On a paper of Dressler and Van de Lune
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ORIGINAL ARTICLE
On a paper of Dressler and Van de Lune P. A. Panzone1 Received: 20 September 2019 / Accepted: 1 February 2020 Ó Sociedad Matemática Mexicana 2020
Abstract If z 2 C and 1 n is a natural number then X ð1 zp1 Þ ð1 zpm Þzq1 e1 þþqi ei ¼ 1; d1 d2 ¼n
pr11 . . .prmm ,
where d1 ¼ d2 ¼ qe11 . . .qei i are the prime decompositions of d1 ; d2 . This is one of the identities involving arithmetic functions that we prove using ideas from the paper of Dressler and van de Lune [3]. Keyword Arithmetic functions Identities Zeta function
Mathematics Subject Classification 11A25 11MXX
1 Introduction and results Recall that the arithmetic function xðnÞ is the number of distinct prime divisors of a positive integer n and XðnÞ is the total number of prime divisors of n. In other words if for a natural number n 2 we write n ¼ pr11 . . .prmm with pi distinct primes (a notation that we keep throughout this paper where p always denotes a prime) then xðnÞ ¼ m, XðnÞ ¼ r1 þ þ rm and Xð1Þ ¼ xð1Þ ¼ 0. Write fðsÞ for the Riemann–Zeta function. One knows that xðnÞ ¼ Oðlog n= log log nÞ which easily implies that P1 xðnÞ s =n is an entire function of z when Rs ¼ r [ . Also n¼1 z UNS-INMABB-Conicet. & P. A. Panzone [email protected] 1
Departmento e Instituto de Matema´tica (INMABB), Universidad Nacional del Sur, Av. Alem 1253, Bahia Blanca 8000, Argentina
123
P. A. Panzone 1 XðnÞ X z n¼1
ns
¼
Y Y z z2 1 1 þ s þ 2s þ ¼ ; p 1 z=ps p p p
is an analytic function of z if 1\r and jzj\2r . In [3] the following remarkable duality relation was proved. Theorem 1 (R. Dressler and J. van de Lune) If jzj\2r and Rs ¼ r [ then ! ! 1 1 XðnÞ X X ð1 zÞxðnÞ z ¼ fðsÞ: ns ns n¼1 n¼1
The aim of this note is to obtain similar formulas using the methods of [3]. To state our results we need some definitions: let xo ðnÞ (xe ðnÞ respectively) be the number of primes in the decomposition of n with odd (even respectively) exponent. Thus xo ð22 35 56 Þ ¼ 1; xe ð22 35 56 Þ ¼ 2. Note that xe ðnÞ þ xo ðnÞ ¼ xðnÞ. b xc is the floor function and l is the Mo¨bius function. The radical of a number n is defined as radðnÞ ¼ p1 pm . Ramanujan’s tau function is defined by (see [2], p. 136) z
1 1 Y X ð1 zn Þ24 ¼ sðnÞzn ; n¼1
n¼1
and its associated Dirichlet series is 1 X sðnÞ n¼1
ns
¼
Y p
1 : 1 sðpÞps þ p112s
ð1Þ
One has the bound jsðpÞj 2p11=2 :
ð2Þ
This result was conjectured by Ramanujan and it was proved by Deligne [2]. The main contribution of this note is the following theorem. Note: in formulas (a)–(g) below it is assumed that in all the sums the term with n ¼ 1 is equal to 1. Theorem 2 (a)
If 1\r ¼ Rs; jzj then 1 X ð1 zp1 Þ ð1 zpm Þ n¼1
(b)
If 1\r; jzj 1 then
123
ns
!
1 p1 r1 þþpm rm X z n¼1
ns
! ¼ fðsÞ:
On a paper of Dressler and Van de Lune 1 p1 r1 þþpm rm X z
ns
n¼1
(c)
¼ fðsÞ
1 X ðzp1 1Þ ðzpm 1Þzðr1 1Þp1 þþðrm 1Þpm : ns n¼1
If 0\jzj\2r and 1\r then 1 XðnÞxðnÞ X z ð1 þ zÞxðnÞ
ns
n¼1
(d)
If z 2 C and 1\r then 1 X ðz þ 2ÞxðnÞ
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