Number Theory: An Introduction to Mathematics

366 IX The Number of Prime Numbers Conversely, supposepn/nlogn→1. Since (n+ 1 )log(n+ 1 )/nlogn=( 1 + 1 /n){ 1 +log( 1 + 1 /n)/l ...

2 Chebyshev’s Functions 367 for complex values ofs. By developing these ideas, and by showing especially that ζ(s)has no zeros o ...

368 IX The Number of Prime Numbers Proof Since ψ(x)= ∑ p≤x logp+ ∑ p^2 ≤x logp+··· andk>logx/log 2 impliesx^1 /k<2, we hav ...

2 Chebyshev’s Functions 369 by Lemma 3, it follows that π(x)=ψ(x)/logx+O(x/log^2 x). Suppose next thatπ(x)=O(x/logx).Foranyx> ...

370 IX The Number of Prime Numbers It follows from (3) 1 –(3) 2 thatR(x)=O(x/logαx)for someα>0 if and only if Q(x)=O(x/logα+^ ...

3 Proof of the Prime Number Theorem 371 This infinite series had already been considered by Euler, Dirichlet and Chebyshev, but ...

372 IX The Number of Prime Numbers It follows at once from Proposition 5 thatζ(s)=0forσ>1, since the infinite product is con ...

3 Proof of the Prime Number Theorem 373 Proof Puts=σ+it,whereσandtare real, and let g(σ,t)=−R{f′(s)/f(s)}. Thus g(σ,t)= ∫∞ 0 e−σ ...

374 IX The Number of Prime Numbers is defined forRs>h, where h> 0. If there exists a constant A and a function G(s), which ...

3 Proof of the Prime Number Theorem 375 of the function k(t)= 1 −|t|for|t|≤ 1 ,=0for|t|≥ 1 , has the properties kˆ(u)≥0for−∞< ...

376 IX The Number of Prime Numbers By the ‘Riemann–Lebesgue lemma’,χ(y)→0asy→∞. In fact this may be proved in the following way. ...

4 The Riemann Hypothesis 377 Since this holds for arbitrarily smallδ>0, it follows that lim x→∞ α(x)≤A. Thus there exists a p ...

378 IX The Number of Prime Numbers Proof Putf(v)=e−v (^2) πy and let g(u)= ∫∞ −∞ f(v)e−^2 πiuvdv be the Fourier transform off(v) ...

4 The Riemann Hypothesis 379 The solution of this first order linear differential equation is g(u)=g( 0 )e−πu (^2) /y . Moreover ...

380 IX The Number of Prime Numbers withF( 1 )=1 which is holomorphic in the half-planeRz >0 and bounded for 1 <Rz<2. It ...

4 The Riemann Hypothesis 381 ∫∞ 0 xs/^2 −^1 e−n (^2) πx dx=π−s/^2 Γ(s/ 2 )n−s. Hence, ifσ>1, Z(s)= ∑∞ n= 1 ∫∞ 0 xs/^2 −^1 e−n ...

382 IX The Number of Prime Numbers Sinceζ(s ̄)=ζ(s), the zeros ofζ(s)are also symmetric with respect to the real axis. Furthermo ...

4 The Riemann Hypothesis 383 and to ψ(x)=x+O(x^1 /^2 log^2 x). Since it is still not known ifα∗<1, the error terms here are s ...

384 IX The Number of Prime Numbers In the language of physics Montgomery’s conjecture says that 1−(sinπu/πu)^2 is thepair correl ...

5 Generalizations and Analogues 385 ζK(s)= ∏ P∈P ( 1 −|P|−s)−^1 holds in this open half-plane. Furthermore the definition ofζK(s ...