The Positivity of a Sequence of Numbers and the Riemann Hypothesis

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1 joural of umber theory 65, (997) article o. NT97237 The Positivity of a Sequece of Numbers ad the Riema Hypothesis Xia-Ji Li The Uiversity of Texas at Austi, Austi, Texas 7872 Commuicated by A. Graville Received September 0, 996; revised December 6, 996 I this ote, we prove that the Riema hypothesis for the Dedekid zeta fuctio is equivalet to the oegativity of a sequece of real umbers. 997 Academic Press. THE RIEMANN ZETA FUNCTION Let [* ] be a sequece of umbers give by for all positive itegers, where (&)! * = d ds [s& log!(s)] s= (.)!(s)=s(s&)? &s2 s 2+ `(s) with `(s) beig the Riema zeta fuctio. Theorem. A ecessary ad sufficiet coditio for the otrivial zeros of the Riema zeta fuctio to lie o the critical lie is that * is oegative for every positive iteger. Proof. Defie.(z)=! &z+ =4 [x 32 $(x)]$ (x &2 x 2(&z) +x &2(&z) ) dx (.2) X Copyright 997 by Academic Press All rights of reproductio i ay form reserved.

2 326 XIAN-JIN LI for z i the uit disk, where Write (x)= = e &? 2x.!(s)=` &s (.3) + where the product is take over all otrivial zeros of the Riema zeta fuctio with ad & beig paired together. It follows that.(z)=` &(&())z. &z A ecessary ad sufficiet coditio for the otrivial zeros of the Riema zeta fuctio to lie o the critical lie is that.$(z).(z) is aalytic i the uit disk. Put for z < 4, where.$(z).(z) = * = _ * + z & & (.4) + & for every positive iteger. O the other had, by (.3) we have (&)! d ds [s& log!(s)] s= =& = _ & k=0 k+ (&)k& & & + ad hece * is also give by the expressio (.). Let We fid that * = l=.(z)=+ (&) l& l j= k,..., k l k +}}}+k l = &, a j z j. (.5) a k }}}a kl

3 THE RIEMANN HYPOTHESIS 327 for every positive iteger. Expadig the right side of (.2) i power series (.5), we fid that a j =2 ( j+)}}}(j+)! (+)! 2 for every positive iteger j. By (.6) we ca write a j =4 = 4 j! d j (+j)}}}(+) j!(+)! 2 + dt j{ t j& [x 32 $(x)]$ (x &2 +(&) + )(log x) + dx (.6) [x 32 $(x)]$ (x &2 +(&) + )(log x) + dx (t2) + (+)! [x 32 $(x)]$ _(x &2 +(&) + )(log x) + dx =t= = 4 j! d j dt j{ t j& [x 32 $(x)]$ (x &2 [e (t2) l x &]+[e &(t2) l x &]) =t= = 4 j! d j dt j{ t j& j =4 l= j& j&l+ l! [x 32 $(x)]$ (x &2 e (t2) l x +e &(t2) l x ) =t= l [x 32 $(x)]$ 2 log x [+(&) + l x &2 ] dx. This expressio implies that a j is a positive real umber for every positive iteger j. Sice the idetity = a z & = i=0 holds, we have the recurrece relatio & * =a & a i z + i j=0 j= * j a &j * j+ z j+ for every positive iteger. By (.), * is a real umber for every positive iteger. If the otrivial zeros of `(s) lie o the critical lie, the &() = for every otrivial

4 328 XIAN-JIN LI zero of `(s). Put &()=exp(i% ) for some real umber %. The by (.4) we have * = (&e i% )= (&cos % ). This implies that the umber * is oegative for every positive iteger. Coversely, if the umber * is oegative for every positive iteger, the * a for every positive iteger. It follows that = * z & = a z & =.$( z )< for z i the uit disk. This implies that.$(z).(z) is aalytic i the uit disk. This completes the proof of the theorem. 2. THE DEDEKIND ZETA FUNCTION Let k be a algebraic umber field with r real places ad r 2 imagiary places. The Dedekid zeta fuctio `k(s) ofkis defied by `k(s)=` (&Np &s ) & p for Re s>, where the product is take over all the fiite prime divisors of k. Put G (s)=? &s2 (s2) ad G 2 (s)=(2?) &s (s). Defie Z k (s)=g (s) r G2 (s) r 2 `k(s). By Theorem 3 of Chapter VII, Sectio 6, of [4], the fuctio Z k (s) is aalytic i the complex plae except for simple poles at s=0 ad s=, ad satisfies the fuctioal idetity Z k (s)= d (2)&s Z k (&s) where d is the discrimiat of k. Its residues at s=0 ad s= are respectively &c k ad d &2 c k with c k =2 r (2?) r 2 hre, where h, R, ad e are respectively the umber of ideal classes of k, the regulator of k, ad the umber of roots of uity i k. Let! k (s)=c & k s(s&) d s2 Z k (s). The! k (s) is a etire fuctio ad! k (0)=.

