DISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P Tsz Ho Chan Dartmnt of Mathmatics, Cas Wstrn Rsrv Univrsity, Clvland, OH 4406, USA txc50@cwru.du Rcivd: /9/03, Rvisd: /9/04, Acctd: 3/4/04, Publishd: 3/5/04 Abstract Lt > b a rim numbr. For ach intgr 0 <n<, dfin n by th congrunc nn (mod with 0 < n<. W ar ld to study th distribution bhavior of n n in ordr to rov th asymtotic formula n n = 3 O( 3/ log 3.. Introduction For any rim and any intgr 0 <n<, thr is on and only on n with 0 < n< satisfying nn (mod. W ar intrstd in how th invrss n fluctuat as n runs from to. In articular, w look at th sum S = n n ( For any intgr k, on can asily show that thr ar at most solutions to th congrunc quation n n k (mod. From this, w hav 8 [/4] 4k S n= [/4] 4k 7 8 7 whr [x] dnots th gratst intgr x. So, S has ordr and th natural qustion is whthr lim S / xists. To this, w hav th following.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (004, #A03 Thorm. For any rim >, S = n n = 3 O( 3/ log 3. Hnc, on avrag, th diffrnc btwn invrss of conscutiv intgrs, n n, is about /3. Mor gnrally, w hav th following rsult. Thorm. For any rim > and λ>0, n n λ = (λ (λ λ O( λ/ log 3. To rov Thorm or, w ar ld to study T (, k =#{n :0<n<,0 < n n k}, T (, k =#{n :0<n<, k n n < 0}, and T (, k =T (, kt (, k =#{n :0<n<,0 < n n k} for intgr 0 <k<. W hav Thorm 3. For any rim > and any intgr 0 <k<, T ± (, k =k k O(/ log 3. Our mthod of roof is motivatd by Profssor W.P. Zhang s ar [5] involving Kloostrman sums and trigonomtric sums. Th roofs of Thorms, and 3 xtnd to airs n and n l for fixd intgr l 0 (mod with slight modifications.. Exonntial and Kloostrman sums First, w start with som notation. Ltting (y = πiy, w dnot th Kloostrman sum S(m, n; q := d (mod q (d,q= ( md nd q Lmma. Lt b a rim numbr. For any intgr a and b 0 (mod, x= ( axb x(x± / whr is ovr all x (mod xct th roots of x(x ± in F..
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (004, #A03 3 Proof. It follows from th Bombiri-Wil bound [] for xonntial sums in th form by Morno and Morno [4, Thorm ] rovidd that axb is not of th form x(x± h(x h(x with h(x F, whr F is th algbraic closur of F. This is tru in our situation. For othrwis, say ax b ( F (x x(x ± = F (x G(x G(x with F (x and G(x olynomials ovr F such that (F (x,g(x =. Thn (ax bg(x = x(x ± F (x(f (x G(x. ( Sinc F (x and G(x ar rlativly rim, w hav G(x x(x ±. This forcs G(x to b a nonzro constant olynomial. Contradiction occurs if on comars th dgrs in (. Lmma. Lt > b a rim numbr. For any intgrs r and s, ( ±n S(r, n; S(s, n; 3/. (3 Hr S(s, n; stands for th comlx conjugat of S(s, n;. Proof.. r 0 (mod. Thn S(r, n; = for n. So, th lft sid of (3 is ( ±n = ( sd = ( sd nd ( n(d. s 0 (mod. Similar to cas. d ±(.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (004, #A03 4 3. r s (mod and (r, ==(s,. Th lft hand sid of (3 ( ±n ( ra na ( sb nb = a= ( ra sb = ( rc sd = c= =( = = b= ( (a b ± n ( (c d ± n ( rd sd ( rc sd ( rd sd ( rc c= c= c d ( sd ( [(r sd ± s]d(d by Lmma. Hr mans summation ovr all d with (d, ==(d,. Not: Th first sum in th scond last lin is a scial cas of Cobli and Zaharscu [, Lmma ] or Cobli, Gonk and Zaharscu [3, Lmma ]. 