關於多項式因式函數的一個漸進公式
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(2) 2. CHIUNG-HSUN TU. The purpose of this paper is to prove an asymptotic expansion for the sum X0 ¿22 (f )¿2 (f ¡ a); deg f =d. where a is an element of the multiplicative group Fq£ . In theorem 3.1, we obtain ³ ´ X0 ¿22 (f )¿2 (f ¡ a) = Sqd d4 + O qd d3 maxf1; ln dg ; deg f =d. where. ¶ µ q ¡ 1 Y0 1 (1 ¡ 1=jP j)2 1 S= + 1¡ 6q jP j jP j (1 + 1=jP j) P. and the implied constant depends on q.. 2. Auxiliary Lemmas Let ¼N denote the number of monic irreducible polynomials in A of degree N . The polynomial prime number theorem is given by (1). qN =N ¡ qN=2 < ¼N · q N =N:. Let Q 2 A+ , the polynomial Euler phi-function Á(Q) denote the order of (A =Q A)£ . Moreover, we have ¶ Y0 µ 1 : Á(Q) = jQj 1¡ jP j P jQ. Using polynomial prime number theorem (see [7] Lemma 2.1), we have (2). jQj · 30Á(Q) ln maxf2; deg Qg:. For any Q 2 A+ , a multiplicative character  modulo Q is a homomorphism of  : (A =Q A)£ ! C£ :. We say that  is primitive if  can not factor through (A =Q0 A)£ for some Q0 jQ with deg Q0 < deg Q. The L-function of  and the zeta function are given by ¶ X0 Â(f ) Y0 µ Â(P ) ¡1 L(s; Â) = = ; 1 ¡ jf js jP js P f ¶ Y0 µ X0 1 qs 1 ¡1 : = = ³(s) = 1 ¡ jf js jP js qs ¡ q f. P.
(3) AN ASYMPTOTIC FORMULA FOR POLYNOMIAL DIVISOR FUNCTIONS. 3. Lemma 2.1. Let k be a positive integer and let Q 2 A+ . We have X0 lnk jP j ¿ maxf1; ln (q deg Q)gk ; jP j P jQ. where the implied constant depends on k. Proof. By calculus, the function [ek ; 1). Let integer N satisfy. lnk x x. increases on [1; ek ] and decreases on. ¼1 + 2¼2 + ¢ ¢ ¢ + (N ¡ 1)¼N ¡1 · deg Q < ¼1 + 2¼2 + ¢ ¢ ¢ + N ¼N :. If q N · ek , then by polynomial prime number theorem we have deg Q < 2qN and X0 lnk ek X0 lnk jP j kk · · deg Q jP j ek ek P jQ. P jQ. · 2q N. kk · 2kk : ek. If q N > ek , then by polynomial prime number theorem, we have deg Q < 2q N and. q N ¡1 4. ·. N X0 lnk jP j X lnk qi lnk q N deg Q · ¼ + i jP j qi qN N i=1. P jQ. N X lnk qi q i. lnk qN 2qN qi i qN N i=1 ´ ³ ´ ³ = O N k lnk q = O maxf1; ln(q deg Q)gk : ·. +. Lemma 2.2. Let N; K be positive integers and X ½ A. Let X(g; N ) = ff 2 Xj jg ¡ f j · qN g. and for any f 2 X(g; N ), af is a ¯xed complex number. Then we have ¯2 ¯ ¯ ¯ X X X0 X ¯ ¯ ¤ jQj ¯ · q supfN +1;2Kg ¯ a Â(f ) jaf j2 ; f ¯ ¯ Á(Q) ¯ ¯ 1·deg Q·K Â (mod Q) f 2X(g;N ) f 2X(g;N ). where the asterisk means summation over all primitive character. Proof. See [7] Theorem 5:3:. Lemma 2.3. Let d be a positive integer, let ² be a real number and let Q 2 A+ ; a 2 A. Suppose that 0 < ² · 1; deg Q · (1 ¡ ²)d; (a; Q) = 1:.
