and that the
n
th Bernoulli number
B
n
is given by
B
n
=
B
n
(0)
.
(a)
Write down the numerical values of
B
0
,B
1
and
B
2
.
Show that
B
n
(1) =
B
n
for all
n
≥
2
,
and that
B
2
k
+1
= 0
for all
k
≥
1
.
(5 marks)
(b)
Prove that
B
n
(
x
) =
n
∑
k
=0
(
n
k
)
x
n
−
k
B
k
,
and hence calculate
B
4
.
(5 marks)
(ii)
Let
f
: (0
,
∞
)
−→
C
be a continuously di erentiable function. Show that
f
(1) +
f
(2) +
...
+
f
(
n
) =
1
2
(
f
(1) +
f
(
n
)) +
∫
n
1
f
(
x
)
d
x
+
∫
n
1
f
′
(
x
)
P
1
(
x
)
d
x.
Here
P
1
(
x
) :=
x
− ⌊
x
⌋ −
1
2
is the rst periodic Bernoulli function.
(7 marks)
(iii)
Prove that the sequence
(
log 1
1
+
log 2
2
+
...
+
log
n
n
−
(log
n
)
2
n
)
∞
n
=1
converges.
(8 marks)
5
If
χ
is a character of
(
Z
/m
Z
)
×
,
de ne the
Dirichlet
L
function
L
(
s,χ
)
, and indicate
its region of convergence.
(4 marks)
The multiplicative group of integers invertible modulo 8 consists of the
elements
{
1
,
3
,
5
,
7
}
.
(i)
List the modular characters modulo 8, indicating which are the nontrivial
characters.
(5 marks)
(ii)
For each nontrivial character
χ
on your list, prove that
0
< L
(1
,χ
)
<
2
.
(6 marks)
(iii)
Prove that there are in nitely many primes congruent to
5 (mod 8)
.
(10 marks)
End of Question Paper
PMA430
4
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 Fall '13
 430
 Statistics, Number Theory, Prime number, Riemann zeta function

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