An Omega Theorem on the General Asymmetric Divisor Problem - Core

Journal of Number TheoryNT2151 journal of number theory 65, 268278 (1997) article no. NT972151

An Omega Theorem on the General Asymmetric Divisor Problem* Manfred Kuhleitner Institut fur Mathematik, Universitat fur Bodenkultur, Gregor Mendel Strasse 33, A-1180 Vienna, Austria Communicated by R. F. Tichy Received May 13, 1996; revised January 15, 1997

This paper deals with a lower estimate for the general asymmetric divisor problem. Continuing on a paper of Nowak, a lower bound for the error term in the asymptotic formula for the corresponding Dirichlet summatory function is established.  1997 Academic Press

1. INTRODUCTION For a given integer p2 and fixed natural numbers a=a 1 a 2  } } } a p , consider the general asymmetric divisor function d(a 1 , ..., a p ; n)=d(a; n)=*[(m 1 , ..., m p ) # N p : m a11 } } } m app =n]

(n # N).

For a large real variable x, we consider the remainder term E(a; x) in the asymptotic formula D(a; x)= : d(a; n)=H(a; x)+E(a; x),

(1)

nx

where H(a; x)=

Res s=s0

: s0 =0, 1a1 , ..., 1ap

\

p

` `(a j s) j=1

xs . s

+

A survey on results concerning upper estimates for the remainder function E(a; x) is given in the textbook of Kratzel [10], supplemented by the recent papers of Menzer [13] and Kratzel [11]. * This paper is part of a research project supported by the ``Austrian Science Foundation'' (No. P 9892-PHY).

268 0022-314X97 25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.

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OMEGA THEOREM

269

Lower estimates, for the special case p=2 were established by Kratzel [9], Schierwagen [17, 18], and Hafner [7]. Concerning lower estimates for the general case p2, a classic theorem of Landau [12] contains the estimate E(a; x)=0 \(x ( p&1)2(a1 + } } } +ap ) ). This result was improved by Ivic [8] and recently by Nowak [16], who showed that E(a; x)=0 (x %(log x) a% (log log x) p&1 ), * where V=\ in the case p4, V=+ in the case p=2, 3 and %=

p&1 . 2(a 1 + } } } +a p )

(2)

2. SUBJECT AND RESULT OF THE PAPER Continuing on the paper of Nowak [16], we improve the 0-theorem for E(a; x) using Hafner's ingenious refinement of Hardy's technique. (See, e.g., [6, 7].) Theorem. E(a; x)=0 (x %(log x) a% (log log x) p&1+ca% exp(&A - log log log x)), * where % is defined in (2), A is a positive constant, c=max[ p log 2&1, p log(a+1)&a],

(3)

and V=\ for p4, whereas V=+ in the case p=2 and 3. Remark. The special case a 1 =a 2 =1, p=2 yields the celebrated result established in Hafner [5].

3. SOME LEMMAS Let a be as above. We say that n is ``a-full '' if for any prime q which divides n, q a divides n, too. For large positive real x we define A(x)=[n # N | nx and n ``a-full ''],

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MANFRED KUHLEITNER

and H(x)=[n # A(x) | w(n)p log log x&A - log log x], where A is a positive constant and |(n) is the number of distinct prime divisors of n. Lemma 1. (a)

*H(x)< >(log q) &a (log t) a (log log t) &ca exp(&A - log log log t). We define B 0 =c 0(log q) &a (log t) a1 (log log t) &ca exp(&A - log log log t) with c 0 sufficiently small such that B 0