(4) 4. CHIUNG-HSUN TU. We have. X0. deg f =d f ´a (mod Q). ¿22 (f ) ¿. qd d3 ; jQj. where the implied constant depends on ² and q. Proof. See [7] Theorem 3:1: Lemma 2.4. Suppose N ¸ 1. We have X0 ¿ r (f ) 2. deg f ·N. jf j. r. ¿ N2 ;. where the implied constant depends on r. Proof. See [7] Theorem 4:1: Lemma 2.5. Let N be a positive integer, s = ¾ + it 2 C and 1=2 · ¾ · 56 . Then we have X X0 1 jL(s; Â)j4 ¿ qN N 21 ; Á(Q) 0·deg Q·N.  (mod Q). where the implied constant does not depend on ¾ and q. Proof. Let L(N ) be the left side, and further let ¤ (mod Q1 ) be the primitive character corresponding to  (mod Q), then we have ¯ ¯ ¯ ¯ ¯ ¯ Y0 Q ¯ ¤ s ¯ (1 ¡  (P )=jP j )¯ jL(s; ¤ )j · ¿2 ( )jL(s; ¤ )j: jL(s; Â)j = ¯ ¯ ¯ Q Q1 ¯ ¯P j Q 1. Hence. L(N ) ·. X0. Q1 ;Q0 0·deg Q1 Q0 ·N. ·. X0. 0·deg Q1 ·N. 1 Á(Q1 )Á(Q0 ). 1 Á(Q1 ). X¤. X. ¿24 (Q0 )jL(s; ¤ )j4. ¤ (mod Q1 ).  (mod Q1 ). jL(s; Â)j4. X0. 0·deg Q0 ·N. ¿24 (Q0 ) ; Á(Q0 ). where the second sum of the ¯rst inequality runs over all the primitive character ¤ (mod Q1 ) and the asterisk of the second inequality means summation over all primitive character. By (2) and lemma 2.4 with r = 4 and N ¸ 1, we have X¤ X0 1 L(N) ¿ N 16 ln maxf2; N g jL(s; Â)j4 Á(Q) 0·deg Q·N  (mod Q) 0 1 X0 X ¤ 1 · N 17 @j³(s)j4 + jL(s; Â)j4 A : Á(Q) 1·deg Q·N.  (mod Q).
(5) AN ASYMPTOTIC FORMULA FOR POLYNOMIAL DIVISOR FUNCTIONS. By ¾ · 56 , we have. 0. L(N ) ¿ N 17 @1 +. X0. 1·deg Q·N. 1 Á(Q). 5. 1. X¤.  (mod Q). jL(s; Â)j4 A ;. where the implied constant does not depend on ¾. Thus, we obtain ¯ ¯4 1 ¯ ¯ 0 X ¯ 1 jQj Â(f ) ¯¯ A 17 @ ¯ 1+ L(N ) ¿ N ¯ jQj Á(Q) jf js ¯¯ 1·deg Q·N  (mod Q) ¯deg f ·deg Q¡1 ¯ ¯2 1 0 ¯ X0 X¤ ¯¯ X0 X 1 ¯ jQj Â(f ) 17 @ ¯ ¯ A; ·N ef 1+ ¯ ¯ K s q Á(Q) jf j ¯ 1·K·N 1·deg Q·K  (mod Q) ¯deg f ·2K¡2 0. X0. X¤. where. X0. ef =. 1 · ¿2 (f ):. gjf deg g·K¡1 deg fg ·K¡1. e. By lemma 2.2 with N = 2K ¡ 2; af = jffjs and g = 0, we have 0 1 supf2K¡1;2Kg 2 (f ) X X0 ¿ q 2 @ A: L(N ) ¿ N 17 qK jf j2¾ deg f ·2K¡2. 1·K·N. Since ¾ ¸ 12 , we have L(N ) ¿ N 17. X. 1·K·N. 0. @q K. X0. deg f ·2K¡2. 1 ¿22 (f ) A : jf j. By lemma 2.4 with N = 2K ¡ 2 and r = 2, we obtain X L(N) ¿ N 17 (3) qK K 4 : 1·K·N. Therefore, we have L(N ) ¿ q N N 21 ; where the implied constant does not depend on ¾ and q. Lemma 2.6. Let s = ¾ + it 2 C and let  be a multiplicative character modulo Q 2 A+ . If ¾ > 1, then we have X0 f. Â(f ). ¿22 (f ) L4 (s; Â) : = jf js L(2s; Â2 ).