4. r s 0 (mod. Similar to cas 3. 3. Thorm 3 whn 0 <k</ Now, w try to xrss T ± (, k as xonntial sums similar to [5]. Lmma 3. For any rim > and any intgr 0 <k</, T ± (, k = ( n k ( mt [ k ( ±ma na a= k b= k ( ±ma na ]. Proof. W shall only rov th formula for T (, k; th roof for T (, k is similar. Obsrv that T (, k =#{a :0<a b k, b a =}. Thus, as ( nr {, r; = (4 0, r,
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (004, #A03 5 T (, k = k a b=t,b a= = k ( m(a b t a>b = ( n(b a ( n k ( mt [ k ( ma na a= k b= k a>b ( ma na a>b ( mb nb ] ( mb nb (5 Not that w do not nd th condition b>a bcaus b a which dos not allow b a = (th only altrnativ bsids may b countd. Now t k. It is valid to dro th condition a>bin both doubl sums within th brackts bcaus k < ( k a b and k < k <( k ( a b rsctivly (not: k</is usd for th scond chain of inqualitis. Hnc, only a b = t is countd vn without condition a>band w hav th lmma. Lmma 4. For any rim > and any intgr 0 <k<, ( n k ( mt =k( ko( 5/ log. ( ±ma na k Proof. W sarat th lft hand sid into thr ics according to: (i m = n =, (ii n = and m, (iii m and n. Th lft hand sid of Lmma 4 =k( ko( ( n k ( mt k ( mt k =k( ko( S S. ( ±ma k ( ±ma na ( mb (6 S k ( mt k ( mb b= sin (πm/ m (7
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (004, #A03 6 by summing th gomtric sris and sin πx x for 0 x /. By (4, = = k ( ±ma na ( ±ma na r= k ( cr r= Hnc, by Lmma, c= ( r(b c k c= S(r m, ±n; S( m, ±n;. S / k r= k ( mt k ( cr 3/ r= c= [ sin (πr/ / 3/ sin (πm/ [[/] / Combining (6, (7 and (8, w hav th lmma. k ( mt 3/ ] 3/ k sin (πm/ ] [/] 3/ m m 5/ log. (8 Lmma 5. For any rim > and any intgr 0 <k</, ( n k ( mt = k O( 5/ log 3. a= k b= k ( ±ma na Proof. Similar to Lmma 4, w slit th lft hand sid into thr ics which =k 3 O( k ( mt ( n k ( mt =k 3 O( S S. a= k b= k a= k b= k ( ±ma ( mb ( ±ma na (9
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (004, #A03 7 Rlacing a by a and b by b, S = = = k ( mt k k k ( ma k k ( ±mb ( ±m(b a t k 3 O(k k k k k 3 O(k = k k 3 O(k b a=t (0 bcaus w may count b a = t or b a = t but th scond cas is not ossibl as t k < k b a. Hr k</ is crucial. Alying (4 twic, a= k b= k = = r= s= c= k r= s= ( ±ma na ( ±ma na ( r(a c d= k ( s(b d K(rK(sS(s m, ±n; S( m r, ±n; whr K(x = n= k ( nx/. Rarranging th sums and alying Lmma, S = k ( mt r= s= K(rK(s ( n k ( mt K(r / S(s m, ±n; S( m r, ±n; r= K(s [ [/] ] 3 k / sin (πm/ (k log / 3 5/ log 3. s= Combining (9, (0 and (, w hav th lmma. (
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (004, #A03 8 To rov Thorm 3 whn 0 <k</, aly Lmma 4 and 5 to Lmma 3 and gt T ± (, k = [k( k ] k O( 5/ log 3 = k k O(/ log 3 which givs Thorm 3 in that rang of k. 4. Thorm 3 whn / <k< Bfor rocding, lt us introduc som notation. U (, k =#{n :0<n<, k n n <}, U (, k =#{n :0<n<, <n n k}, V (, k =#{n;0<n<,n n t (mod for t k}, V (, k =#{n;0<n<,n n t (mod for t k}. Thn on can asily s that V (, k =T (, ku (, k and V (, k =T (, ku (, k. ( Lmma 6. For any rim > and any intgr 0 <k<, V ± (, k = ( n k ( mt ( ±ma na Proof. Us (4. Not: it is asir than Lmma 3 bcaus on dos not nd to worry about a>bor a<b. Lmma 7. For any rim > and any intgr 0 <k<, ( n k ( mt =k O( 5/ log ( ±ma na. Proof. Similar to Lmma 4, w slit th lft hand sid into thr ics which =k O( k ( mt ( n k ( mt =k O( S S. ( ±ma ( ±ma na ( mb (3
