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Homework 3. Due by March, 7.
1. Show that if 0< a <1, then
1 2πi
Z b+iT
b−iT
asds
s =O( ab T|log(a)|).
2. Let k > 0be an integer. Prove that Z x
2
dt
log t = x
log x + 1!x
log2x +. . .+ (k−1)!x
logkx +O( x logk+1x) 3. Let T(x) =P
n≥1Λ(n)[x/n]. Show that T(x) =P
n≤xlog n.
4. Compute ζ(0).
1
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