(6) 6. CHIUNG-HSUN TU. Proof. For ¾ > 1, by lemma 2.3, we know that X0. Â(f ). f. ¿22 (f ) jf js. converge absolutely and 1+. Â(P )¿22 (P ) Â2 (P )¿22 (P 2 ) + + ¢¢¢ = 60; jP js jP 2 js. hence X0 f. ¶ Y0 µ ¿22 (f ) Â(P )¿22 (P ) Â2 (P )¿22 (P 2 ) Â(f ) = + +¢¢¢ 1+ jf js jP js jP 2 js P ¶ Y0 µ 4Â(P ) 9Â2 (P ) 1+ + + ¢¢¢ : = jP js jP 2 js P. Since for jxj < 1, 1+x = 1 + 4x + 9x2 + 16x3 + ¢ ¢ ¢ ; (1 ¡ x)3 we have X0 f. Â(f ). Y0 1 + Â(P )=jP js ¿22 (f ) L4 (s; Â) : = = jf js (1 ¡ Â(P )=jP js )3 L(2s; Â2 ) P. Lemma 2.7. Let Q 2 A+ , s 2 C and let ÂQ be the principal character modulo Q. If ln deg Q · d, then we have L4 (s; ÂQ ) ds q ¡ 1 Y0 (1 ¡ 1=jP j)3 d 3 q = q d s=1 L(2s; ÂQ ) 6q ln q (1 + 1=jP j) P jQ ½ µ ¾ ¶ 1 Á(Q) d 2 max ; logq deg Q q d ; +O jQj ln q. Res. where the implied constant does not depend on q and Q. Proof. De¯ne. ¶ Y0 µ 1 1¡ ; fQ (s) = jP js P jQ. we have L(s; ÂQ ) = fQ (s)³(s) and L4 (s; ÂQ ) ds ³ 4 (s) q = f 4 (s)q ds : L(2s; ÂQ ) fQ (2s)³(2s) Q.
(7) AN ASYMPTOTIC FORMULA FOR POLYNOMIAL DIVISOR FUNCTIONS. 7. Since 1 2 11 1 ¡4 ¡2 (s ¡ 1)¡1 + ¢ ¢ ¢ ; + 3 (s ¡ 1)¡3 + + 4 (s ¡ 1) 2 (s ¡ 1) ln q ln q ln q 6 ln q 2 1 q ¡ 1 2 ln q 2 ln q 4 ln3 q = + (s ¡ 1) ¡ (s ¡ 1)2 + (s ¡ 1)3 + ¢ ¢ ¢ ³(2s) q q q 3q and 1 1 = + O(1)(s ¡ 1) + O(1)(s ¡ 1)2 + O(1)(s ¡ 1)3 + ¢ ¢ ¢ ; fQ (2s) fQ (2) ³ 4 (s) =. where the implied constants do not depend on q and Q, by fQ1(2) · ³(2) · 2, we have ¶ µ ³ 4 (s) q¡1 1 1 1 ¡4 (s ¡ 1) + O (s ¡ 1)¡3 = fQ (2s)³(2s) q fQ (2) ln4 q ln3 q (4) µ µ ¶ ¶ 1 1 ¡2 +O (s ¡ 1)¡1 + ¢ ¢ ¢ ; (s ¡ 1) + O ln q ln2 q where the implied constants do not depend on q and Q. Since 0. 3 (1)fQ (1)(s ¡ 1) fQ4 (s) =fQ4 (1) + 4fQ ³ ´ 0 2 00 2 + 6fQ (1)fQ (1) + 2fQ3 (1)fQ (1) (s ¡ 1)2 ³ ´ 000 0 00 0 3 3 + 2fQ (1)fQ (1)=3 + 6fQ2 (1)fQ (1)fQ (1) + 4fQ (1)fQ (1) (s ¡ 1)3. and. + ¢¢¢. Á(Q) ; jQj X0 ln jP j 0 fQ (1) =fQ (1) ; jP j ¡ 1. fQ (1) =. P jQ. 00. fQ (1) = ¡ fQ (1). X0 P jQ. X0 ln jP j ln2 jP j 0 ; jP j + fQ (1) 2 (jP j ¡ 1) jP j ¡ 1. 13 X0 ln jP j A fQ (1) =fQ (1) @ jP j ¡ 1 000. 0. P jQ. P jQ. X0 ln jP j X0 µ ln jP j ¶2 ¡ 3fQ (1) jP j jP j ¡ 1 jP j ¡ 1 P jQ P jQ X0 µ ln jP j ¶3 + fQ (1) jP j(jP j + 1); jP j ¡ 1 P jQ.
