Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)

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1 Z[i] Z[i] π 4, (x) π 4,3 (x) x (x ) 2 log x π m,a (x) x ϕ(m) log x. ( ). π(x) x (a, m) = π m,a (x) x modm a π m,a (x) ϕ(m) π(x) ϕ(m) x log x ϕ(m) m f(x) g(x) (x α) lim f(x)/g(x) = x α mod m (a, m) > π m,a (x) (a, m) =

2 (Cauchy ). {a n } ϵ > 0 n 0 p > n 0, q > n 0 a p a q < ϵ 2.2 (). {a n } s n = a + a a n lim s n = s n a n = a + a a n + s lim s n = ± Cauchy 2. ϵ > 0 R n,m = s m s n = a n+ + a n a m R n,m < ϵ 2.3 (). a n a n a n R n,m = a n+ + a n a m a n+ + a n a m n a n

3 2.4 ( ). x f (x), f 2 (x),..., f n (x),... x f(x) ϵ > 0 n 0 n > n 0, x [a, b] f(x) f n (x) < ϵ x {f n (x)} [a, b] f(x) a n = a n (x) x n s n (x) = a v (x) v= s n (x) s(x) s(x) s n (x) = v=n+ a v (x) = R n (x) R n (x) a n (x) c n, c n c n a n (x) 2.6. f(x) f(x) = O(g(x)) f(x) g(x) g(x) 0 ( x, ) f(x) = o(g(x)) f(x) ( ) g(x) f(x) g(x) ( ). U C a U ϵ > 0 z a < ϵ z U 2

4 a a a a 2.8. f U f U C / 2.9 ( ). U U f a f(z) = a 0 + a (z a) + a 2 (z a) Taylor Taylor 2.0 ( ). f, g f = g f = g 2.. {z C r < z a < R} f f(z) = a n (z a) n Laurent 2.2 (). a a Laurent a n 0 n a a n 0 n m m 0 f a a m > 0 a m U U

5 2.3 ( ). z = z 0 f(z) f z = z 0 Laurent z z 0 a f z = z 0 Res z=z0 f(z) = a z = z 0 Res z=z0 f(z) = lim z z 0 {(z z 0 )f(z)} 2.4. f, g f ± g, fg, f f, g g 0 f/g f, f 2,... U f f f (z) = lim n f n(z) 2.6 ( ). 2 D, D 2 f (z) D D D 2 f (z) D 2 f 2 (z) f 2 (z) f (z) D D 2. f (z) = n=0 z n z < z z < f (z) = ( z < ) z f 2 (z) = (z ) z C {} 2 z < f = f 2 f 2 f z = C 3

6 (Abel ). G + : G G G G a, b, c (G) (G4) G Abel G a + b = b + a G2 a + (b + c) = (a + b) + c G3 G e G a a + e = e + a = a G4 G b b + ( b) = ( b) + b = e G ( b) G Abel 2.8 ( ). G G card(g) 2.9 ( ). G G C χ G a, b χ(ab) = χ(a)χ(b) G Ĝ χ χ(a) = a card(g) = e() 2.2. χ(a ) = χ(a) = χ(a) 2.22 (). χ, χ 2 Ĝ (χ χ 2 )(a) = χ (a)χ 2 (a) Ĝ Abel G H G H G card(g) = card(ĝ) n = card(g), χ Ĝ { n (χ = ) χ(a) = 0 (χ ) a G

7 2.26. n = card(g), a G { n (a = ) χ(a) = 0 (a ) χ Ĝ G (Z/mZ) (a, m) > a χ(a) = 0 χ Z C modm χ(n) modm [2] p s, z n m a 3 [] 3.. ζ(s), Φ(s), θ(x) ζ(s) = n s, Φ(s) = p log p p s, θ(x) = log p (s C, x R) p x 3.2 (Euler ). ζ(s) = p ( p s ) (Re(s) > ) 4

8 . 3.3 ( ). ζ(s) Re(s) > 0 s. ϕ n (s) = s = n+ n t s dt = n+ n t s dt (n s t s )dt ϕ(s) = ϕ n (s) Re(s) > 0 ϕ n (s) Re(s) > 0 k ϕ n (s) Re(s) > 0 ϕ n (s) n+ n n s t s = n s t s dt sup n s t s n t n+ t n t = s s dt ts+ s dt t s+ n t n s n x+ dt t x+ (x = Re(s)) ϕ n (s) s S Re(s) > 0 nx+ s S s Re(s) M, m S ϕ n (s) M k n m+ 2.5 ϕ n (s) ϕ(s) Re(s) > θ(x) = O(x). n N 2 2n = ( + ) 2n ( ) 2n n n<p 2n p = e θ(2n) θ(n)

