华罗庚与中国的数论.pdf
华罗庚与中国的数论.pdf
, , , , (1982), , – !, ! ) , , 1928 ( (L. E. Dickson)# # , (x − 1)x(x + 1) 6 % 1929 , 6, + + 21931 <(8 , + 1936 , , &/,8 , * (G. H. Hardy) * (L. J. Mordell) 1938 , .2 , 2 , @f 1935 #, , -, +, 2, ,, . , @f#,2 , , , *– (J. E. Littlewood)–( (S. Ramanujan) +@ , (I. M. Vinogradov) , !=3,, +, , , 3, ,&', T, , O , , 268 | q f (x) # f (x) = ak xk + · · · + a1 x, (ak , · · · , a1 , q) = 1/ q S(q, f (x)) = e x=1 f (x) q , e(x) = e2πix S(q, x2 ) ) (C. F. Gauss) , S(q, x2 ) ≤ 3 f (x) = x2 , √ 2q, S(q, f (x2 )) @ :3, # 1940 , : & % 1 |S(q, f (x))| = O q 1− k +ε , (1) ε O /(7 k ε @ (1) N, + q = pk , p , p = k, S(q k , xk ) = 1 (pk )1− k , (1) 5 2 ; , 1958 , (a, q) = 1 b = 0 , q (2) e x=1 ε >0 O axk + b q & % 1 = O q 2 +ε (q, b) , (7 ε k, (E. Waring) (H. Weyl) @, + % *# , % (A. Walfisz) $ (E. Ch. Titchmarsh) * (K. Prachar) , , 1 1770 , 7 ; N = xk1 + · · · + xks , N > 0, x1i s ≥ 0 (D. Hilbert) 1900 s(k) 1920 , * s % k , (3) 7 , k ≥ 2, /(7 k s(k), ! —— , (3) , G(k) ' , \N g(k) ' s ! (3) N g(k) s! G(k) , G(k) @ rs,k (N ) ' (3) # , * + 5 , 1 s Γ 1+ s k % s & N k −1 rs,k (N ) ∼ σ(N ) Γ k , s ≥ (k − 2)2 σ(N ) %*, k−1 ! N G(k) ≤ (k − 2)2k−1 + 5, 1938 *– G(k) ≤ 2k + 1 (4) , s ≥ 2k + 1 , rs,k (N ) &# % )-#, 2k 1 p k (5) e(αxk ) dα = O(P 2 −k+ε ) 0 x=1 k , - # Vaughan) : “ , &-#” 20 " 30 , ! * (H. Davenport) xk & k # @ (1) , fi (x)(1 ≤ i ≤ s) s k * # 1940 (3) N = f1 (x1 ) + · · · + fx (xs ) (6) (3) rs,k (N ) & ' , (R. C. s ≥ 2k (k 3) s ≥ 7 · 2k−3 + 1 (k 6), B, (L. G. Shnirel’man) (Yu. V. Linnik) " –/" (D. R. Heath-Brown) , # %& , , )– ) (C. Goldbach) , x ,+ * , s≥ (7) 2k + 1, k ≤ 10, 2k (log k − log log k + 2.5), 2 k > 10 , N = P1k + · · · + Psk (8) &#, p + , @ , (7) # -# , @ 10k 2 log k (8) %**, s ≥ 269 270 | ≡ s (mod K) ' s k ' &#, s (7), K (7 k ' &#, s C s0 s0 ∼ 4k log k 8: k=1 , K = 2; k = 2 , K = 24, : (9) ( ); (10) ≡ 5 (mod 24) 5 \N +@ (11) \N , "$ # 1947 , " 1960—1970 , -2 , +@ / , , /, , (C. L. Siegel) L- +@, 20 " 40 , \N , , ) * (H. Halberstam) b: “ 1947 \N ” Gs ' s-- Gs = {x : x(x1 , · · · , xs ), 0 ≤ xi ≤ 1, 1 ≤ i ≤ s} f (x) Gs , 1, f (x) ( 1 (J. Fourier) # C(m1 , · · · , ms )e2πi(m1 x1 +···+ms xs ) , f (x) = m ( 1 @ |C(m1 , · · · , ms )| ≤ C > 0 α > 1 C , (m1 · · · ms )α , m = max(1, |m|) a = (a1 , · · · , as ) n > 0, n f k=1 as k a1 k ,··· , n n n = m = e(a1 m1 +···+as ms )k/n C(m) k=1 n C(m) m a1 m1 +···+as ms ≡O( mod n) C(0, · · · , 0) = 1 ··· 1 0 f (x1 , · · · , xs )dx1 · · · dxs = 0 1 f (x)dx − Gs n n f k=1 ≤ f (x)dx, Gs