(8) 8. CHIUNG-HSUN TU. by lemma 2.1, we get 4 (s) fQ. =. ¶ Á(Q) maxf1; ln (q deg Q)g (s ¡ 1) jQj µ ¶ Á(Q) 2 maxf1; ln (q deg Q)g (s ¡ 1)2 +O jQj ¶ µ Á(Q) maxf1; ln (q deg Q)g3 (s ¡ 1)3 + ¢ ¢ ¢ ; +O jQj. 4 fQ (1) + O. (5). µ. where the implied constants do not depend on q and Q. Combining (4), (5) with ln deg Q · d and q ds =q d + q d ln q d (s ¡ 1) +. q d ln2 q d q d ln3 q d (s ¡ 1)2 + (s ¡ 1)3 + ¢ ¢ ¢ ; 2 6. we obtain the lemma. Lemma 2.8. Let a 2 Fq£ . Then we have 0 X0. 0·deg Q· d2 ¡33 ln d. B B @. X0. deg f =d f ´a (mod Q). ¿22 (f ) ¡. 1. ³ ´ L4 (s; ÂQ ) ds C ln q Res q C = O qd ; Á(Q) s=1 L(2s; ÂQ ) A. where the implied constant does not depend on q. Proof. By lemma 2.6 and a 2 Fq£ , we have 1 Á(Q). X. Â(a).  (mod Q). = =. L4 (s; Â) L(2s; Â2 ) 1 Á(Q) 1 Á(Q). =. 1 Á(Q). =. 1 X d=0. X. Â(a). X0. Â(a). 1 X0 X. f.  (mod Q). X. Â(f ). ¿22 (f ) jf js Â(f ). ¿22 (f ) q ds. d=0 deg f =d 2 X ¿2 (f ) Â(f a¡1 ) q ds d=0 deg f =d  (mod Q).  (mod Q) 1 X0 X. X0. ¿22 (f )q ¡ds :. deg f =d f ´a (mod Q). Thus we know that X0. deg f =d f ´a (mod Q). ¿22 (f ).
(9) AN ASYMPTOTIC FORMULA FOR POLYNOMIAL DIVISOR FUNCTIONS. is the q ¡ds -term coe±cient of the power series 1 Á(Q). X. Â(a).  (mod Q). L4 (s; Â) L(2s; Â2 ). in q ¡s . Since (6). ln q 2¼i. Z. ¼i 2+ ln q. 1 Á(Q). ¼i 2¡ ln q. X. Â(a).  (mod Q). L4 (s; Â) ds q ds = L(2s; Â2 ). X0. ¿22 (f ). deg f =d f ´a (mod Q). and Z. 1 + 1 + ¼i 2 3d ln q. ¼i 2+ ln q. (7). =. Z. X. Â(a).  (mod Q). 1 + 1 ¡ ¼i 2 3d ln q. ¼i 2¡ ln q. X. L4 (s; Â) ds q ds L(2s; Â2 ). Â(a).  (mod Q). L4 (s; Â) ds q ds; L(2s; Â2 ). we have ln q Res Á(Q) s=1. (8). X.  (mod Q). X0. =. Â(a). L4 (s; Â) ds q L(2s; Â2 ). ¿ 2 (f ). deg f =d f ´a (mod Q). ln q 1 ¡ Á(Q) 2¼i. Z. 1 + 1 + ¼i 2 3d ln q 1 + 1 ¡ ¼i 2 3d ln q. X.  (mod Q). Â(a). L4 (s; Â) ds q ds: L(2s; Â2 ). When  6 = ÂQ and Re s > 1=2, we have X0. jL(s; Â)j = j. deg f <deg Q. Â(f )jf j¡s j < 1;. ¯ ¯ X0 ¯ ¯ 1 ¯ ¯=j ¹(f )Â2 (f )jf j¡2s j < 1: ¯ L(2s; Â2 ) ¯ f. Thus ln q Res Á(Q) s=1. X.  (mod Q). Â(a). L4 (s; ÂQ ) ds L4 (s; Â) ds ln q q Res q : = L(2s; Â2 ) Á(Q) s=1 L(2s; ÂQ ). 9.