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (004, #A03 9 By (4, Aftr rarranging summations, S k. (4 S = 3/ k ( mt ( n k ( mt 3/ [k [/] S( m, ±n; S( m, ±n; ] 3/ (k log 5/ log sin (πm/ (5 by Lmma, summing th gomtric sris and sin (πx x. W hav th lmma by (3, (4 and (5. Lmma 8. For any rim > and any intgr 0 k</, U ± (, k = k O(/ log 3. Proof. First not that whn k =0,U ± (, k = 0. Now assum k > 0. From (, U ± (, k = V (, k T (, k. Thus, by Lmma 6, 7 and Thorm 3 in th rang 0 <k</, U ± (, k =k O( / log (k k O(/ log 3 which givs th lmma. To rov Thorm 3 whn / <k<, obsrv that 0 k </ and ( T ± (, k =T ±, ( [U ±, ] U ± (, k = ( [ ( ( k ] O( / log 3 = k k O( O( / log 3 =k k O(/ log 3 by Thorm 3 in th rang 0 <k</ and Lmma 8.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (004, #A03 0 5. Proof of Thorms and By Thorm 3, T (, k =k k /O( / log 3 for intgr 0 <k<. Mor gnrally, T (, u =#{n :0< n n u} =u u O(/ log 3 (6 for any ral numbr 0 u. Using th Rimann-Stiltjs intgral, intgration by arts, and (6, n n λ = 0 u λ dt (, u =T (, λ λ = λ O( λ λ u λ uλ O(u λ / log 3 du 0 [ λ = λ λ λ λ ] λ λ O( λ/ log 3 = (λ (λ λ O( λ/ log 3 which givs Thorm and hnc Thorm. 0 u λ T (, udu Numrical calculations (by a C rogram suggst that th rror trm in Thorm 3 is robably bst ossibl xct for th xtra logarithms. Lt m ± = max k< T ± (, k (k k. 0 03 07 09 3 7 3 37 m / 0.495 0.603 0.9794 0.7504 0.4595 0.9 0.633 0.640 m / 0.595 0.7098 0.6993 0.4956 0.48 0.7976 0.9067 0.546 550 5503 5507 559 55 557 553 5557 m / 0.5786 0.6384 0.70 0.6350.337 0.458 0.698 0.750 m / 0.5945 0.653 0.734 0.5888.0775 0.46 0.79 0.7530 30 33 39 343 347 373 377 379 m / 0.549 0.6978 0.866.046.5 0.6540 0.87 0.694 m / 0.55 0.6745 0.80.0346.069 0.6789 0.844 0.6778 6003 6007 6009 60037 6004 60077 60083 60089 m / 0.57 0.7776 0.5806 0.865 0.844 0.9457 0.4469 0.59 m / 0.53 0.7789 0.579 0.84 0.8489 0.954 0.4373 0.5897
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (004, #A03 W conclud with th following Conjctur. m ± / and m m = o( /. Not: Aftr finishing this ar, w cam across Cobli and Zaharscu [], and found that it is far mor gnral and would giv T ± (, k =k k O(5/6 log /3. Hnc thir mthod givs our rsults xct for a biggr rror trm. Rfrncs [] E. Bombiri, On xonntial sums in finit filds, Amr. J. Math. 88 (966, 7-05. [] C.I. Cobli and A. Zaharscu, Th ordr of invrss mod q, Mathmatika 47 (000, 87-08. [3] C.I. Cobli, S.M. Gonk and A. Zaharscu, Th distribution of attrns of invrss modulo a rim, J. Numbr Thory 0 (003, 09-. [4] C.J. Morno and O. Morno, Exonntial sums and Goa cods. I, Proc. Amr. Math. Soc. (99, no., 53-53. [5] W.P. Zhang, On th Distribution of Invrss Modulo n, J. Numbr Thory 6 (996, 30-30.