Number Theory: An Introduction to Mathematics

386 IX The Number of Prime Numbers whereAais a positive constant which can be explicitly described. (The expression for Aawhich ...

5 Generalizations and Analogues 387 This seems quite similar to the number field case, but the function field case is actually s ...

388 IX The Number of Prime Numbers The Weil conjectures have a topological significance, since the integerein (ii) is the Euler ...

6 Alternative Formulations 389 arbitrary closed orbit of the geodesic flow andλ(P)its least period, one shows that the zeta-func ...

390 IX The Number of Prime Numbers ζ(s− 1 )/ζ(s)= ∑∞ n= 1 φ(n)/ns. From the property by which we defined the M ̈obius function w ...

6 Alternative Formulations 391 M(x)=O(xα ∗+ε ) for everyε> 0. It follows thatthe Riemann hypothesis holds if and only if M(x) ...

392 IX The Number of Prime Numbers Conjecture B lim x→∞ M(x)( 2 xlog logx)−^1 /^2 = √ 6 /π =−lim x→∞ M(x)( 2 xlog logx)−^1 /^2. ...

7 Some Further Problems 393 Table 2. x π 2 (x) L 2 (x) π 2 (x)/L 2 (x) 103 35 46 0.76 104 205 214 0.96 105 1224 1249 0.980 106 8 ...

394 IX The Number of Prime Numbers Bateman and Horn gave a heuristic derivation of their formula. However, the only case in whic ...

9 Selected References 395 For a proof thatπ(x)−Li(x)changes sign infinitely often, see Diamond [17]. Estimates for values ofxsuc ...

396 IX The Number of Prime Numbers [4] R. Ayoub,An introduction to the analytic theory of numbers, Math. Surveys no. 10, Amer. M ...

9 Selected References 397 [32] N. Katz, An overview of Deligne’s proof of the Riemann hypothesis for varieties over finite field ...

398 IX The Number of Prime Numbers [62] D.V. Widder,The Laplace transform, Princeton University Press, Princeton, 1941. [63] D. ...

X A Character Study........................................... 1 PrimesinArithmeticProgressions Letaandmbe integers with 1≤a< ...

400 X A Character Study also equivalent to πj(x)∼x/φ(m)logx (j= 1 ,...,φ(m)). The validity of the second conjecture in this form ...

2 Characters of Finite Abelian Groups 401 The functionχ 1 :G→Cdefined byχ 1 (a)=1foreverya∈Gis obviously a character ofG,thetriv ...

402 X A Character Study Proposition 2Let G be a finite abelian group of order g andG its dual group. Thenˆ (i) ∑ a∈G χ(a)= { gif ...

3 Proof of the Prime Number Theorem for Arithmetic Progressions 403 3 Proof of the Prime Number Theorem for Arithmetic Progressi ...

404 X A Character Study Proof Any positive integerNcan be written in the formN=qm+r,whereq≥ 0 and 1≤r≤m.Sinceχ(a)=χ(b)ifa≡bmodm, ...

3 Proof of the Prime Number Theorem for Arithmetic Progressions 405 Proposition 4L(s,χ)=Πp( 1 −χ(p)p−s)−^1 forσ> 1 , where th ...