(10) 10. CHIUNG-HSUN TU. Combining this with (8), we have X0 L4 (s; ÂQ ) ds ln q ¿22 (f )¡ Res q Á(Q) s=1 L(2s; ÂQ ) (9). deg f =d f ´a (mod Q). ln q = 2¼iÁ(Q) When s =. 1 2. Z. 1 + 1 + ¼i 2 3d ln q 1 + 1 ¡ ¼i 2 3d ln q. X. Â(a).  (mod Q). 1 3d. +. + it, we have ¯ ¯ ¯ ¯ ¯ X0 ¯ 2 ¯ ¯ ¯ 1 1  (f ) ¯¯ X0 ¯=¯ ¯ ¹(f ) 2s ¯ · 2 ¯ L(2s; Â2 ) ¯ ¯ jf j ¯ jf j1+ 3d ¯ f f =. 1 X0 X. 1. k=0 deg f =k. 2 (qk )1+ 3d. 1. =1+ q. 2 3d. < 4d: Combining this with (9) and by lemma 2.5 with ¾ · we have (10) 0 X0. 0·deg Q·N. =. X0. B B @. deg f =d f ´a (mod Q). X0. 0·deg Q·N. ¿ ln q. Z. 1 + 1 + ¼i 2 3d ln q 1 + 1 ¡ ¼i 2 3d ln q d. d 1 + 3 +N 2. ¿22 (f ) ¡. ln q 2¼iÁ(Q). 1. ¿ (ln q)q 2 + 3 d ¿q. L4 (s; Â) ds q ds: L(2s; Â2 ). Z. Z. 1 + 1 ¡ ¼i 2 3d ln q. 0·deg Q·N 1 + 1 + ¼i 2 3d ln q. 1 + 1 ¡ ¼i 2 3d ln q. and N =. d 2. 1. X. Â(a).  (mod Q). 1 Á(Q) X0. X.  (mod Q). 0·deg Q·N. 1 Á(Q). L4 (s; Â) ds q ds L(2s; Â2 ). jL4 (s; Â)j ds jq jj dsj jL(2s; Â2 )j X.  (mod Q). jL4 (s; Â)j jdsj. dN 21. ¿ qd ; where the implied constant does not depend on q. Lemma 2.9. We have X0 1 Y0 (1 ¡ 1=jP j)3 Á(Q) (1 + 1=jP j) deg Q=N P jQ ¶ µ ³ N´ Y0 1 (1 ¡ 1=jP j)2 1 + + O q¡ 2 ; 1¡ = jP j jP j (1 + 1=jP j) P. ¡ 33 ln d,. L4 (s; ÂQ ) ds C ln q Res q C Á(Q) s=1 L(2s; ÂQ ) A. 1 + 1 + ¼i 2 3d ln q. X0. 5 6. ¡1.
(11) AN ASYMPTOTIC FORMULA FOR POLYNOMIAL DIVISOR FUNCTIONS. 11. where the implied constant does not depend on q and N . Proof. We de¯ne. ¢(s) =. X0 Q. X0 1 Y0 (1 ¡ 1=jP j)2 Y0 (1 ¡ 1=jP j)3 1 = : Á(Q)jQjs (1 + 1=jP j) jQjs+1 (1 + 1=jP j) Q. P jQ. P jQ. This series converges absolutely for Re s > 0. Moreover, we have. X0. 1 Y0 (1 ¡ 1=jP j)2 jQjs+1 (1 + 1=jP j) Q P jQ ¶ µ Y0 1 (1 ¡ 1=jP j)2 1 (1 ¡ 1=jP j)2 + 2 s+1 + ¢¢¢ 1+ = jP js+1 (1 + 1=jP j) jP j (1 + 1=jP j) P à ¶! ¶¡1 µ Y0 µ 1 1 1 (1 ¡ 1=jP j)2 1¡ = 1¡ + jP js+1 jP js+1 jP js+1 (1 + 1=jP j) P ¶ ¶¡1 Y0 µ Y0 µ 1 1 1 (1 ¡ 1=jP j)2 = 1¡ 1¡ + jP js+1 jP js+1 jP js+1 (1 + 1=jP j). ¢(s) =. P. P. =³(s + 1)¤(s); where Y0 µ 1¡ ¤(s) = P. 1 (1 ¡ 1=jP j)2 1 + jP js+1 jP js+1 (1 + 1=jP j). ¶. :. Since ¯ X0 ¯¯ 1 1 (1 ¡ 1=jP j)2 ¯¯ ¯ ¯ jP js+1 ¡ jP js+1 (1 + 1=jP j) ¯ P. converges absolutely for Re s > ¡1, ¤(s) converges absolutely for Re s > ¡1. Thus we have. (11). Res¢(s)q N s = s=0. ¶ µ 1 (1 ¡ 1=jP j)2 1 1 Y0 + : 1¡ ln q jP j jP j (1 + 1=jP j) P.