9 ( ) n < p 2n p 2n p = (2n)!/(n!) 2 n 2n log 2 θ(2n) θ(n) θ(x) x log x θ(2x) θ(x) = θ( 2x ) θ( x ) θ(2 x + ) θ( x ) θ(2 x ) + log(2 x + ) θ( x ) 2 x log 2 + log(2x + ) 2x log 2 + log(2x + ) C > 2 log 2 x 0 = x 0 (C) x > x 0 θ(2x) θ(x) Cx x/2 r+ < x 0 x/2 r r x, x/2, x/4,..., x/2 r θ(x) < 2Cx+θ(x 0 ) θ(x) = O(x) 3.5 (ζ ). ζ(s) 0(Re(s) ) Φ(s) s. σ > Euler log ζ(σ + it) = ( ) log ζ(s) + log ζ(s) 2 = ( log( p s ) + log( p s ) ) 2 p = (p s p 2s + 3 p 3s + + p s + p 2 s + ) 2 p = p = p (Re(p s ) + 2 Re(p 2s ) + ) k= cos(kt log p) kpkσ Euler Re(s) > ζ(+it) = 0

10 log ζ(σ) 3 ζ(σ + it) 4 ζ(σ + 2it) = k,p (3 + 4 cos(kt log p) + cos(2kt log p)) kpkσ ζ(σ) 3 ζ(σ + it) 4 ζ(σ + 2it) σ ζ(σ) 3 = O( (σ ) 3 ) ζ(σ + it) 4 = O((σ ) 4 ) ζ(σ + 2it) = O() = 2 k,p kp kσ ( + cos(kt log p))2 0 ζ(σ) 3 ζ(σ + it) 4 ζ(σ + 2it) 0 (σ ) Φ(s) Euler ζ ζ (s) = p log p p s = Φ(s) + p log p p s (p s ) Re(s) > /2 Φ(s) s = ζ(s) Re(s) ζ(s) s = ζ(s) /(s ) = ϕ(s) ϕ(s) Re(s) ζ ζ (s) = ϕ (s) (s ) 2 ϕ(s) + (s ) s (s ) s = Re(s) > θ(x) x x 2 dx Φ(s) = p log p dθ(x) θ(x) p s = x s = s dx = s xs+ 0 e st θ(e t )dt

11 Stieltjes 5 θ(x) Φ(s) Stieltjes θ(x) s x s+ dx = k= = pk+ p k θ(p k )( p s k= k = = k= k= s x s+ θ(p k)dx ( p k k ) p s ) k+ (θ(p k ) θ(p k )) p s k log p k p s k = Φ(s) ( p 0 = ) 3.7. f(t) t 0 a, b R 0 b a f(t)dt g(z) = f(t)e zt dt(re(z) > 0) Re(z) 0 0 f(t)dt g(0) [] f(t) = θ(e t )e t, g(z) = Φ(z + )/(z + ) /z f(t) 3.4 g(z) = (Φ(z + ) /z )/(z + ) 3.5 Re(z) 0 g(z) = 3.8. θ(x) x 0 0 f(t)e zt dt f(t)dt = θ(x) x x 2 dx. θ(x) 3.6 5

12 θ(x) > λx x λ > λx x dt θ(t) t t 2 dt > λx x λx t t 2 dt = λ log λ > 0 θ(x) 3.6 λx θ(t) t x t 2 dt 0(x ) θ(x) < λx x λ < x λx θ(t) t t 2 dt < x λx λx t t 2 dt = (λ log λ) < 0 θ(x)/x (x ) θ(x) = p x log p x p x log x = π(x) log x ϵ > 0 θ(x) log p x ϵ p x x ϵ p x ( θ(x) x ϵ ) x( ϵ) + O log x x ( ϵ) log x = ( ϵ) log x(π(x) + O(x ϵ ) π(x) log x x θ(x) x x, ϵ L(s, χ) = χ(n) n s, Φ m,a(s) = χ(a) χ(p) log p p s p,χ

