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Show that L(s, χ

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Homework 1. Due by February, 7.

1. Let S ={1,11,21,31, . . .} be the set of positive integers which have the last digit 1when written in base 10. Prove that S has a Dirichlet density (in the set of all integers{1,2,3, . . .}), and compute it.

2. Show that

ζ0(s)

ζ(s) =−X

p

X

m≥1

log p pms

for Re(s)>1, whereζ(s) is the Riemann zeta function.

3. Determine all Dirichlet characters modulo 8 and modulo 12.

4. Let χ be a Dirichlet character modulom,χ(2) 6= 0. Show that

L(s, χ) = (1−2−sχ(2))−1

∞

X

n=0

χ(2n+ 1) (2n+ 1)s.

1

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