UPPER BOUNDS FOR RESIDUES OF DEDEKIND ZETA FUNCTIONS AND CLASS NUMBERS OF CUBIC AND QUARTIC NUMBER FIELDS

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1 MATHEMATICS OF COMPUTATION Volume 80, Number 275, July 20, Pages S Article electronically published on January 25, 20 UPPER BOUNDS FOR RESIDUES OF DEDEIND ZETA FUNCTIONS AND CLASS NUMBERS OF CUBIC AND QUARTIC NUMBER FIELDS STÉPHANE R. LOUBOUTIN Abstract. Let be an algebraic number field. Assume that ζ s/ζs is entire. We give an explicit upper bound for the residue at s =ofthe Dedekind zeta function ζ s of. We deduce explicit upper bounds on class numbers of cubic and quartic number fields.. Introduction Let be an algebraic number field of degree m = r +2r 2 >, where r is the number of real places of and r 2 is the number of complex places of. Let κ be the residue at s = of the Dedekind zeta function function ζ s of. Letd be the absolute value of the discriminant of. Leth be its class number. Then see [Lan, Chapter XIII, Section 3, Theorem 2]: w d h = 2 r 2π r 2Reg κ, where w 2 is the number of complex roots of unity in and Reg is the regulator of. To get upper bounds on h we need lower bounds on Reg e.g., see [Sil] and upper bounds on κ e.g., see [Lou00]. If is a real quadratic number field, then 2 h 2 d [Le] and [Ram, Corollary 2]; if is a real cyclic cubic number field, then 3 h 2 3 d see [MP], and use [Lou93] instead of [MP, Lemme 3.2] to obtain that this bound is valid for real cyclic cubic number fields of not necessarily prime discriminants. With e = exp, it is known that m e log d 4 κ 2m Received by the editor November 25, 2009 and, in revised form, June 5, Mathematics Subject Classification. Primary R42; Secondary R6, R29. ey words and phrases. Dedekind zeta function, number field, class number. 83 c 20 American Mathematical Society

2 84 STÉPHANE R. LOUBOUTIN [Lou00, Theorem ] and [Lou0, Theorem ]. If is abelian, wehaveabetter bound: m log d + mλ 5 κ, 2m where λ =0if is real and λ =5/2 log 6 if is imaginary use [Ram, Corollary ] and notice that if is imaginary, then m/2 ofthem characters in the group of primitive Dirichlet characters associated with are odd. For some totally real number fields, an improvement on 4 is known see [Lou0, Theorem 2]: if ranges over a family of totally real number fields of a given degree m> for which ζ s/ζs is entire, there exists C m computable such that d C m implies m 6 κ logm d 2 m m! e log d. 2πm 2m It is known that ζ s/ζs is entire if is normal see [MM, Chapter 2, Theorem 3], or if the Galois group of its normal closure is solvable see [Uch], [vdw] and [MM, Chapter 2, Corollary 4.2], e.g., for any cubic or quartic number field. This paper generalizes 6 to not necessarily totally real number fields: Theorem. Let r and r 2 be given, with r +2r 2 3. Thereexistsd r,r 2 effectively computable such that for any number field of degree m = r +2r 2 with r real places and r 2 complex places, we have 7 κ logm d 2 m m!, provided that i d d r,r 2 and ii that ζ s/ζs is entire. For given r and r 2, we will explain how to use any mathematical software, we use Maple, to compute such a d r,r 2. It appears that for the small values of r +2r 2 = m, sayfor3 m 6, this bound 7 holds true with no restriction on thesizeofd in fact, we have an even better bound, see Theorem 3, the reason being that these computed d r,r 2 s are less than or equal to the least discriminants of number fields of degree m = r +2r 2 6 with r real places and r 2 complex places. However, even in the simplest situation where we assume that is totally real, we could not in [Lou05] obtain beforehand a C>0such that 7 holds true for s of root-discriminants ρ = d /m greater than C. Set m γ = lim m k log m = k= Euler s constant and λ r2,m =2+r 2 log 4 m log4π γ. Since λ r2,m < 0form 3, Theorem follows from the bound m log d + λ r2,m 8 κ 2 m + O r2,mlog m 3 d, m!

