Small Class Numbers and Extreme Values

Transcription

1 MATHEMATICS OF COMPUTATION VOLUME 3, NUMBER 39 JULY 9, PAGES 8-9 Small Class Numbers and Extreme Values of -Functions of Quadratic Fields By Duncan A. Buell Abstract. The table of class numbers h of imaginary quadratic fields described in ( ] was placed on magnetic tape. This tape was then processed to find the occurrences of h < and to find the successive extreme values of the Dirichlet -functions (, X_r>). X_n the Kronecker symbol of the field Q(-J D ) of discriminant D. A comparison was made between the observed extrema and the bounds obtained for the -functions by Littlewood ] assuming Riemann hypotheses. Introduction. Recently, a computation was made of the class numbers and class groups of imaginary quadratic fields of discriminant - D, for 0 < D < In [], we discussed that computation and summarized some of the results. Since then, the data have been put on magnetic tape for permanent storage and ready access. At the suggestion of Daniel Shanks, we undertook to reexamine the data with a view to confirming and extending the results in a review of Lakein and Kuroda [], in Shanks [3] and [8], and in Lehmer, Lehmer, and Shanks []. In [], the first and probably last appearances of odd class numbers h < 9 are listed. In Section of this paper, we confirm this list with our own tables and extend it to all h <. The list seems to be complete. We also list, in the tables at the end, the first occurrences of specific class groups of these small orders. The rest of our paper concerns the successive extreme values of the -functions of real characters of these imaginary quadratic fields and extends some of the tables of [3] and []. These two topics are connected: A maximum of the -function can only occur at the first appearance of a given class number; and, unless minima occur for different discriminants having the same class number, the minimum can only occur at the last appearance of a given class number. All of the computations were done on the IBM 30/8 computer at the University of Illinois at Chicago Circle, Chicago, Illinois; we thank the University for the use of the computer facilities.. Small Class Numbers and Class Groups. In [], the first and last occurrences of odd class numbers h = fc(ô(v~ D)) less than 0 are exhibited in two tables, one for D = (mod 8), and one for D = 3 (mod 8), for 0 < D< 0, D prime. In Tables,, and 3, we present, for 0 < D < (D no longer necessarily prime), the first and last occurrences of all class numbers h <, for even D as well as the two congruence classes modulo 8. We also include the number of examples of each Received December 0, 9; revised November 3, 9. AMS (MOS) subject classifications (90). Primary A, A0, H. 8 Copyright 9, American Mathematical Society

2 -FUNCTIONS OF QUADRATIC FIELDS 8 Table Small class numbers for even discriminants Class number H, first D, last D, number of D, with H(- D) = H FIRST D LAST D NO. FIRST D LAST D NO class number in this range of discriminants. Up to h =9, our tables coincide completely with those of []. Although mere computation cannot supply an answer, one can ask whether these tables are complete are the last examples observed actually the last examples? To assume h < for D > would imply that L(\, x_d) < r/000 =.93. For D = l (mod 8), the observed minimum of the -function for D < (see Section ) is three times that. To find such a large D of this congruence class with so small a class number would be quite surprising, but the nonexistence of such D has only been proved for h < 3 [8]. For D = 3 (mod 8), the minimum of the -function is.988, which occurs for the very unusual discriminant - 99, of class number 3 (see [3], [] ). It does not seem completely unlikely, therefore, for other examples with h < to occur. However, as a glance at Table 3 shows, it is likely that few more, if any, will exist; the largest D in this table is smaller than 3. million. For the even discriminants, it is much more likely that this list is complete. The minimum of the -function, in Table are smaller than. million..30, occurs as early as D = 008, and all entries

3 88 DUNCAN A. BUELL Table Small class numbers for discriminants mod 8 Class number H, first D, last D, number of D, with H(- D) = H H FIRST D LAST D NO. FIRST D LAST D NO ID X X

4 -FUNCTIONS OF QUADRATIC FIELDS 89 H Table 3 Small class numbers for discriminants mod 8 Cass number H, first D, last D, number of D, with H(- D) = H FIRST D LAST D NO. H FIRST D LAST D NO

5 90 DUNCAN A. BUELL Table First occurrence of small noncyclic -groups Group x x8 xx x x 8x8 xx8 xxx x0 x xx x8 Even D D= *#* DS Table First occurrence of small noncyclic 3-groups Group 3x3 3x 3x9 3x x 3x 3x8 3x 3x x xx 3x 9x9 3x30 3x33 3x3 x8 3x39 Even D ** **# D= D= II Table First occurrence of small noncyclic - and -groups Group x x x x0 x x x x Even D I9 II8 8 D= D=

6 -FUNCTIONS OF QUADRATIC FIELDS 9 Table Successive minima of () for even discriminants Negative discriminant D, D/A, (), and LLID D d/ L(l) ll h Table 8 Successive minima of () for discriminants mod 8 Negative discriminant D, (), D(l), and LLID L(l) yi) LLID In Tables,, and we list the first occurrences of noncyclic class groups of orders <. In these, the notation N x M signifies a group C(N) x CXM), with C(N) a cyclic group of order N. The symbol means that no example occurred. Also, as in [], the term "noncyclic -group" is used for a group whose principal genus is noncyclic. It was proved in [8] that, for D = (mod 8), the groups 3x3, x,x and x do not occur. If our list of small class numbers is complete, then neither x nor xx occur for imaginary quadratic fields. The groups 3x3x3 and xx and 3 x 3 x 3 x 3 are also missing here and are even more likely to be nonexistent for imaginary quadratic fields since any such example would have an improbably small value of (, x).

