3. Applications to theorems of the Schnee- Landau type

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1 3. Applicatios to theorems of the Schee- Ladau type Objekttyp: Chapter Zeitschrift: L'Eseigemet Mathématique Bad (Jahr): 15 (1969) Heft 1: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am: Nutzugsbediguge Die ETH-Bibliothek ist Abieteri der digitalisierte Zeitschrifte. Sie besitzt keie Urheberrechte a de Ihalte der Zeitschrifte. Die Rechte liege i der Regel bei de Herausgeber. Die auf der Plattform e-periodica veröffetlichte Dokumete stehe für icht-kommerzielle Zwecke i Lehre ud Forschug sowie für die private Nutzug frei zur Verfügug. Eizele Dateie oder Ausdrucke aus diesem Agebot köe zusamme mit diese Nutzugsbediguge ud de korrekte Herkuftsbezeichuge weitergegebe werde. Das Veröffetliche vo Bilder i Prit- ud Olie-Publikatioe ist ur mit vorheriger Geehmigug der Rechteihaber erlaubt. Die systematische Speicherug vo Teile des elektroische Agebots auf adere Server bedarf ebefalls des schriftliche Eiverstädisses der Rechteihaber. Haftugsausschluss Alle Agabe erfolge ohe Gewähr für Vollstädigkeit oder Richtigkeit. Es wird keie Haftug überomme für Schäde durch die Verwedug vo Iformatioe aus diesem Olie-Agebot oder durch das Fehle vo Iformatioe. Dies gilt auch für Ihalte Dritter, die über dieses Agebot zugäglich sid. Ei Diest der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz,

2 exactly as (2.2) (a) implies (2.7). Sice ow y><7 r >p, the above coditio i its tur implies By Theorem A with hypothesis (1.2) (a) ad a = b = 0, it follows that l~ s is coverget for ay g such that G^y>Gr ad therefore g0 <(Tr. But, i ay case, Go^G k^gr for o</:<r ad sowe hâve the coclusio (2.5). I the precedig argumet we hâve supposed that <r r <oo sice aa r =oo implies trivially aa k = 00. Ia 3. Applicatios to theorems of the Schee-Ladau type Theorem II give ext is the simplest of the theorems of the type me tioedabove ad it is a direct combiatio of Theorems I, B. Theorems V, VI are geeralizatios, respectively of Aada-Rau's ad Gaapathy Iyer's extesios of the Schee-Ladau theorem ([2], Theorem 9; [7], Theorem 10), as give by Chadrasekhara ad Miakshisudaram ([6], pp. 88-9, Corollaries 3.73, 3.74). Theorems 111, IV are apparetly ew couterparts of Theorems V, VI, the ewess cosistig i the replacemet of the two sidedtauberia coditios of the latter pair of theorems by aalogous oe-sided coditios suitably supplemeted. Suppose that (i) the Dirichlet séries, Theorem IL is summable (R,, q) for some g>o whe <J>p, (ii) the sum-fuctio thiis defied is regular for G> whe rj<p, ad satisfies the coditio /,/ f (s) (iii) the coefficiets a of the Dirichlet séries satisfy oe of the two alter ~ atives(a), (b) of (2.2), but with 6 (x) = xx (p //)/ '\ The the Dirichlet séries 1 is summable (R, /?, k), Proof for a>rj ~(~ o</c<r, for f By Theorem B, the Dirichlet séries is summable (R,, r'), rr >r, is the évidet Therefore hece from ad it proof of o> <^<p. /,/

3 Theorem I (A) edig with (2.10) that the Dirichlet séries is summable (R,U),o<kr', for whece the desired coclusio follows whe we let r'-±r. Theorem 111. I Theorem 11, let pbe replaced by a+ l i hypothèses (i) ad (ii) ; also let hypothesis (iii) be replaced by (3.1) The the coclusio is that Ia l \ s= cr+h, is summable (R, l 9 k), 0 <k<r,for (3.2) Proof. As i the proof of Theorem 11, the séries Ia ls is summable (R, /?, r'), rr f >r, for a>rj where ow rj<a+l, so that cr r,<^<a+l. We begi by choosig y ad correspodigly 6 (x) as follows : The, sice r'>r ad y>, we hâve (3.3) Ad so (3.1) gives us, as «->oo, (3.4) Also, if l <lm <l +eo(l ), (3.1) agai gives us as -^oo, so that, whether <x>o or a<o, (3.5)

4 I (3.4) ad (3.5), Hece, combiig (3.4) ad (3.5), we get (3.6) (3.6) ad the fact, followig from Theorem B, that Ia ls is summable (R, /?, r'), eables us to use (2.10) i the proof of Theorem I (A) with r, is summable p replaced by r\ p' respectively, so as to ifer that Ia l~ s (R,lH9 k), o<k<r\ for This yields (3.2) as required whe we let r'-*r ad recall that y (>t]) ca be take arbitrarily close to rj. Theorem IV. I Theorem 111, (3.1) aloe ca be chagea to (3.7) with the coclusio chagea i coséquece to the assertio that Ia ls is summable (R, /?, /c), o</c<r, for (3.8) Proof. We observe that Theorem 111 may be viewed as the limitig case p = oo of Theorem IV. The proof itself is similar to that of Theorem 111 with the différece that the choice of y ad 9 (x) i (3.3) is ow altered as below: /0/ i) We suppose that 0 =0.

5 Ad furthermore the step correspodig to (3.6) is obtaied as follows. Writig 1-1/p =W,we get, for /?</,?</?+ e9 (/?), (3.9) where we hâve used the hypothesis (3.7) i the passage to the step precedig (3.9). Takig m = -\-\ i the step precedig (3.9), we get also (3.10) From (3.9) ad (3.10) with +l chaged to, we obtai, istead of (3.6) i the proof of Theorem 111, where After this the proof is completed exactly like that of Theorem 111 subséquet to (3.6).

6 It may be observed that the assumptio a+l+/? *>() ivolves o loss I <o makes successively a +\a sice 0 0, a of geerality ad so a r = a+l+/?"?oo for ail r>o. \ = = Theorem V. I Theorem 11, /e hypothesis (i) &e omitted o accout beig implicit {with q = 0, p = a+l) i hypothesis (iii) modified as uder. Let hypothesis (ii) Z?e retaied with p chagea to a+l, ad hypothe of its sis(iii) replaced by (3.11) 77ze«/7ze coclusio is that Ia ls is summable (R, /,/, k), o<lk<r,for a satisfyig (3.2). Theorem VI. If 9 i Theorem V, (3.11) aloe is chagea to will become the assertio that Ia ls is summable (R, /?, /<), o o</c<r, for satisfyig (3.8). The proofs of Theorems V, VI are omitted, beig obvious simplifica tiosof those of Theorems 111, IV, ivolvig the use of Theorem I (A) with hypothesis (2.2) (b) istead of (2.2.) (a) as formerly. Theorems V ad VI, as poited out by Chadrasekhara ad Miakshisudaram, yield Aada Rau's ad Gaapathy Iyer's extesios of the Schee-Ladau theorem whe a->+o. the coclusio 4. Further applicatios Theorem I (A) is a base which, combied with Theorem B, produces Theorem 11, ad i this sesé Theorem I (A) may be said to correspod to Theorem IL There are results correspodig to each of Theorems 111-VI i the same sesé. For istace, Déductio 1 below correspods to Theo rem111 ad shows how other déductios correspodig to Theorems IV- VI may be formulated. Déductios 2,3 are further examples of results based o Theorem L Déductio 1. (A) I Theorem I (A), suppose that oo r< a+l ad that (2.2) (a) is replaced by (4.1)