(12) 12. CHIUNG-HSUN TU. Now we consider (ln q)Res¢(s)q. Ns. s=0. ln q = 2¼i. Z. +. ¢(s)q. ¼i 1¡ ln q ¡1 ¼i ¡ ln 2 q. ln q + 2¼i ln q = 2¼i. ¼i 1+ ln q. Z. Z. ¡1 ¼i + ln 2 q. ¼i 1+ ln q. ¼i 1¡ ln q. ln q 2¼i. Z. Ns. Z. ¼i 1¡ ln q. ¢(s)q N s ds. ¡1 ¼i ¡ ln 2 q ¡1 ¼i + ln 2 q. ln q ds + 2¼i. Z. ¢(s)q N s ds. ¼i 1+ ln q. 1 Y0 (1 ¡ 1=jP j)3 N ¡deg Q s (q ) ds Á(Q) (1 + 1=jP j). Q. ¡1 ¼i + ln 2 q. ln q ds + 2¼i. ¢(s)q. X0. ¡1 ¼i ¡ ln 2 q. Ns. P jQ. ³(s + 1)¤(s)qN s ds. X0. 1 Y0 (1 ¡ 1=jP j)3 Á(Q) (1 + 1=jP j) deg Q=N P jQ Z ¡1 ¡ ¼i 2 ln q ln q ³(s + 1)¤(s)qN s ds: + 2¼i ¡1 + ¼i. =. 2. Since. ¯ ¯ ¼i ¯ ¯ ln q Z ¡1 ¡ ln 2 q ¯ ¯ ³(s + 1)¤(s)q N s ds¯ ¯ ¡1 ¼i ¯ ¯ 2¼i + 2. ln q. ln q < 2¼. Z. ln q 2¼. Z. ¿. ¿q. ¡N 2. ¡1 ¼i ¡ ln 2 q. ¯ ¯ j³(s + 1)j j¤(s)j ¯q N s ¯ jdsj. ¡1 ¼i + ln 2 q ¡1 ¼i ¡ ln 2 q ¡1 ¼i + ln 2 q. (. q. ¡1 2. 1¡q. ¡1 2. )(q. ¡N 2. ) jdsj =. ¡N 1 ln q 2¼ ( 1 ) )(q 2 )( 2¼ q 2 ¡ 1 ln q. ;. we obtain (12). ln q. X0. deg Q=N. ³ ¡N ´ 1 Y0 (1 ¡ 1=jP j)3 = ln q Res¢(s)q N s + O q 2 ; s=0 Á(Q) (1 + 1=jP j) P jQ. where the implied constant does not depend on q and N . Combining this with (11) we complete the proof.. 3. Main Theorem The purpose of this section is to prove.
(13) AN ASYMPTOTIC FORMULA FOR POLYNOMIAL DIVISOR FUNCTIONS. 13. Theorem 3.1. Let a 2 Fq£ . Then we have X0. deg f =d. ³ ´ ¿22 (f )¿2 (f ¡ a) = Sqd d4 + O qd d3 maxf1; ln dg ;. where ¶ µ 1 (1 ¡ 1=jP j)2 1 q ¡ 1 Y0 + 1¡ S= 6q jP j jP j (1 + 1=jP j) P. and the implied constant depends on q. Proof. We have X0. deg f =d. ¿22 (f )¿2 (f ¡ a) =. X0. ¿22 (f ). X0. ¿22 (f ). deg f =d. +. X0. 1. 0·deg Q· d2 ¡33 ln d. (13) X0. 1. Qjf ¡a. deg f =d. =2. X0. f ´a (mod Q). X0. ¿22 (f ). deg f =d. d ¡33 ln d<deg Q< d2 +33 ln d 2. f ´a (mod Q). = 2S1 + S2 : By lemma 2.3 with ² = 0:1, we have X0. S2 =. d ¡33 ln d<deg Q< d2 +33 ln d 2. (14). ¿ q d d3. X0. X0. ¿22 (f ). deg f =d f ´a (mod Q). d ¡33 ln d<deg Q< d2 +33 ln d 2. 1 jQj. ¿ q d d3 ln d; where the implied constant depends on q. Now, we just have to estimate S1 . By lemma 2.8, we obtain S1 =. X0. 0·deg Q· d2 ¡33 ln d. (15) =. X0. 0·deg Q· d2 ¡33 ln d. X0. ¿22 (f ). deg f =d f ´a (mod Q). L4 (s; ÂQ ) ds ln q Res q + O(q d ); Á(Q) s=1 L(2s; ÂQ ). 1.
