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# Show that T(x) =P n≤xlog n

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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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