Some Problems on the Distribution of the Zeros

Transcription

1 Some Problems on the Distribution of the Zeros of the Riemann Zeta Function 1

2 ~.. J;, 'J: x'v ~) -'" f ""'-~ ~-~--, :>~ - R;~.Q..." )\A,("',! ~~~~ ~ ~~~~ G... r-o r~ --

3 Riemann \ 2

4 00 1 (( s) == 2:: ~ n=l == II (1 - p 1 ps -1 for 7f +00 L n=-oo e -- -'K~21: 1 II!.fi ~ +cx:> n=-cx:> e -7rn2fx Thus we have <X> X 28-1 L n=l ( X2 ls-1 + lcxj x 2 2 e Ln=l 3

5 for for except at and we have 1 28(8 1 s( -e 2 tco -1+ ~ lo,g( 47i"» II p where Co is the Euler constant and we denote the nontrivial zeros by p -,8 + 'l1'.

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7 The first forir.1ul(!). represents an explicit relation between the primes an~ the zeros. For X > 1, we have and - L pn<x 6

8 7r Li(X ~dt+ for 0->1- c 7

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10 p 1= 100 co L logp L (f(pm) + j*(pm)) p m=l =T(f], Weil's criterion on C~(R+ 100 Yoshida Bombieri 8

11 The second formula is an explicit formula for the number N(T) of the zeros,b + i, in 0 <, :::; T and 0 <,B < 1. It states that II w here ~~ib~ 4.. I II..- ~d t9( T) is t:i:ntinuosiy differentiable function defined T 2 7r

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13 XP T > To and let if 0 otherwise. LO<,~T Thus the above sum picks up the prime powers. It is interesting to refine Landau's theorem. Under the Riemann Hypothesis (R.H.), we could find the first and the second oscillating terms: for any X > 1 L O<,~T Xti = + 27r A further oscillation must be hidden in the remainder term. Another important problem is to clarify the explicit dependence on X in Landau's theorem. Hlawka Gonek Fujii 10

14 Remark. a,n Rademacher Elliot Hlawka Fujii a"yn ~definition lim ~~{n ~ Njainmod one E [O,t})} = 1] N-tooN for any.{==?-weyl critfrion l=: ')'~T 11

15 Hlawka 1 Dr( -log X 27r == 0 (ll~gl~g-t ) ( On R.H. DT ( LlogX = O(~IOgX.) for all integers X > 1 21[".0 T. ~~fjt1 -",J ~. cell. J'~~~"~.....{i>.~, """-"~-~ J...~ ',; -..A mean value Itheorem on S(t + h To see what should be the correct order, we have assumed R.H. and prpved, among others, the following results. I 12

16 !~~~~~! 1 Dr(2; log X ~\2 -~ T log 2-;:e vx log JX-=l + O(!~ ~~~.! log(x + 3)) T2 log T 1 log T + O(T"fi log(i;gx 1 1 ~T.1oglogTlog(rog-T.l~glogT + 3) If X = 1 + OCT " logt 1 log T + OC T " log( ~ log(x +13}) I 1 1 ~T.log logt log( r~l~giog-t + 3) 13

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18 1 DT(2; - p o( iait ) 15

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20 L( We shall collect our results related with this problem LI "Y~T I 1 -}-- 27r 2 L,~T al}2 27r -{a2; I '"Y~T integer n(:;:::: 1) such that e na is a prime power, or ~ if such n does not exist.,

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22 , the most interest- L O<f~T X-ib'eib,log ~ This kind of sum was studied first by Hardy and Littlewood under R.H. I l=: O</~T e ~ O<'Y~T eib'y log -2:1-2 b I.!:...;a ~ea *-e4.~ b e(-ant 19

23 L 2:1 o<""(~~ This implies, by Weyl criterion, that bin log ~ is uniformly distributed mod 1 for any b > 0 and for any Dositive Q. II ~~ ~ ~~l ~

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25 Cor.4 (On R.H. The Generalized Riemann Hypothesis for all Dirichlet L-function L( s, X) with a Dirichlet character X mod q ~ 2 is equivalent to the relation L O<'Y~T i,log "1:-;~ e 00 L xc n ) n=l ns 22

26 Infinite series version was obtained by Sprindzuk by extending Linnik. On R.H. The Generalized IRiemann Hypothesis for all Dirichlet L-function L( s, X) with a Dirichlet character X mod q ~ 2 is equivalent to the relation L"Y 'Y1(x + 27ri~',-~-i'Y == -,j q e ), Thus we see that the vertical distribution of the zeros of «( s) is really important for the Generalized Riemann Hypothesis for Dirich~et L-functions. 23