7 9 DUNCAN A. BUELL Table 9 Successive minima of L(\) for discriminants mod 8 Negative discriminant D, (), LD(\), and LLID D L(l) LLI Lpd) LLIp Table Successive maxima of L(\) for even discriminants Negative discriminant D, D/A, (), and ULID D D/ L(l) ULI^ b

8 -FUNCTIONS OF QUADRATIC FIELDS 93 Table Successive maxima of L(\) for discriminants mod 8 Negative discriminant D, (), ULI, LD(\), and ULID L(l) ULI Lpd) ULL Extreme Values of -functions. Let - D be the discriminant of an imaginary quadratic number field, hence D = 0 or 3 (mod ), and D is squarefree except for the possible factor of. For the real character Xß(") = (- D/n) = the Kronecker symbol modulo D, the -function is () L(s, X- -ifëw- n( n = l p prime \ ps - ( D/p)

9 9 DUNCAN A. BUELL Table Successive maxima of () for discriminants mod 8 Negative discriminant D, (), D(l), and ULID D L<) Iajjd) ULIp Then, as is well known, () L(l)=L(l,X-D) = hir/ws/d, where h is the class number of the field Q(\J- D), and w is the number of roots of unity in the field. An exceptionally large or small class number, relative to \JD, corresponds to a large or small value of (). Certain restrictions on the size of () may exist, however. Assuming the validity of the Riemann hypothesis for the - functions, Littlewood [] deduced, for large values of D, the bounds (3) { -r-o(l)}(/v)eloglog > < () < { - o(l)}e log log D. It is clear from the Euler product that, for odd D, the factor of or /3 for the prime is the most significant single factor. This accounts for the great difference in magnitudes of D in Tables and 3. Since the. values of () are, in a sense, a measure

10 FUNCTIONS OF QUADRATIC FIELDS 9 of the quadratic residue symbol distribution for - D, it is desirable to remove this strong dependence on the prime. In [3] and[] the following approach was used: Define, for any positive integer D, () MD-C^V- H,^^FÖF) n=l \ / p prime \ v I For square free D we note: d() = (,X_d)> > =, (mod ), () - (/)(, X_ß). 0 = (mod 8), = (3/)(, X_D). D = 3 (mod 8). Shanks [3] then strengthens the bounds (3) for this modified -function and gives ()- < D(l) < { + o(l)}eloglog AD. { +o(l)}(8/r)e?loglog > This can be interpreted in terms of class numbers as follows: When we compare various D(l), for odd D = 3 (mod ), we are comparing the class numbers, not of the maximal orders of discriminant - D, but of the orders of discriminants - AD [, pp. 8ff.]. approached. It is of some interest to examine how closely the Littlewood bounds are For this reason, we define, for both (, x_d) and D(l), the upper and lower Littlewood indices as () ULI = L(l)/(e'rloglogD), / = (l)(/r)e'loglog ), (8) ULID=LD(\)/(e^\og\ogAD), /D = D(l)(8/VVloglog D. (Remark on notation: Our () is Shanks' (3), page, and our (8) is his (3), page 0 [3].) If the Riemann hypothesis holds, then the bounds () and () become inequalities (9) ULK I, LLI>\, ULID<\, LLID>\, with some allowance, of course, for the o(l). We have computed the successive minima and maxima of () for the three types of fundamental discriminants within the range of our tables. These extreme values, with the appropriate Littlewood indices, are listed in Tables -. Not surprisingly, we notice that only three examples, D = 3,, and 3, violate (9). Shanks has shown [3] that for D = 3 the >() is probably large enough to accommodate the Littlewood bounds. One would expect the other

11 9 DUNCAN A. BUELL two cases to be even less suspect because the bounds are, after all, asymptotic. Thus, our computation gives no reason to believe that the Riemann hypothesis is false for these -functions. Department of Mathematics Carleton University Ottawa, Ontario KS SB, Canada. DUNCAN A. BUELL, "Class groups of quadratic fields," Math. Comp., v. 30, 9, pp. -.. D. SHANKS, Review of Richard B. Lakein and Sigekatu Kuroda, "Tables of class numbers h(- p) for fields Q{yJ-p),p < 0," UMT 39, Math. Comp., v., 90, pp DANIEL SHANKS, Systematic Examination of Littlewood's Bounds on (, x). Proc. Sympos. Pure Math., vol., Amer. Math. Soc, Providence, R.I., 93, pp MR 9 #9.. D. H. LEHMER, EMMA LEHMER & DANIEL SHANKS, "Integer sequences having prescribed quadratic character," Math. Comp., v., 90, pp. 33-_._MR #889.. J. E. LITTLEWOOD, "On the class-number of the corpus P(\/- k)," Proc. London Math. Soc, v. 8, 98, pp G. B. MATHEWS, Theory of Numbers, nd ed., Chelsea, New York, 9. MR #A398.. DANIEL SHANKS, "Calculation and applications of Epstein zeta functions," Math. Comp., v. 9, 9, pp DANIEL SHANKS, "On Gauss's class number problems," Math. Comp., v., 99, pp. -3. MR #8.