3 UPPER BOUNDS FOR RESIDUES AND CLASS NUMBERS 85 where the implied constants are effective and depend on r 2 and m only. To prove 8, we generalize the method introduced in [Lou96]. Set m log k γ = lim m k 2 log2 m = k= and μ r2,m =3+r 2 π 2 /2 m π 2 /8 γ 2 2γ. The error term in 8 is less than or equal to zero if μ r2,m > 0andd is large enough. Now, μ r2,m > 0 if and only if we are in one of the following cases: Table m r 2 λ r2,m μ r2,m d γ log4π = γ log π = γ 2log4π = γ 2log2π = γ log6π 3 = γ log4π 3 = γ 4log2π = γ log6π 5 = It will follow that we have a pleasingly explicit bound: Theorem 2. Assume that we are in one of the eight cases of Table. Then, m log d + λ r2,m κ 2 m, m! provided that d is large enough, as given in the last column of Table. The results in [Lou93] and [Lou96] are the case m = 2 of Theorem 2 above. However, in the quadratic case we have an even better bound see [Ram]. Finally, by taking constants slightly less than these λ r2,m, we have a the fully explicit following result where we do not have any restriction on d compare with Theorem : Theorem 3. Let be a number field of degree m {2, 3, 4, 5, 6} for which ζ s/ζs is entire. Then, log d + λ m κ 2 m, m! where λ is as in Table 2:

4 86 STÉPHANE R. LOUBOUTIN Table 2 m r 2 =0 r 2 = r 2 =2 r 2 = Corollary 4. If is a totally real cubic number field, then 9 h 2 d. If is a totally real quartic number field which contains no quadratic subfield, then 0 h 5 0 d. 24 We refer to [Dai] for examples of number fields with very large class numbers. 2. Proof of the bound 8 We adapt [Lou00, Proof of Theorem 7]. Let be a number field of degree m = r +2r 2 >. Assume that ζ s/ζs is entire. Set A /Q = d /4 r 2 π m, Γ /Q s =Γ r s/2γ r 2 s = 2r 2s π r 2/2 Γr +r 2 s/2γ r 2 s +/2 notice that r + r 2 0andr 2 0 and F /Q s =A s /Q Γ /Qsζ s/ζs. Then, F /Q s isentireandf /Q s =F /Q s. Let S /Q x := c+i F /Q sx s ds c >andx>0 2πi c i denote the inverse Mellin transform of F /Q s. Then, 2 S /Q x = x S /Q x notice that F /Q s is entire, shift the vertical line of integration Rs =c> in leftwards to the vertical line of integration Rs = c<0, then use the functional equation F /Q s =F /Q s to come back to the vertical line of integration Rs =c>. For Rs >, F /Q s = S /Q xx s dx 0 x is the Mellin transform of S /Q x. Using 2, we obtain 3 F /Q s = S /Q xx s + x s dx x

5 UPPER BOUNDS FOR RESIDUES AND CLASS NUMBERS 87 on the whole complex plane. Now, write ζ s/ζs = n a /Qnn s and ζ m s = n a m nn s Rs >. Then, a /Q n a m nsee[lou0, 55] and S /Q x = a /Q nh /Q nx/a /Q, n where H /Q x = c+i Γ /Q sx s ds. 2πi c i Since H /Q x > 0forx>0 see [Lou0, Theorem 20], we have S /Q x n a m nh /Q nx/a /Q. Plugging this into 3, we obtain d 2π r 2 κ = F /Q = n a m n = n a m n 2πi = a m n 2πi n c+i = 2πi Therefore, we have c i S /Q x + /xdx H /Q nx/a /Q +/xdx nx/a /Q s +/xdx Γ /Q sds c+i c i c+i c i s + s s + s Γ /Q sζ m sa s /Q ds. c+i Γ /Q sn/a /Q s ds 4 κ I s := f sds c >, 2πi c i where f s = Γ r 2 sλ m s s + d s /2, s Λs =π s/2 Γs/2ζs and Γs =Γs+/2/ Γs/2/Γ/2. Recall that Λs has only two poles, both simple, at s =ands = 0, and satisfies the functional equation Λs =Λ s. Moreover, /Γs/2 is entire whereas Γs +/2 has a simple pole at each odd negative integer. It follows that f s has a pole of order m>ats =, a pole of order m r 2 = r + r 2 ats =0,andapoleoforder r 2 0 at each negative odd integer. Now, as in [Lou0, Page 207], in the range σ σ σ 2 and t, we have Γσ + it =O t andthereexistsm 0 such that Λσ + it =O t M e π t /4. Hence, we are allowed to shift in 4 the Notice the misprints in [Lou00, page 273, line ] and [Lou0, Theorem 20] where one should read M M 2 x = M x/tm 2 t dt 0 t.

