part, we proceed as in Lemma 7.1 to prove that
$\displaystyle\sum_{m\le M}\log M=M\log M-M+\mathcal{O}(\log(2M))$.
This leads to
$$
\sum_{\ell \le \sqrt{x}}\mu(\ell)\sum_{m\le x/\ell}\log m
=
x\sum_{\ell \le \sqrt{x}}\frac{\mu(\ell)}{\ell}
\Bigl(\log\frac{x}{\ell}-1\Bigr)
+\mathcal{O}(\sqrt{x}\log x)\;.
$$
.
P. 240, line 9.
``value fo $F'(1)$''
should be
``value of $F'(1)$''
.
P. 240, Exercise 23-9.
$\mathscr{C}_1$
should be
$\sqrt{2}\mathscr{C}_1$.
.
P. 243, line 6.
``due to the K.~Iseki'' should be``due to K.~Iseki''.
P. 244, line 4.
``$F_2(x)=x-\gamma$''
should be
``$F_2(x)=x-\gamma-1$''
.
P. 245, proof of Proposition 24.1.
Several $\pm$ signs are not what they should be and some more details
are called for. Here is a corrected
version:
``We use Theorem 6.2 with $h(t)=(-\log t)/t$, $f(n)=\mu(n)$ and $g(n)=1$. On recalling Lemma 7.1
as well
as Eq. (7.1), we find that
$\displaystyle
\sum_{n\le t}\frac{\log(t/n)}{n}
=\frac12\log^2t+\gamma\log t-\gamma_1+\mathcal{O}\Bigl(\frac{\log(2t)}{t}\Bigr)\;.
$
This leads to our choice of $H$, as the main term above should be $H'(t)$. Since
$$
(at\log^2t+bt\log t+ct)'=
a\log^2t+(2a+b)\log t+b+c
$$
we select $H(t)=\tfrac12t\log^2t-(\gamma+1)t\log
t+(\gamma_1+\gamma+1)t$. Theorem 6.2 with these choices gives us
\begin{multline*}
\sum_{n\le x}\frac{\mu(n)}{n}\Bigl(
\frac12\log^2\frac xn-(\gamma+1)\log \frac{x}n+(\gamma_1+\gamma+1)
\Bigr)
=(\gamma_1+\gamma+1)\frac{M(x)}{x}
+\log x
\\-1+\frac1x
+\frac1x\int_1^x M(x/t)
\biggl(
\frac12\log^2t+\gamma\log t-\gamma_1-\sum_{n\le t}\frac{\log(t/n)}{n}\biggr)
dt\;.
\end{multline*}
We then plug in the estimates $|M(t)|\le t$, $|\sum_{n\le x}\mu(n)/n|\le1$ as well as the von Mangoldt estimate
$$
\sum_{n\le x}\frac{\mu(n)}{n}\log \frac{x}n\ll1
$$
to conclude the proof.''
.
P. 251, line 7.
sticked
should be stuck.
P. 251, line 7 from bottom.
``to every
integers'' should be
``to every integer''.
P. 251, last displayed formula.
$\zeta(2 -
a)$
should be $C \zeta(2 - a)$.
P. 252, line 4.
Exr. 3-4
should be Exer. 3-6.
P. 252, Exercise25-2 part 1.
any nonnegative $\sigma$
should be any
positive $\sigma$,
as $\sigma=0$ causes a division by 0.
P. 255, line 13.
$\sum_{\genfrac{}{}{0 pt}{}{d \le D}{d | P(z)}} \mu(d)
$
should be
$\sum_{\genfrac{}{}{0 pt}{}{d \le D}{d | Q}} \mu(d)
$
.
P. 258, line 1.
``The sieve problematic''
should be
``The sieve method''
.
P. 258, line 1.
``The sieve problematic''
should be
``The sieve method''
.
P. 258, reference [5].
This reference is to a book and should be formatted as a book.
P. 260, line 3.
$(\rho - \rho^{-1})\rho^{-2k}$
should be
$(\rho + \rho^{-1})\rho^{-2k}$
, (wrong sign).
P. 260, line 8 from bottom.
$N_k \ge (1 + \sqrt{2})^k / 2$
should be
$N_k \le (1 + \sqrt{2})^k / 2$
.
P. 260, line 2 from bottom.
``coprime positive integer''
should be
``coprime positive integers''
.
P. 261, line 5.
``the points $h \alpha$ are distant from one another by at least $1/(2q)$''
should be
``the points $h \alpha$ modulo 1 are distant from one another by at least $1/(2q)$''
.
P. 262, line 9.
``loose a logarithmic factor''
should be
``lose a logarithmic factor''
.
P. 262, line 9.
``loose a logarithmic factor''
should be
``lose a logarithmic factor''
.
P. 264, line 7 from bottom.
``forcibly larger than or equal to $z$''
should be
``larger than or equal to $z$''
.
PP. 264--265.
The symbol
$Q$
should be
$P(z)$
, twice.
P. 265, proof of Theorem 26.3.
The function $g$ to use in the application of Theorem 26.2
is not stated; it should be $g(n) = e(\rho m)$.
P. 266, line 13.
$Q\le X/n$
should be
$Q\le x/n$
, (lower case $x$).