13 θ m,a (x) = ϕ(m) p x p a (mod m) log p 4.2 (Euler ). L(s, χ) = p ( χ(p) p s ) (Re(s) > ) 4.3. χ L(s, χ) Re(s) > 0 s = ϕ(m)/m. χ Re(s) > 0 f k (s) = m χ(n) (km + n) s L(s, χ) = f 0(s) + f k (s) f k (s) = m χ(n)( (km + n) s m (km) s ) (χ χ(n) = 0) = m χ(n) ( km+n s km x s+ dx) m χ(n) n s (km) s+ m s k σ+ k= ( Re(s) = σ ) s = m k σ+ f k (s) Re(s) > 0 k= χ L(s, χ) Euler L(s, χ) = (n,m)= n s = ζ(s) ( p s ) p m ζ(s) s = ( p ) = ϕ(m) m p m 4.4. θ m,a (x) = O(x)

14 ( ). Re(s) L(s, χ) 0, Φ m,a (s) s. Euler L(s, χ) Re(s) > L(+ it, χ) = 0 ζ(s) log L(s, χ) = ( (χ(p) ) ) k k Re p s = kp kσ cos(k( c p + t log p)) k,p k,p c p χ p = e icp χ ζ(s) χ log L(σ, ) 3 L(σ + it, χ) 4 L(σ + 2it, χ 2 ) = k,p kp kσ (3 + 4 cos(k( c p + t log p)) + cos(k( 2c p + 2t log p)) = 2 k,p L(σ, ) 3 L(σ + it, χ) 4 L(σ + 2it, χ 2 ) kp kσ ( + cos(k( c p + t log p)) 2 0 σ L(σ, ) 3 L(σ + it, χ) 4 L(σ + 2it, χ 2 ) 0 σ L(σ + it, χ) 4 = O((σ ) 4 ) L(σ, ) = O((σ ) 3 ), L(σ + 2it, χ 2 ) = O() χ 2 L(, χ) = 0 t = 0 L(σ + 2it, χ 2 ) = O(/(σ )) L(σ, ) 3 L(σ + it, χ) 4 L(σ + 2it, χ 2 ) 0 L(, χ) χ L(, χ) 0 Dirichlet 6 χ 2 χ χ

15 [3, 4, 5] L(s, χ) Re(s) L(s, χ) Euler L (s, χ) = L p χ(p) log p p s χ(p) = p χ(p) log p p s + χ(p)2 log p p s (p s χ(p)) Re(s) > /2 χ(p) log p p s L(s, χ) p Φ m,a (s) = χ(a) χ(p) log p p s Φ m,a (s) p,χ s = L (s, ) L L ζ (s, ) = L ζ (s) + + p s log p p s p m s = Φ(s) Φ m,a (s) = p,χ = θ m,a (x) x x 2 dx = s 0 χ(a) χ(p) log p p s dθ m,a (x) = s x s e st θ m,a (e t )dt = ϕ(m) θ m,a (x) x s+ dx p a (mod m) log p p s 3.6 Φ m,a (s) mod m a L(s, χ) Φ m,a 4.8. θ m,a (x) x

16 5 Z[i] Z[i] x π Z[i] (x) Z[i] π Z[i] (x) = 2π 4, (x) + π 4,3 ( x) + x log x 6 modm p A p s log s k R (s ) k A modm x A π(x) k(x ) A k

17 m= m π(x/m ) = Li(x) ρ Li(x ρ dt ) + x t(t 2 ) log t log 2 dt Li(x) = 2 log t x ρ ζ(s) log x ζ(s) Re(s) < π m,a (x) L(s, χ) ζ(s) π(x) Li(x) ζ(s) Re(s) = /2 HP( ) 5 4 [] D.Zagier Newman s Short Proof of the Prime Number Theorem (the American Mathematical Monthly,Vol.04,No.8, ) [2] G.H.Hardy,E.M.Wright I,II, 200 [3] T.M.Apostol Introduction to Analytic Number Theory (Springer,976) [4] Jean-Pierre Serre A Course in Arithmetic (Springer,973)

18 [5],, I -Fermat,2005 [6] 983 [7] 974 [8] R.v.Churchill,J.W.Brown,989

, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,,

, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,, 6,,3,4,, 3 4 8 6 6................................. 6.................................. , 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p,

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