3.2 $L_p(s,\tau;\lambda)$ FOR $\tau \in C_p$, $ \tau _p \leq 1$

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1 3.2 $L_p(s,\tau;\lambda)$ FOR $\tau \in C_p$, $ \tau _p \leq 1$ Objekttyp: Chapter Zeitschrift: L'Enseignement Mathématique Band (Jahr): 46 (2000) Heft 3-4: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am: Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz,

2 and thus The uniqueness of L p{s,t\x) follows from Lemma 2.5. D At this point we hâve not completed our goal of showing that the /?-adic function L p (s,t\x) exists for each rg C p, \r\ <1. In order to prove this, we will need to study the coefficients, a n(r), of the power séries expansion of L p (s,t',x) f r eacn r Q/?> t < 1. From the results of this we will show that the function L p (s,r;x) exists for each rg C p, \r\ <1, and for any séquence {r/}g 0 in Q p, with tv p <1, converging to r, the values L p (l -n, T/; x) converge to L p (l -n,r;x) for each n ez, n>\. 3.2 $L_p(s,\tau;\lambda)$ FOR \in C_p $\tau $, $ \tau _p \leq1$ Our previous work has been for rg Q p, [r^ <I.To extend this resuit to ail rg C p, \r\ p <I,we need to find a way to express a n (r) so that it can be defined for thèse values of r. For k G Z, fc > 0, the Stirling numbers of the first kind, s(n,k), are defined by the generating function (14) Since the power séries expansion of log(l +0 lacks a constant term, we must hâve s(n, k) = 0 whenever 0 < n < k. We also hâve s(tî, n) = 1 for ail n > 0. The s(n,k) are integers, where n,fc G Z, n>o, &>O, and they satisfy the relation (15) For additional information on Stirling numbers of the first kind we refer the reader to [6], pp LEMMA 3.6. Let re Q p, \r\ p <1. For n ez, n> -1, Proof. From Corollary 2.8 we can write

3 where secp such that \s\ p < \p\ l / (p l) \q\ p l- Now, expanding the quantity (^) according to (15) yields where s(m,ri) G Z is a Stirling number of the first kind. At this point we wish to switch the order of summation in this expression, but before doing so we must show that the ternis in the summation converge to 0 at a sufficient rate. Let e>o and let Ce C p such that \ \ p < \p\\ /{p ~l~ l) l \q\~. Then there such that exists 6eR, o<6< 1, Let N,M ez,n>o,m>o, such that if n>n then \pqf x \ ll 6 n <e, and if m > M then \pqf x \~ l \p\~ m/{p ~ l) \q\^ < e (such an M exists since Let m,n G Z, m > 0, n>o. If n> m, then s(m,ri) = 0, and so Thus we can assume that m= max{m, n}. Consider Utilizing Proposition 3.3 and the fact that v p (m\) < m/(p - 1), we can write Suppose that m> M + N. If m- n < M, then so that n>n. Thus If m?n > M, then

4 Either case implies that Therefore, whenever max{ra,rc} > M + N, this bound must hold, implying that by Proposition 2.4 Writing we hâve from (13) which implies the lemma, since we must hâve convergence for the inner sum.? Since we hâve only derived L p (s,r;x) f r T QQ P > \ T \ P <hwe cannot say that a n (r) is deflned for ail rg C p, \r\ p <1. For n ez, n>?l, let us define (16) for thèse values of r Note that in the proof of Lemma 3.6, the only influence. generated by the value of r is in the bound of the value of c m(r), which was determined in Proposition 3.3. However, this proposition holds for ail rg C p, \r\ <1. Thus this sum converges and a n{r) is well-defined for ail rec p, \r\ p <l.

5 Q p, THEOREM 3.7. Let rg C p, \r\ p <1, and let {77}^ with \n\ p <1, such that 77 -> r. Then for nez, n>-\, be a séquence in Proof. By définition, for rgc p) r p <1, and for ngz,n> -1, we hâve the expansion and as we hâve seen, regardless of the value of r, as m_> oo # Therefore such that given e >0 there must exist some ra0 G Z, ra0 > n+l, Thus for any séquence {77} ^ in Q p, with \n\ p <1, such that 77 -> r Since t; -^ r and c m(r) is a polynomial in r, we see that for ail m with n + 1 < m < mo when z is sufficiently large, which implies that for such /. Therefore the theorem must hold. n The purpose of the following three lemmas is to build an upper bound for the value of \a n{r)\ p. After doing so we can derme L p {s,r\x) f r au re C p,

