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Show that L(s, χ

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MATH-GA 2210.001: Homework Analytic Number Theory 1

1. Compute the set of Dirichlet characters modulo 8 and modulo 12.

2. Let S ={1,11,21,31, . . .} be the set of positive integers which have the last digit1when written in base 10. Prove thatS has a natural Dirichlet density, and compute it.

3. 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. 4. 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.

5. Let γ be the Euler constant:

γ = limN→∞1 + 1 2 +1

3 +. . .+ 1

N −logN.

Verify that

lims→1ζ(s)− 1

s−1 =γ.

1

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