as k a1 k ,··· , n n C m a1 m1 +···+as ms ≡O( mod n) 1 (m1 · · · ms )α = CΩ(a) ( ), Σ 'O m = 0 * Ω(a) + a ! ! (N. M. Korobov, 1959) (E. Hlawka, 1962) ! , n = p , a ! αsp log |Ω(a)| = O pα 9, “” p, ' O(p2 ) ' a, # “ $” s = 2 (Fn )(n = 0, 1, · · · ) !R (L. P. Fibonacci) ', [ # F0 = 0, F1 = 1, Fn+1 = Fn + Fn−1 (n ≥ 1) ' 1960 Fn log Fn |Ω(1, Fn−1 )| = O , Fnα (12) # 1 1 (13) 0 1 f (x1 , x2 )dx1 dx2 − Fn 0 Fn f k=1 k Fn−1 k , Fn Fn =O log Fn Fnα (13) : √ 2π 2π 5−1 Fn−1 = 2 cos ; 3, Q 2 cos 1. Fn 2 5 p , p > 5 , ! $f % & () 1, h2 , · · · , h() , n , s 1 < n1 < n2 < · · · , s= p−1 , 2 $ s- # n () () % − α − α +ε & 1 k h2 k hs k (14) f (x)dx − f , ,··· , = O n 2 2(s−1) n n n n Gs k=1 271 272 | 2. (s) !R ', [ # (s) (s) (s) (s) = · · · = Fs−2 = 0, Fs−1 = 1, ( ' (s) # ' Fn , = F1 F0 Fn(s) s Fn+s = Fn+s−1 +· · ·+Fn(s) (s) (s) Fn+1 k F f (x)dx − (s) f , , · · · , n+s−1 (s) (s) Fn(s) Gs Fn k=1 Fn Fn α α − −α− (s) 2 2s+1 log 2 22s+3 = O Fn (15) 1 k (n ≥ 0) : (16) (), , 1981 1742 7 , (L. Euler) $ , ) n−3 ! 7 : (17) ! ≥6 ; (18) ≥9 (18) (17) , 3 (17) , n ≥9, ≥6 (17), n − 3 = p1 + p2 , (18) )7 (17) (17) “L 5 ”L (Eratosthenes) L1919 , / (V. Brun) )7 6 /" (9, 9), “ 250 !& ! 9 ” : (7, 7) (H. Rademacher, 1924), (6, 6) (T. Estermann, 1932), (5, 7), (4, 9), (3, 15), (2, 366) (G. Ricci, 1937), (5, 5) (A. Buchstab, 1938), (4, 4)(A. Buchstab, 1940) 1953 . )7 , . (A. Selberg) L , : (19) (3, 4) ( L , 1956) ,.' (a, b)(a + b ≤ 6) , , ! 1957 (P. Kuhn) 1954 (a, b)(a + b ≤ 5), : (20) (2, 3) ( /L , 1957) (Yu. V. Linnik) L ., (A. Rényi) (1, c), c %3, Riemann) 7 , (1, 6), ! , (T. Estemann) *, (G. F. B. % , (1, 4)R , (1, 6)R , 1957 (21) (1, 3)R *,7 , * , *# x lix 2 μ (D) max π(x, D, ) − , =O φ(D) ( mod D) logA x 1 (22) D≤x 3 −ε (,D)=1 μ(D) φ(D) &'G x dt , π(x, D, ) = lix = 2 log t (A. F. Möbius) ε>0A>0 1, p≤x p≡( mod D) (22) (23) (1, 5) (* ,, 1962) 1 3 & η π(x, D, ) '& 1, , 1 3 * ,) (M. B. Barban) ! (22) , & , 3 8 , (1, c) *#, (22) (24) (1, 4) (* ,, Barban, 1963) , 1962 (22) (1, 4), +3 (22) 1 1000 3 , , (1, 3)1965 , & (E. Bombieri) 3 2475 8 1 1 !& (22) , 3 2 *,7 (1, 3) 1966 , ' & *#, /& (25) (1, 2) ( /, 1966) / 3 : (1, 3) @! n R(n) / %3 ! 1 3 R(n) @ R(n) > r(n), /@ r(n) , ' / )7 #, & ! (1, 2) “!&” @ 1. , , /, # ?, 9 , (G. Tarry) , (E. C. Catalan) , *, ζ- , ',, 273 274 | , , * @, #, + 9 ,D /, , @, : , , , & &5 , , ' 2. , ;, / *#, F: “ (leading) , , , , , # , $ , —— 50 , ”1) 1) 4 H. Halberstam. Loo-keng Hua, Biographical Memoirs, Vol. 81, 2002, U. S. National Academy Press.