6 88 STÉPHANE R. LOUBOUTIN vertical line of integration Rs =c> leftwards to the vertical line of integration Rs =/2. We pick up one residue and obtain: 5 κ Res s= f s + I /2 = Res s= f s + O r2,md /4. The bound 8 now follows from Lemma 5 below. 3. Computation of some residues To prove Theorems 2 and 3, we need a better approximation to I s. By shifting in 4 the vertical line of integration Rs =c> leftwards to the vertical line of integration Rs = 2, we pick up three residues and we obtain: 6 κ Res s= f s + Res s=0 f s + Res s= f s + I 2, where Res s= f s is a polynomial of degree m inlogd with real coefficients, d Res s=0 f s is a polynomial of degree r + r 2 inlogd with real coefficients, and d Res s= f s is a polynomial of degree r 2 inlogd with real coefficients. This section is devoted to computing these residues. Lemma 5. Set f k,l s = Γ k sλ l s s + e s X. s Then, Res s= f k,l s is a polynomial of degree l in X with real coefficients and l X + Ak,l Res s= f k,l s = l =, l! l X + Ak,l X l 2 Res s= f k,l s = C k,l l =2 l! l 2! and l X + Ak,l X l 2 Res s= f k,l s = C k,l l! l 2! + O k,lx l 3 l 3, where A k,l = 2+klog 4 llog4π γ /2 and C k,l = 3+kπ 2 /2 lπ 2 /8 γ 2 2γ /2. Proof. We have Γs =+as + bs 2 + Os 3, with a =log2and b = log 2 2 π 2 /2/2, and s Λs =+cs + ds 2 + Os 3, with 7 c = log4π γ 2 and d = 2log4π γ2 + π 2 8γ 2 6γ. 6 For k 0andl 0, it holds that +az + bz 2 + Oz 3 k +cz + dz 2 + Oz 3 l +z z 2 + Oz 3 =+A k,l z + B k,l z 2 + Oz 3, where A k,l = ka+ lc+ and B k,l = klac+ k b + k 2 a2 + l d+ l 2 c2 + ka+ lc.

7 UPPER BOUNDS FOR RESIDUES AND CLASS NUMBERS 89 Hence, the desired results hold true with C k,l = A 2 k,l/2 B k,l = ka 2 2b+lc 2 2d+3 /2. We have a 2 2b = π 2 /2 and c 2 2d = γ 2 +2γ π 2 /8. Lemma 6. Set r = l k, letf k,l s and A k,l be as in Lemma 5, andset C k,l = 3 kπ 2 /2 lπ 2 /8 γ 2 2γ /2. If r =0or r =,then r X Res s=0 f k,l s = l π/2 k Ak,l e X. r! If r =2,then r X Res s=0 f k,l s = l π/2 k Ak,l C k,l X r 2 e X. r! r 2! If r 3, then Res s=0 f k,l s = l π/2 k X Ak,l r r! X r 2 C k,l r 2! + O k,lx r 3 e X. Proof. Here, Γs = πs 2 as + bs 2 + Os 3, with a =log2andb = log 2 2+ π 2 /2/2, and sλs = cs + ds 2 + Os 3, with c and d as in 7. Lemma 7. Let f k,l s be as in Lemma 5. We have Lemma 8. It holds that Proof. Using and Res s= f,l s = 3 2 I 2 5 Γr 2 /2+ 4π 2 3 m Γ 2+it = π le 2X. 6 r2 /2+ 4 d 3/2 m. 4+t 2 /2 πt +t 2 2 tanhπt 2 2π t Λ 2+it = Λ3 it ζ3 ζ3 Γ3 it/2 = π3/2 2π +t 2 coshπt/2 3 e π t /7, we obtain: d 3/2 I 2 5 6π 0 5 2π3 m and the desired bound. Γ 2+it r 2 Λ 2+it m dt 0 2πt r 2/2 e πm t/7 dt,