P. 266, line before last displayed equation.
``On gathering our estimates, we have reached''
should be
``On selecting $z=\exp\frac12\sqrt{\log x}$ and on gathering our estimates, we have reached''
.
P. 268, line 6.
``if only finitely primes $p$ where such that $\{\rho p\} \notin [a,b]$''
should be
``if there are only finitely primes $p$ such that $\{\rho p\} \notin [a,b]$''
.
P. 273, lines 1--2.
``the sequence $(\cos 2 \pi \rho n)$ where $n$ ranges the irregular integers dense in $[-1, 1]$''
should be
``the sequence $(\cos 2 \pi \rho n)_n$ where $n$ ranges the
irregular integers is dense in $[-1, 1]$''
.
P. 273, reference [3].
''Distjointness''
should be
''Disjointness''.
P. 274, reference [19].
This reference is to a book and should be formatted as a book.
P. 275, line 11.
''E. Cohen''
should be
''H. Cohen''.
P. 276, line 4.
$x \ge 2 \cdot 10^6$
should be
$x \le 2 \cdot 10^6$.
PP. 275--277, proof of Theorem 27-1.
The proof claims that a verification up to $2\cdot 10^6$ is
enough, but we need this bound to be $\ge (3500)^2$ and $2\cdot
10^6$ is not enough, it should be $13\cdot 10^6$. The same
simplistic script proposed works well, but may take some
minutes, about three and a half on my laptop.
The computations are speeded in the next script.
{Checkbis(upperlimit) =
my(somme=1.0, bound = 4/log(upperlimit+1));
forfactored(K = 2, upperlimit,
somme += moebius(K)/eulerphi(K);
if(abs(somme) > bound, print("Problem at ", K[1])))};
This script uses forfactored and avoids computing
several logarithms, but still takes more than two minutes. It verifies the claimed property for $k \ge
14$ and the first script concludes immediately.
P. 278, line 4.
$\frac{2}{25} + \frac{27 \log x}{x^{1/3}} \le 4$
should be
$\frac{2\cdot 3}{25} + \frac{27 \log x}{x^{1/3}} \le 4$,
(add factor for $D(|h_6|,0)$).
P. 279, line 5.
``the large sieve. and stems from''
should be
``the large sieve, and stems from''
,
(replace period with comma).
P. 281, Exercise 28-3.
$\sum_j|f(i,j)|$
should be
$\frac12(\sum_j|f(i,j)|+\sum_j|f(j,i)|)$
.
The initial inequality is correct if we add the additional
hypothesis $f(i,j)=f(j,i)$, which is satisfied in practice.
P. 285, Exercise 28-4.
The notation $S(q;b)$ should better be $S(b;q)$
to be coherent with P.291.
P. 285, line 10.
$(W^*(d) = q \sum_{\delta | q}$
should be
$W^*(q) = \sum_{\delta | q}$
.
P. 285, line 13.
$V^*(d) = q^2 \sum_{\delta | q}$
should be
$V^*(q) = q \sum_{\delta | q}$
.
P. 285, last displayed equation.
$\frac{N \log^2 Q}{Q \sqrt{p}}$
should be
$\frac{N \log Q}{Q \sqrt{p}}$
.
P. 286, line 6 from bottom.
``almost every pairs''
should be
``almost every pair''
.
P. 292, Exercise 29-2, part 2 and 6.
$|\pi^2T \hat{K}(t)|\ge1$ when $|t|\le T$
should be
$|\pi^2T \hat{K}(t)|\ge1$ when $|t|\le T/2$
.
As a consequence, the integral in part 6 should be
$\int_{-T/2}^{T/2}$ and not $\int_{-T}^{T}$.
P. 293, Exercise 29-3.
The statement of part 2 is not clear. It should be
``Enumerate the triples $(n, n+2, n+4)$ where $n \le 3^{100}$
and each element of the ordered triple is a prime power.''
P. 294, Exercise 29.8 title.
Erd
should be Erdős.
P. 295, reference [4].
$2^k \ p$
should be
$2^k + p$
.
P. 295, reference [8].
$p \ 2^k$
should be
$p + 2^k$
.
P. 295, references [9] and [12].
These references are to books and should be formatted as books.
P. 295, reference [14].
``14 O. Ramaré''
should be
``O. Ramaré''
.
P. 318, Hint for 23-3.
Lemma 19.2 should be Lemma 19.1.
P. 318, Hint for 23-8.
Lemma 12.2.1 should be Lemma 12.2.
P. 320, line 13 from bottom.
for(y = 1, x,
should be
for(y = begx, endx,
.
P. 324, line 12.
$\hat{K}(t)$
should be
$\hat{K}(t) \pi^2 T$
.
P. 325, line 1.
``ofenly''
should be
''often''
.
P. 330, item 8.
Erd
should be Erdős.
P. 330, item 9.
``All the paper of
may''
should be ``All the papers of Ramanujan may''.
P. 331, Name Index.
Bertand, Joseph
should be Bertrand, Joseph.
P. 332, Name Index.
Geršhgorin, Sëmen Aranovič
should be
Geršhgorin, Semën Aranovič
.
P. 332, Name Index.
Meissel, Daniel Friedrich Ernst Meissel
should be only
Meissel, Daniel Friedrich Ernst
.
