Showing papers in "Crelle's Journal in 1977"

On the linear diophantine problem of Frobenius.

Abstract: The problem of Frobenius consists in determining the largest integer g(a1,a2,...,ak) with no such representation. To the author, a particularly nice aspect of this problem is the ease with which it can be explained to a non-mathematician : Given coins (in sufficient supply) of denominations ai9 a2, . . ., ak. Determine the largest amount which cannot be formed by means of these coins. It is well known that (1. 1) g(ai,a2) = ala2-al-a2. For k > 2, formulas for g have been proved only in special cases. More or less accurate bounds for g have also been given. One of these bounds, due to Erd s and Graham, is improved below. In later years, some authqrs have also examined the number

Fourier coefficients of the resolvent for a Fuchsian group.

Abstract: *~T + ~T~2~)~2ifc.y-r— acting on a Hubert space §k of automorphic forms dx dy) dx of weight k e IR. In this paper, we present the basic eigenfunction expansions of Gs k(z, z') and discuss applications to conditionally convergent Poincare series and series of Dirichlet type for Fuchsian groups of the first kind, and to the spectral decomposition of §* for groups of the second kind. The outline of the paper is äs follows: in § l we set up the basic eigenfunctions of Dk and the spectral decomposition of f>fc for the trivial group, used in the construction of an automorphic \"prime-form\" and automorphic functions with prescribed automorphic eigenfunctions are introduced in § 2 and are used to give summation methods for the classical kernel functions on a compact Riemann surface, äs well äs for the construction of an automorphic \"prime-form\" and automorphic function with prescribed singularities. The Fourier coefficients of the resolvent at a parabolic cusp are worked out in § 3 and include many special cases of historical interest; the use of the resolvent here explicates certain multiplicative relations of Hecke type and \"expansions of zero\" associated to analytic forms of positive dimension for the modular group. Finally, in § 4 we consider Fourier developments at the hyperbolic fixed-points, including a summation method for the period matrix of a compact Riemann surface; for a Fuchsian group of the second kind with a single free side, the continuous spectral measure for §k is given in terms of a Poisson kernel and Fourier coefficients analogous to the Eisenstein series for horocyclic groups.

Darstellungen halbeinfacher Gruppen und kontravariante Formen.

Abstract: Sei Gk eine universelle Chevalley-Gruppe ber einem unendlichem K rper k. Die irreduziblen, algebraischen, endlichdimensionalen Darstellungen von Gk werden durch die dominanten Gewichte des zugeh rigen Wurzelsystems R klassifiziert. Die irreduzible Darstellung zu einem h chsten Gewicht λ kann man so erhalten: Man zeichnet in F(A)C, der irreduziblen Darstellung zum h chsten Gewicht λ der entsprechenden Gruppe ber C, einen Z-Modul ν(λ)ζ aus und bildet Υ(λ\\ = Υ(λ)ζ ® k. Dies ist auf nat rliche Weise ein Gfc-Modul und besitzt einen gr ten echten Untermodul M. Dann ist Lk(/l) = V(^)kjM die irreduzible Darstellung von Gk zum h chsten Gewicht λ. Der Untermodul M l t sich nach [14] auch anders beschreiben: Es gibt auf F(A)Z eine symmetrische Bilinearform (,), die kontravariante Form, mit Werten in Z. Diese Bilinearform definiert eine Bilinearform auf Υ(λ\\ und M ist genau das Radikal dieser Form. F r (,) sind Gewichtsr ume zu verschiedenen Gewichten orthogonal. Wir w ten also f r alle K rper k den (formalen) Charakter ch (Lk(X)] von Lfc(A), wenn wir f r alle Gewichtsr ume von ν(λ)τ die Elementarteiler von (,) w ten. Eine gewisse Information geben uns aber schon die Determinanten von (,) f r die verschiedenen Gewichtsr ume von F(A)Z. Sei Ολ(μ) die Determinante f r den Raum zum Gewicht μ. Sei char(fc) =p φ 0 und sei vp die /?-adische Bewertung von Q. Sei νρ(ϋλ) das Element des Gruppenrings der Gruppe der Gewichte von R, dessen μKomponente gleich νρ(Όλ(μ)} f r alle Gewichte μ ist. Man kann nun vp(DA) und ch(F(/l)k), den Charakter von F(A)k, durch die Charaktere der Lk(ju) mit μ dominant ausdr cken: ch(K(A)k) = Z (A,//)ch(LfcO£)) und vp(Dλ) = Σb(λ,μ)ch(Lk(μ)) mit α(λ,μ), μ μ 6(λ,μ)€Ζ. Ist Λ, Φ μ, so gilt dann α(λ,μ)^(λ,μ) und es ist genau dann α(/1,μ)Φθ, wenn />(Α,μ)Φθ ist (Satz 5). Sind zum Beispiel f r ein λ alle 6(λ, μ) e {0, 1}, so ist ch(Lk(A)) = ch(F(A)k)~ νρ(Ώλ). Dies l t sich beim Typ B2 benutzen. Doch auch, wenn einige (wenige) b(A, μ)^2 sind, lassen sich, etwa beim Typ G2, die ch(Lk(A)) mitunter berechnen.