27 We can add soine more connections in our present context. (On R.H. mod q > 1, 21(" - lim T-+oo T ~ O<'Y~T 1+ ' q2 'l'y f(x) L( ~ + i" x) -1 2 lim T-+oo 27r T L 0 <,( t/j) ~T the zeros of 24

28 -- ~ ~ ~~~ 1o, ' at """- GrR

29 6. The distribtion Dirichlet L-functions. of the zeros of different llo g.t 47r 27r We start with Gonek's result (under R.H.) for L 0 <'Y"::;T «(~ + i(, + 27r sin 7r0:' )2} ~ log2 27r T 27r Our refinement under R.H. L O<'Y~T 1 log2 - ((2 + i(, + ~! + 2[ -1Lt- Co + (1-2Co) sin(~7ra) I ~ -" ))] T T 27rCY. 27rG: +?RCtCl Ii' + i log L -2 log - 27ra: log.t )) 12- -{I 2 I -( 7ra fll ~c 2 7r 7r 27r ~~- 27r 25

30 ) A further extension under the Generalized Riemann Hypothesis For any primitive Dirichlet character X mod q? 1 and 1jJ mod k? 1, for any a > 0 and for any X? 1, we have II lim T T-..oo - 27r 1 27ra 2 L xi'y(tjj) log T O<'Y(t/J)~T /2 x(a)x(q) VA'Q if 1ra x = ~ with integers A ~ Q ~ 1, 0 if (A,Q)=l X is irrational, 0 otherwise.

31 We recall another example. lim T-+cx:> l~~:~~:!~j:~~:~~~:~~~:~:1 The distribution of the zeros of different Dirichlet L- functions are independent. Dirichlet L-functions have no com- LE~~~~ Different primitive mon zeros. a positive proportion of the zeros of two different primitive Dirichlet L-functions are non-coincident. 27

32 ing S(T). We shall describe some of the known results concern- S(T) Littlewood On R.H.) Landau, Bohr-Landau, Littlewood 28

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34 Mean value theorems it.).,"-jj. This enables us to determine the distribution function of S(t). Under R.H., Selberg also showed that for each j == 1,2,3,..., it 2 " J. 1 30

35 , it + T 21[2 cx:> LL m=2 p + _.!.-+ m is the F( a) =: F( a, T) =:,, and " running ov~r the imaginary parts of the zeros of I (( s), a ~ 0 and II t2' 31

36 (S(t Mean value theorems for short intervals For 0 < 1~~- ~ T log '2-;:- it 27ra + T log- 27r, (2~~~kf2kT(~~) =. Ci(27ra) + C~k +.J(T(Ak)k((log(27ra) -Ci(27ra) + Co)k-t + kk) I if 0 < a~lo~,t I «fi il!:t)'~~k +O(T(Ak)k((loglogT -log I ((I + i~ g7rl) I 2'K if Ci( x) == 100 ~dt. 32

37 Whep a < a ~ 1, this formula does not give an asymptotic formula. However, we can recover it, applying Goldston, for the case of k --: 1, under the Riemann Hypothesis, as follo!ws. Then we have it L 7r2 VJO 27r0:' log~ fjr~7rci 1 ~cos a a if if 33

38 1 a + 0 ( r~~~~~~! v~)+ + 0 (r!~~~~~! T V ~ ))(2; T is bounded. for a~l 34

39 Thus if we assume R.H. and the Montgomery's conjecture on F( a), then we get a finer asymptotic formula for the case for 0 < a = o(log T), which should be co~pared with Berry's conjecture which will be described belpw.,under the Riemann Hypothesis and Montgomery's Conjecture) I Suppose that 0 < 127r~ <::g: T. og fi Then we have it ~~ 27rG' T' log 2'1r"1 if 0 < a = o(log T) ~ ~O~- I;::g I «1~~i:t ~+ 0 ( 1 if logt ~ a ~ TlogT. Here we shall describe a conjecture proposed bjt Berry in (19) of p.402 of Berry, with a slight change of notlations, I as follows.5'\),*9 :.I,cc;f 35

40 + 7r2a -cos(27ra Co} 11o -) 1 sin2( 7rar1~~og 2~ +-{2 7r2 L P r «2:;:- T ) 7"*,r->1 r2pr 1+ Ci(27raT* where p runs over the prime numbers, r runs Qver the integers, B( t) being the inverse function for t > tb of the function.i I satisfies and Si(x) = 36

41 (1 (~)2) We see that th~ right hand side is exactly the right hand side of the mean value of S(t+ ~)-S(t) descrlibed above L O<-YI-y/~T O<-y_-y/~ ~r log 2'K 1= T 27r log T.{1Q' - This is equivalent to L O<l~T 3('+ 1 27ra..T.- og 27r T 27f 10' 7rt dt+o(l)} We have shown, without assuming any unproved hy~othesis, that L S(~+ O<"Y~T 27ra T log 2""1r" 37