(14) 14. CHIUNG-HSUN TU. where the implied constant does not depend on q. Therefore, by lemma 2.7, we have X0 L4 (s; ÂQ ) ds ln q Res q Á(Q) s=1 L(2s; ÂQ ) d 0·deg Q· 2 ¡33 ln d. X0. =. 0·deg Q· d2 ¡33 ln d. 0. q ¡ 1 Y0 (1 ¡ 1=jP j)3 d 3 q d 6qÁ(Q) (1 + 1=jP j) P jQ. 1. X0. B +O@. 0·deg Q· d2 ¡33 ln d. =. X0. q¡1 d 3 q d 6q. 1 C max f1; ln deg Qg q d d2 A jQj. 0·deg Q· d2 ¡33 ln d. 1 Y0 (1 ¡ 1=jP j)3 Á(Q) (1 + 1=jP j) P jQ. + O(qd d3 ln d); by lemma 2.9, =. X. 0·N · d2 ¡33 ln d d 3. 0. B Sq d d3 + O @q d d3. X. 0·N · d2 ¡33 ln d. 1. NC q¡ 2 A. + O(q d ln d). =. X. Sq d d3 + O(qd d3 ln d). 0·N · d2 ¡33 ln d. 1 = Sqd d4 + O(qd d3 ln d); 2 where the implied constant does not depend on q and S. Thus, by (15) and the above inequality, we have ³ ´ S S1 = q d d4 + O q d d3 ln d ; 2 where the implied constant does not depend on q and S. Combining this with (13) and (14), the proof is complete. References [1] L. Carlitz, The Arithmetic of Polynomials in a Galois Field, American Journal of Mathematics 54 (1932), 39{50. [2] K. Chandrasekharan, Introduction To Analytic Number Theory, Spring-Verlag (1968). [3] H. Davenport, Multiplicative Number Theory, Springer-Verlag, GTM 74 (1980)..
(15) AN ASYMPTOTIC FORMULA FOR POLYNOMIAL DIVISOR FUNCTIONS. 15. [4] G. W. E±nger and D. R. Hayes, Additive Number Theory of Polynomials Over a Finite Field, Clarendon Press Oxford (1991). [5] Chih-Nung, Hsu, Estimates for Coe±cients of L-functions for Function Fields, Finite Fields and Their Applications 5 (1999), 76{88. [6] Chih-Nung, Hsu, A Large Sieve Inequality for Rational Function Fields, J. Number Theory 58 (1996), 267{287. [7] Chih-Nung, Hsu, A Polynomial Additive Divisor Promblem. [8] R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and Its Applications Vol. 20, Cambridge University Press Cambridge UK (1997). [9] Y. Motohashi, An Asymptotic Formula in Theory of Numbers, Acta Arith. 16 (1970), 255{264. [10] P. Shiu, A Brun-Titchmarsh theorem for multiplicative functions, J. reine angew. Math. 313 (1980), 161{170.. Department of Mathematics National Taiwan Normal University 88 Sec. 4 Ting-Chou Road Taipei, Taiwan E-mail address: redpony4783@ms84.url.com.tw.
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Particularly, combining the numerical results of the two papers, we may obtain such a conclusion that the merit function method based on ϕ p has a better a global convergence and
Arbenz et al.[1] proposed a hybrid preconditioner combining a hierarchical basis preconditioner and an algebraic multigrid preconditioner for the correc- tion equation in the