Continued fractions and density results for Dedekind sums.

Abstract: For integers h and k with k^.\\ and (A, fc) = l, let s (h, k) be the Dedekind sums, äs defmed and discussed in [2]. According to [2], p. 28, it is unknown whether the points (h/ k, s (h, k)) are dense in the plane. It is the object of this paper to show that they are. In the process, a formula will be given for s (h, k) in terms of the simple continued fraction (s.c.f.) expansion of h/k. As in [2], it is convenient to consider s (h, k} äs a function of a single rational argument, by defming s(h/k) = s(h, k). That this function is well defined is clear from the conditions on h and k. Note that s is periodic with period l . The reader is referred to [1], eh. 7, for the elementary properties of continued fractions which will be used here. Theorem 1. Let

On the distribution of amicable numbers. II.

Abstract: Let σ (ή) denote the sum of the divisors of n and let s(ri) = a(ri) — n. Two natural numbers n, m are called an amicable pair if s(ri) = m and s (m) —n. The least such pair with n Φ m is n = 220, m = 284. We say a natural number n is an amicable number if it is a member of an amicable pair. An equivalent defmition is s(s(ri)) = n and still another is σ (η) = σ($(ιϊ)). Amicable numbers have a very long history; they were mentioned (in fact, defined) by Pythagoras, they were investigated by the Arabs during the European Dark Ages, and they were studied by Fermat, Descartes, and Euler.

On the (p, q)-type and lower (p, q)-type of an entire function.

Abstract: be a nonconstant entire fimction where A0 = 0 and {An}*=1 is the strictly increasing sequence of positive integers such that an Φ 0 for n = l, 2, 3, . . . . The rate of growth of/(z) is studied in [8] in a general manner in terms of its (/?, #)-order and lower (p,

The generalized Kronecker function ring and the ring R(X).

Abstract: This paper extends the concept of Kronecker function rings from integral domains to rings with zero divisors. To accomplish this we use Marot's property (P) and a new property which we label äs (A). The first part of the article is devoted to developing results on properties (P) and (A), and showing that the class of rings satisfying these properties is quite large. The main theorem, a generalization of a result about integral domains due to J. Arnold, is given in §3. We prove that if R is a ring having properties (P) and (A), and if R is the Kronecker function ring of R, then the following are equivalent: (1) R is a Prüfer ring; (2) R(X) = R; (3) R(X) is a Prüfer ring; and (4) R is a regulär quotient ring of R[X]. The class of rings for which this result holds is large. For example, it properly contains the set of commutative semihereditary rings. In the final section we construct an example, using the idealization of an -module, to show that the main theorem does not hold if property (A) is deleted from the hypothesis.

The 2-class group of certain biquadratic number fields.

Abstract: Methods for determining the exact power of 2 dividing the class number of a quadratic number field have been developed by Hasse [3], Bauer [1] and others. In view of numerical evidence recently obtained by Parry [9], we are now convinced that these methods can be applied in more general settings. In this paper, we give methods for determining the power of 2 dividing the class number of a certain class of cyclic biquadratic fields.

When do Beurling generalized integers have a density

Abstract: Let Ν0,π0, etc. denote the corresponding functions for the natural integers and primes. Suppose that one of the sequences Jf , & is distributed rather like the corresponding sequence of natural numbers or primes. We want to know whether the other sequence i s also distributed like its classical counterpart. Beurling proved that π(χ)~ χ /log χ, i.e. the prime number theorem holds for ̂ , if N(x) = ex 4O(x log\" je), o 0, y > 3/2 , and he showed by example that the theorem can fail for γ = 3/2. (cf. [1], [2], [4], [6].) We are seeking weak conditions on π which enable us to deduce that N(x}~cx for some positive c, i. e. that Jf has a density. A closely related problem is the determination of when a multiplicative arithmetic function has a mean value. (cf. [3], [7], [8], [12], [13].) We shall give conditions which guarantee that N has a density, and we show by example

On minimal centers of attraction and generic points.