5 THE RIEMANN HYPOTHESIS 329 Let [* ] be a sequece of umbers give by (&)! * = d ds [s& log! k (s)] s= for all positive itegers. The aim ow is to prove the followig theorem. Theorem 2. A ecessary ad sufficiet coditio for the otrivial zeros of the Dedekid zeta fuctio `k(s) to lie o the critical lie is that * is oegative for every positive iteger. Lemma PROOF OF THE THEOREM 2 The idetity * = & & + + holds for every positive iteger, where summatio is take over all otrivial zeros of the Dedekid zeta fuctio `k(s) with ad & beig paired together. Proof. By Theorem 2 of Barer [], we have the formula (cf. Chapter 2 of [2])! k (s)=` &s +, (3.) where the product is take over all zeros of! k (s) with ad & beig always paired together. A argumet similar to that made for the Riema zeta fuctio i Chapter 2 of [2] shows that the covergece of the product (3.) is uiform o compact subsets of the complex plae. Sice! k (s)=! k (&s), we have d ds [s& log! k (s)] s= =(&) d ds [(&s)& log! k (s)] s=0. (3.2) Sice `k(s) does ot vaish at s=0, we ca write log! k (s)=& m= &m m sm (3.3) where s <= for a sufficietly small positive umber =, where ad & are paired together i the summatio over. Sice the product (3.)

6 330 XIAN-JIN LI coverges uiformly, the series (3.3) coverges uiformly for s <=. It follows that (&)! d ds [(&s)& log! k (s)] s=0 =& m= (&) &m This formula together with (3.2) implies the stated idetity. Defie.(z)=! k &z+ m+ &m K. for z i the uit disk. Sice the fuctio! k (s) is aalytic i the complex plae of s, the fuctio.(z) is aalytic i the uit disk. Lemma 3.2. Let.(z)=+ j= a j z j. The the coefficiet a j is a positive real umber for every positive iteger j. Proof. Defie = v to be oe whe v is a real place of k ad to be two whe v is a imagiary place of k. Let x=> x v be the variable i the half space R r +r 2. Deote by x the product + >x= v v, which is take over all ifiite places of k. IfN=r +2r 2, the the Hecke theta fuctio 3 k (x) is defied by 3 k (x)= exp &? d &N (Nb) 2N b v = v x v+ where the summatio over b is take over all ozero itegral ideals of k ad where the summatio over v is take over all ifiite places of k. Put dx=>dx v. It follows from Theorem 3 of Chapter XIII, Sectio 3, i [3] that! k (s)=+c & k s(s&) x 3 k (x)( x s2 + x (&s)2 ) dx x. (3.4) Let 3 k (x)( x 2(&z) + x 2 x &2(&z) ) dx x x = b m z m. (3.5) m=0

7 THE RIEMANN HYPOTHESIS 33 It is clear that b 0 is a positive umber. We have b m = (m+)}}}(m+)!(+)! 2 + x 3 k (x)(+ x 2 (&) + )(log x ) + dx x for every positive iteger m. By computatio, we fid that b m = m! (+m)}}}(+) (+)! 2 + _ 3 k (x)(+ x 2 (&) + )(log x ) dx + x x d m dt m tm& 3 k (x)(e (t2) log x + x 2 e &(t2) log x ) dx. x x +t= = m! It follows that m b m = l= m& m&l+ l! l 3 k (x) x 2 log x ( x + 2 +(&) l ) dx x (3.6) for every positive iteger m. Sice 3 k (x) is positive for every x i R r +r 2 +, it follows from (3.6) that the coefficiets b m are positive real umbers for all oegative itegers m. The idetity z (&z) 2= q= holds for z i the uit disk. It follows from (3.4) ad (3.5) that j& c k a j = m=0 qz q ( j&m)b m (3.7) for every positive iteger j. Sice b m are positive umbers for all oegative itegers m, we see that a j is a positive real umber for every positive iteger j. K Proof of the Theorem. Sice! k ()= ad! k (s)=! k (&s), it follows from the product formula (3.) that.(z)=` &(&())z. (3.8) &z

8 332 XIAN-JIN LI Sice! k (s) does ot vaish at s=, we ca write.$(z).(z)= * + z (3.9) by usig the formula (3.8) whe z <= for a sufficietly small positive umber =. Sice we have = a z & = i=0 & * =a & a i z + i j=0 j= * j+ z j+, * j a &j (3.0) for =2, 3,..., where * =a ad a 0 =. If the otrivial zeros of `k(s) lie o the critical lie, it follows from Lemma 3. that the umbers * are oegative for all positive itegers. Coversely, assume that the umber * is oegative for every positive iteger. It follows from (3.0) ad Lemma 3.2 that * a for every positive iteger. This iequality together with Lemma 3.2 implies that = * z & = a z & =.$( z ) (3.) for z i the uit disk. Sice.$(z) is aalytic i the uit disk,.$( z ) is fiite for z i the uit disk. It follows from (3.9) ad (3.) that.$(z).(z) is aalytic i the uit disk. It is clear that a ecessary ad sufficiet coditio for the otrivial zeros of the Dedekid zeta fuctio `k(s) to lie o the critical lie is that.$(z).(z) is aalytic i the uit disk. Therefore, the otrivial zeros of the Dedekid zeta fuctio `k(s) lie o the critical lie. This completes the proof of the theorem. K Remark. We kow from the proof of Theorem 2 that * =a, which is a positive umber by Lemma 3.2. A explicit expressio for * is implicit i the recurrece relatio (3.0) together with formulas (3.6) ad (3.7).

9 THE RIEMANN HYPOTHESIS 333 ACKNOWLEDGMENT The author thaks the referee for valuable suggestios o improvig the presetatio of the origial versio of this paper. REFERENCES. K. Barer, O A. Weil's explicit formula, J. Reie Agew. Math. 323 (98), H. M. Edwards, ``Riema's Zeta Fuctio,'' Academic Press, New York, S. Lag, ``Algebraic Number Theory,'' AddisoWesley, Readig, MA, A. Weil, ``Basic Number Theory,'' Spriger-Verlag, Heidelberg, 967.