8 820 STÉPHANE R. LOUBOUTIN Table 3. Minimal discriminants m r 2 =0 r 2 = r 2 =2 r 2 = Proof of Theorems 2 and 3, and contents of Tables and 2 We use 6, the previous lemmas and Table 3 above see [Odl].. If is a real quadratic field, then κ log d +2+γ log4π 2 log d 2 + γ log4π 2 d + is less than or equal to log d +2+γ log4π /2ford If is an imaginary quadratic field, then κ log d +2+γ log π 2 π 2 + π π d 4d 36π 2 d 3/2 is less than or equal to log d +2+γ log π /2ford If is a totally real cubic number field, then 35 8π 2 d 3/2 log d +2+2γ 2log4π 2 κ 3/2+γ 2 +2γ π 2 /8 8 log d 2+2γ 2log4π /2+γ2 +2γ π 2 / d d 08π 2 d 3/2 is less than or equal to log d +2+2γ 2log4π 2 /8ford 46, and less than or equal to log d /8ford If is a not totally real cubic number field, then κ log d +2+2γ 2log2π 2 3/2+γ 2 +2γ π 2 /2 8 + π 4 log d 2 2γ + 2 log2π + π π d 24d 26π 2 d 3/2 is less than or equal to log d +2+2γ 2log2π 2 /8ford 4.

9 UPPER BOUNDS FOR RESIDUES AND CLASS NUMBERS The other cases are easily dealt with by using any software for symbolic computation, e.g., Maple, to compute the residues which appear in Proof of Corollary 4. Let be a totally real cubic field. Then, Reg 6 log2 d /4 see [Cus, Theorem ] or [Nak, Section 2.3]. Hence, d h = κ log d Reg 2log 2 d d. d / Let be a totally real quartic number field which contains no real quadratic subfield. Then, Reg 80 0 log3 d see [Cus, Theorem 2]. By 7 see also [Lou0, Theorem 2, point 3], we have κ 48 log3 d. Hence, by, we obtain d h = κ 5 0 d. 8Reg 24 References [Cus] T. W. Cusick. Lower bounds for regulators. Lecture Notes in Math., 068, Springer, Berlin 984, MR k:052 [Dai] R. Daileda. Non-abelian number fields with very large class numbers. Acta Arith , MR k:86 [Lan] S. Lang. Algebraic number theory Second Edition. Graduate Texts in Mathematics 0, Springer-Verlag, New York, 994. MR f:085 [Le] M. Le. Upper bounds for class numbers of real quadratic fields. Acta Arith , MR j:0 [Lou93] S. Louboutin. Majorations explicites de L,χ. C. R. Acad. Sci. Paris Sér. I Math , 4. MR m:084 [Lou96] S. Louboutin. Majorations explicites de L,χ. Suite. C. R. Acad. Sci. Paris Sér. I Math , MR k:23 [Lou00] S. Louboutin. Explicit bounds for residues of Dedekind zeta functions, values of L- functions at s =, and relative class numbers. J. Number Theory , MR i: [Lou0] S. Louboutin. Explicit upper bounds for residues of Dedekind zeta functions and values of L-functions at s =, and explicit lower bounds for relative class numbers of CM-fields. Canad. J. Math , MR d:67 [Lou05] S. Louboutin. Explicit upper bounds for the residues at s = of the Dedekind zeta functions of some totally real number fields. Séminaires & Congrès 2005, 7 78; AGCT 2003 SMF. [MM] R. M. Murty and V.. Murty. Non-vanishing of L-functions and applications. Progress in Mathematics 57. Birkhäuser-Verlag, Basel, 997. MR h:06 [MP] C. Moser and J. J. Payan. Majoration du nombre de classes d un corps cubique cyclique de conducteur premier. J. Math. Soc. Japan 33 98, MR a:2006 [Nak]. Nakamula. Certain quartic fields with small regulators. J. Number Theory , 2. MR h:28 [Odl] A. M. Odlyzko. Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results. Sém Théor. Nombres Bordeaux , 9 4. MR i:54 [Ram] O. Ramaré. Approximate formulae for L,χ. Acta Arith , MR k:44 [Sil] J. H. Silverman. An inequality relating the regulator and discriminant of a number field. J. Number Theory 9 984, MR c:094

10 822 STÉPHANE R. LOUBOUTIN [Uch]. Uchida. On Artin L-functions. Tohoku Math. J , MR :5558 [vdw] R. W. van der Waall. On a conjecture of Dedekind on zeta functions. Indag. Math , MR :344 Institut de Mathématiques de Luminy, UMR 6206, 63, avenue de Luminy, Case 907, 3288 Marseille Cedex 9, France address: loubouti@iml.univ-mrs.fr