Abstract: The restrictions of dynamical Systems to minimal centers of attraction are characterized topologically s those dynamical Systems which admit an invariant measure positive on all open sets. The minimal centers of attraction of an Axiom A diffeomorphism are just the supports of invariant measures lying in some basic set. By a dynamical System we shall understand a homeomorphism T from a compact metric space X onto itself. A set Ca X is said to be a center of attraction (CA) for χ e X if C is closed and lim^r Σ \\v(Tx) = \\ N n=0 for every open F=>C (1F denotes the indicator function). Thus the \"probability of sojourn\" of χ in any neighborhood of C is one. It can be shown that the smallest such set, the so called minimal center of attraction (MCA), exists and is equal to the set of points which are \"interesting for je\", i.e. the set j N-l ̂ : P Tf Σ lu(Tx) > 0 for every open U 3 y) ^ «=o I(x) is nonempty, closed and invariant. MCA's were introduced by Hilmy in [6]. They are discussed in [7] and [9], for example. In general χ won't belong to I(x), and a MCA is therefore defined by the behaviour of orbits which may lie outside of it. Thus one can ask for a criterion allowing to decide whether a given closed T-invariant subset of X is a MCA without looking at anything exterior of this set. A closely related question, with which we deal first, is that of characterizing all those dynamical Systems which may occur s restrictions of some System to one of its MCA's. The same questions for co-limits have been studied by Dowker and Friedlander [5] and by Bowen [1] (see remark 5). A dynamical System T:X —>X is said to be an abstract MCA if there exists a dynamical System T: X' —> X and an x' e X' such that the restriction of T to I(x'} is topologically conjugate to T: X —» X. Theorem 1. T\\X —> X is an abstract MCA iff there exists a T-invariant probability measure on X which is positive on all nonempty open sets. Sigmund, Minimal centers of attraction 73 We shall prove this by a series of propositions. Let M(X) denote the set of all Borel probability measures on X9 provided with the weak topology for measures. Then M(X) is compact and μη — * μ in M (X) iff §/άμη — > J/d/z for all continuous functions / from X to f?, or equivalently iff μ(€)^\\ϊΐη$\\λρμη(€) for all closed Cd X. For VaM(X) we denote by Supp V the support of V, i.e. the smallest closed set C such that μ(€) = l for all μ e F. Clearly Supp V= U Supp {μ}. For χ e JT μεΚ let (5(.x) be the measure with support x. The mapping χ-^>δ(χ) is a homeomorphism from X onto the extremal points of the convex set M(X). Let d be a metric on X with diam X^l. d can be extended to the Prohorov metric on M( X}, which will again be denoted by d. Thus ά(μ, ν) inf {ε : μ(Α) ^ v (A*) + ε and ν(Λ) ̂ μ(Λ) + ε for all Borel A c ̂ } , (here ^4 = {.x:d(/i, χ) ̂ ε}), and d induces the weak topology on M(X) . The transformation T: X— > X induces an affine homeomorphism from M(X) onto itself, which will again be denoted by Γ Thus T μ(Α) = μ(Τ~ ι A) for all Borel sets A c: X and all μ E M (T). MT(X) will denote the set of Γ-invariant measures. A dynamical System T \\X' — > -ST' is said to be an extension of T: X -* X if T: X — » Jf is topologically conjugate to the restriction of T to some closed Γ'-invariant set. For example, the map from M(X) onto itself which is induced by the map T:X-^X is an extension of T\\X-+X. The System T\\X— -» X is an abstract MCA iff it is an MCA in some extension. Note that any Γ-invariant measure can be considered s an