6 LEMMA 3.8. Let p be prime. If i,n GZ with I<i<n, then For igz such that I</<n, (8) implies that v p (i\) <i-i,or Proof. equivalently, \i\\ p > ~ l \p\ ll. Therefore by combining this with the resuit will follow. D LEMMA 3.9. Let p be prime. Then for m,n GZ, m>n>o, Proof From (14), the generating function for the s(m,n) 9 we obtain Thus we wish to evaluate the power of p that divides the coefficient of t m in the expansion of (log(l + qt)) n The. power séries expansion of the logarithm function (10) yields and by factoring qt out of the sum For i>2,we see that p^q1'1/i G Z p. Therefore where /(/) G Z p [[>]]. Now, this can be written and from Lemma 3.8, the p-adic absolute value of the coefficients of the terms in the sum on the right must be bounded above by \np\ g ". Thus, for m>n, the coefficient of tm must also be bounded above by this quantity, implying the resuit.?

7 LEMMA Lé* rg C^, r p <1. Then for nez, n>o, Proof. From (16), we see that for ngz, n>o, Proposition 3.3 implies that By Lemma 3.9, when m > n + 2, Thus We are nearing our goal of deflning L p (s,t;x) for ail tg C p, t <1. The final step before doing so is proving the following lemma on the convergence of a spécifie infinité sum. LEMMA Let rg C p, \r\ p <1. Then the sum converges for ail s G ). Proof. Let G S). Then - l p < Ipl^^l^!; some 6GR, o<<s< 1, such that 1. Thus there must be Let n G Z, n > 0. From Lemma 3.10 and from Proposition 3.3,

8 Therefore which implies that Thus Now, so that Since o<6<l,we see that \a n (r)(? - l) w?? oasn?? 00. Thus the sum must converge. D (17) Note that from this proof we hâve obtained the bound for each ngz,n> -1, and for ail rg C p, \r\ p <1. Now let us define for te C p, \r\ p <1, and 5?S, lifx=l. This définition is consistent with what we already hâve for rg Q p, \r\ <I.We will now show that, for ail rg C p, r <1, this function satisfies for each n G Z, n > 1. To do this, we prove the following:

9 LEMMA Let rg C p, \r\ p <1, and let {77}^ be a séquence in Q p, with \Ti\ p <1, such that r,- -> r. Then for each ngz, re >1, Froo/ We can write where the power séries converges for each s G 5). Let e > 0, and let n?z, n> 1. Then we must hâve -n 1 thus the power séries converges for s? 1? n. Also, by (17) G S), and independently of r as m-» 00. Therefore, for ra0 GZ sufflciently large, For rg C p, Consider \r\ p <1, let {77} ^ be in Q^, with \n\ p <1, such that n-?r. Since ûm(r/)ûm(r/)?> a m(r) as r t -^ r, we hâve for / sufficiently large. Thus the lemma must hold.? At this point we hâve finally proven THEOREM For each rg C p, with \r\ p <1, there exists a unique p-adic, meromorphic function L p(s,r;x) that satisfies for each n G Z, n > 1. Furthermore, this function can be expressed in the form

10 where the power séries converges in the domain T>, and Since L p(s,t\x) is defined for each recp such that \r\ p < 1, we now hâve a p-adic function of two variables, L p {s,t;x)i where GS,s lif X=l, and tgcp with f <1. 4. Properties of $L_p(s,t;\lambda)$ Most of the properties that follow are direct conséquences of similar properties that hold for the generalized Bernoulli polynomials. In ail of the following we will take p prime and x a Dirichlet character with conductor fx. 4.1 A SYMMETRY PROPERTY IN t The first property we obtain regarding L p (s, t\ x) is a direct conséquence of the generalized Bernoulli polynomials being either odd or even functions, except when x = 1. Recall that L p (s,t;x) interpolâtes the values (18) for ngz,n>l, and tec p, \t\ p <1, where (19) and we define (20) LEMMA 4.1. For ail n ez, n>o,we hâve Proof. This holds for n=o since #0,1 (0 =1? Now assume that n>l. Because B n< \ =0 for odd n>3,we can write (2) in the form

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