文档介绍：Math 259: Introduction to Analytic Number Theory
Proof of the Prime Number Theorem;
the Riemann Hypothesis
We finally have all the ingredients that we need to assemble a proof of the Prime
Number Theorem with an explicit error bound. We shall give an upper bound
on |(ψ(x)/x) − 1| that decreases faster than any power of 1/ log x as x→∞,
though slower than any positive power of 1/x. Specifically, we show:
Theorem. There exists an effective constant C > 0 such that
ψ(x) = x + O(x exp(−C log x)) (1)
p
for all x ≥ 1.
Proof : There is no difficulty with small x, so we may and shall assume that
x ≥ e, so log x ≥ 1. We use our integral approximation
1 0
1+ log x +iT 2
1 ζ s ds x log x
ψ(x) = Z −(s) x + O (T ∈[1, x]) (2)
2πi 1 −ζ s T
1+ log x iT
to ψ(x). Assume that T ≥ e, and that T does not coincide with the imaginary
part of any ρ. Shifting the line of integration leftwards, say to real part −1,
yields
xρζ0 x log2 x
ψ(x) − x −! = I1 + I2 −(0) + O , (3)
X ρζ T
| Im(ρ)|<T
0 s
in which I1, I2 are the integrals of −(ζ(s)/ζ(s))x ds/s over the vertical line
σ= −1, |t| < T and the horizontal lines σ∈[−1, 1 + 1/ log x], t = ±T respec-
tively. We next show that I1 is small, and that I2 can be made small by adding
O(1) to T . The vertical integral I1 is clearly
log T ζ0 log2 T
sup (−1 + it) .
x | | ζ x
t <T
The horizo