invariant measure for any extension of T'.X— > X. A bilateral sequence {xf} (oo < / < + oo) is said to be a pseudo orbit for T: X-^> X if d(TXi, xi+1)— » 0 for |/| — > oo. By δ(χ1, . . . ,x„) we denote the measure — (<Η*ι) + * * * + <5CO) i ^Υ ^[X] ^he set of accumulation points of (xiy . . . ,x„), s n— > oo. If {χα is a pseudo orbit then V[x^\\ is nonempty (since M(X} is compact), closed (obviously), connected (since d(6(xl, . . ., χη),δ(χ1, . . ., x n + 1))— > 0)andasubsetof MT(X) (nncQd( (Tx1, . . ., 7X,),<5(*2, · · · » ^«+1))-* 0 and J ((5 (x2 , ., Λ:Β + Ι),Ο(Λ:Ι , . . . ,x„))->0 and therefore ^T^iX , . . . , xn), (xl9 . . . , xn)} — > 0). If F ·̂] consists of a single element μ, one says that {jcj generates μ. One says that χ e X is a generic point for μ e Mr(T) if the orbit {Tx} generates μ. By the ergodic theorem, every ergodic measure (i.e. every extremal point of MT(X)) has generic points (cf. [3] for example). For a pseudo orbit {xt} define C j JVl ^[*i] = ° f° y °P ^ 9 l N i = o This set is called the MCA of {xj. It is Γ-invariant and the smallest closed set such that j Nl lim — Σ V(X) = l holds for every neighborhood V N i=0 Proposition l . Let {jcj £>e α pseudo orbit. Then /M = Supp K[X·]. 7w particular I(x) = Supp KIT*] / r // χ e JT. 74 Sigmund, Minimal centers ofattraction Proof. 1) Let y e /[je,·] be given. If y is not in Supp F[jct], there exists an open U such that y e Uc D aX \\Supp K[jcJ. One has l \"hmsup— Σ lu(Xi) ^ i = 0 Thus there exists a sequence 7Vfc f oo such that for all Nk. There exists a subsequence Nk of Nk such that δ(χί, . . . , xNli) converges to some a μ E F[xf]. Hence μ(£/)>—-. But μ(8ηρρ F[xJ)= l, a contradiction. 2) Let Um be the open —-neighborhood of /[xj, and let μ e F[XJ] be given. For some sequence ./Vk t °° one has (xl9 . . . , xNJ —* μ. Since obviously j Nl \" Τ7 Σ * m(i) = l 5 one has and hence μ(0τη)= 1. Lettingm— * oo one obtains μ(/[χΙ]) = l, i.e. Supp F[jcJ d/[^]. Proposition 2. T/* {xf} w a pseudo orbit for T:X -+ X there exists αμέ ΜΤ(Χ) with Supp {μ} = /Μ. oo Proof. Let v l 5 v2, . . . be dense in V[x^\\ and let μ = Σ 2~ n v n . v„ e K[xJ implies « = i therefore μ(/[Χί]) = l, i.e. Supp {μ} c /[xj. Consider a neighborhood U of some >> e /[jcj. By proposition l, <7 intersects Supp {v}, i.e. there exists a ve F[JC,·] with v(C/) = a>0. Since v„ k — > v for some subsequence nk one has vN(U)>~— for some 7V and so μ(υ)>α2~\". It follows that je Supp {μ}. Proposition 3. For any μ E MT(X) there exists a pseudo orbit for T: M(X) — > M(X) which generates μ. Proof. By the Krein-Milman theorem, μ = 1ίηιρη, where ρη is a finite convex combination of extremal points of MT(X). We may assume that with α\"^0, Σ j l· Since μ\" is ergodic it has some generic point z\". We shall construct a pseudo orbit {jcj for Γ: Af(JQ— ̂ Af(AT) with F[jcJ = {/x}. For the construction we subdivide N into adjacent blocks of integers of the form {a + l, 0 42, . . . , 0 + w}, namely N = fit 52 . . . . The block Bn is further subdivided into blocks B (n, k), Bn = Α(Λ, 1) . . . B(n, kn), and each B(n, k) further subdivided into n blocks J(n, k, s), B(n, k) = J(n, /:,!).. J(n, k, ri), where J(«, A:, s) has length 7V„? s + w. Both 7V„ s and kn will be specified later. To distinguish Sigmund, Minimal centers ofattraction between these three kinds of blocks, we shall call the B„ \"maxiblocks\", the B(n, k) \"blocks\" and the J(n, k, s) \"miniblocks\". Defme xt e M(X) by δ(Τζ{}, for /^0, and for / > 0 s follows: / is the t-th element, say, of some miniblock J(n,k,s). If t

SK1 von Algebren über vollständig diskret bewerteten Körpern und Galoiskohomologie abelscher Körpererweiterungen.

Abstract: der Faktorkommutatorgruppe A* von A* zu untersuchen. Diese Gruppe spielt sowohl in der algebraischen Ä^-Theorie (vgl. etwa [3], Ch. V, §9) als auch in der Theorie der algebraischen Gruppen (vgl. etwa die Einleitung von [20] sowie die dort zitierte Literatur) eine Rolle. Die Untersuchungen von S KI (A) — in der russischen Literatur das „TannakaArtin-Problem\" genannt — konzentrierten sich rund 30 Jahre lang darauf, die Vermutung

On power.free numbers and polynomials. II.

Abstract: A füll history and background of this subject having been furnished by both I and two earlier memoirs of similar title (Hooley, [6] and [7]), the briefest of introductions serves to usher in the preliminary Statement of the results we shall obtain. Suffice it then to repeat that problems involving /-free numbers and polynomials of degree r are almost inevitably easy to resolve when /^r, while the successive work of Erdös [3] and the author ([6], [7], and [8]) has fully covered the question of the representation of /-free numbers by fixed polynomials of degree r with integer arguments when / = r — 1 . Nevertheless, this work did not settle the general Situation relating to the case / = r — l , since, for instance, it shed no light whatsoever here on either the important problem regarding polynomials with prime arguments or the conjugate problem involving the representation of large numbers äs the sum of an rth power and an (r— l)-th power-free number. The latter are the problems, briefly alluded to in I, that will now occupy our attention.

A selection theorem for Boolean correspondences.

Abstract: In a fundamental theorem M. H. Stone [16] has shown that for every Boolean algebra 9l there is a Boolean space X(Wj such that 9l is isomorphic to the field of all subsets of 1X91), which are both open and closed. The algebra 9l uniquely determines Jf(9l) to within a homeomorphism. In this setting every Boolean homomorphism from a Boolean algebra 9l to another one, say 33, is induced by a unique continuous map from ^(23) to X(9l). Conversely every continuous map from X(3$) to Jf(9l) induces a unique Boolean homomorphism from 9l to S.

On the estimation of Schnirelman's constant.

Abstract: I [H]? [12] Schnirelman introduced an approach to Goldbach's problem in the additive theory of prime numbers whereby it can be shown that every integer larger than unity can be written s the sum of a bounded number of prime numbers. Until quite recently the best bound was 6 · 10 given by Klimov [4]. However, Klimov, Pil'tai and Sheptitskaya [5], by using sharper sieve estimates, have replaced this by 115.

A criterion for the class number of a pure quintic field to be divisible by 5.

Abstract: Let 0 denote the field of rational numbers, and let Z be the ring of rational integers. Let K be a pure quintic extension of 0; i.e., K= 0(J/ra) with m a positive, 5-th power free rational integer. Let ζ be a primitive 5-th root of unity. Let fc = Q(Q and L = Let hK (resp. AL) denote the class number of K (resp. L). A necessary condition and sufficient one for hK to be divisible by 5 were given by C. Parry [4] s follows. Theorem. Let ALfk denote the order of the group of ambiguous ideal classes of L/ k. Then in order (hat 5\\hK, (i) the condition 5\\AL/k is necessary, (ii) the condition 5 \\AL/k is sufficient. Remark. Both the conditions that 5\\AL/k and that 5\\hL are equivalent. The computation of the exact value of AL/k can be easily done by knowing all rational prime divisors of m. In this paper we deal only with the case 5\\\\AL/k and give a criterion for hK to be divisible by 5. And then we show, using this criterion, that there are infinitely many fCs both with 5J(hK and 5\\hK.

A criterion for rational places over local fields.

Abstract: By this we mean a place P of Fover Ä^whose residue field FP equals K. For any place P of F\K, the residue field FP is an extension of K; the field degree [FP'.K] is usually called the degree of P. Thus we are looking for places of degree 1. In German language, function fields which admit a place of degree l are called "aufgeschlossen" [5] but there does not seem to exist an adequate and accepted English translation.

A remark about zeta functions of number fields of prime degree.

Abstract: Here linearly disjoint means that [K· K':Q]=p. When this maximum degree/? is not attained, it very frequently happens that K and K' turn out to be isomorphic. An example with/? = 7 showing that this is not always the case may be found in [5]. However, if three fields K, K\\ K\" of degree p have the same zeta function, there is good reason to believe that at least two of them are isomorphic: this is known to be true for all p up to two million, and would follow in general from the validity of a conjecture made by Wielandtinl955.See[l],p. 12.

On a conjecture of Issai Schur.

Abstract: Improving results of Vinogradov, A. Brauer, and the author, we find by purely elementary arguments an integer n0 such that for every prime p > nQ the maximum number of consecutive quadratic non-residues (mod/?) is always less than p. In some special cases our bpunds are slightly better than/?, and our arguments hold also for the maximum number of consecutive quadratic residues (mod/?) provided p is l (mod 24). The author notes that the non-elementary bound of D. A. Burgess, which is much sharper for large primes, can be combined with the elementary arguments in this paper and Computer data for small primes leading to a value for «0, possibly äs small äs 13. However the content of this paper is purely elementary since Issai Schur, who proposed this problem would, we feel, have strongly preferred such a proof.