Generalized divisor problem

Transcription

1 Generalized divisor problem C. Calderón * M.J. Zárate The authors study the generalized divisor problem by means of the residue theorem of Cauchy and the properties of the Riemann Zeta function. By an elementary argument similar to that used by Dirichlet in the divisor problem, Landau [7] proved that if α and β are fixed positive numbers and if α β, =ζ(β/α)x /α + ζ(α/β)x /β + O(x /(α+β) ). m α n β x In 952 H.E. Richert by means of the theory of Exponents Pairs (developed by J.G. van der Korput and E. Phillips ) improved the above O-term ( see [8] or [4] pag. 22 ). In 969 E. Krätzel studied the three-dimensional problem. Besides, M.Vogts (98) and A. Ivić (98) got some interesting results which generalize the work of P.G. Schmidt of 968. In 987 A.Ivić obtained Ω-results for T 2 k(a,...,a k,x)dx where k (a,...,a k,x) is the error-term of the summatory function of d(a,...,a k,n). Moreover, he proved that his results hold for the function ζ q (b s)ζ q 2 (b 2 s)ζ q 3 (b 3 s)..., b <b 2 <b 3... q j, being some positive integers. In 983 E.Krätzel studied the many-dimensional problem m a...m an n, m b...m bn using the properties of the Riemann Zeta-function. M. Vogts (985) also studied that problem but using elementary methods. Here, we will analyse some sums of the above kind by means of non-elementary methods. Supported by Grant PB of the DGICYT and the University of the Basque Country Received by the editors November 993 Communicated by Y. Félix AMS Mathematics Subject Classification : N37, 30B50 Bull. Belg. Math. Soc. 2 (995), 59 63

2 60 C. Calderón M.J. Zárate Theorem. Let a j >, > 0, (j =,...,n), (n 2) be real numbers. Suppose that the fractions +a j belong to the gap (c,c+min{0, }],j =,...,n 2 and If c = +a b +a j =max. j n () n m( (c ) a j ) where we have m b...m bn When the fractions +a j and we have When Res s= +a k m(σ) =sup{m : T m a...m an n = Res s= +a k ζ(σ + it) m T +ɛ, ɛ >0} ζ( s a j ) xs s + O(xc 2b +ɛ ). are different, the point s = +a j is a simple pole of ζ( s a j ) ζ( s a j ) xs n s = k= j k +a r b r = +a r+ b r+ ( r<r+ q n), thesum ζ( +a k =...= +a r+q b r+q a j )x +ak. r+q k=r j k ζ( +a k a j )x +ak is replaced by x +ar br P r,q (log x) where P r,q (log x) is a polynomial in log x of degree q. Proof. The function n ζ( s a j ) can be expresed as a Dirichlet series absolutely convergent for σ>c m a...m an n N s. N= m b...m bn n =N

3 Generalized divisor problem 6 By Lemma 3.2 [0] if x isthehalfofanoddintegerwehave m b...m bn m a...man n = c+ɛ+it 2πi c+ɛ it ζ( s a j ) xs ds + O(xc+ɛ s T ). We will consider the rectangle R T with vertices c ± it, c + ɛ ± it. Therefore, = Res s= +a k c+ɛ+it 2πi c+ɛ it ζ( s a j ) xs s ds = ζ( s a j ) xs s + 2πi (I + I 2 + I 3 ) where I,I 2,I 3 are the integrals along the other three sides of the rectangle I + I 3 c+ɛ c ζ( σ a j + i T ) x σ T dσ. Let µ(σ) =inf{ξ, ζ(σ + it) =O( t ξ ), as I + I 3 c+ɛ T c t } n µ(σ a j )+ɛ T x σ dσ T + n µ((c+ɛ) a j )+ɛ x c+ɛ + T + n µ((c ) a j )+ɛ x c. From hypothesis () we can deduce + n µ( (c ) a j ) 0 therefore For the integral I 2 we have I + I 3 x c T ɛ + xc+ɛ T. I 2 x c + x c T x c + x c T T ζ( (c ) a j + t) dt t ζ( (c ) a j + t) dt. Calling m j = m( (c ) a j )

4 62 C. Calderón M.J. Zárate by the Hölder inequality and hypothesis () we get I 2 x c T ɛ. Then m b...m bn m a...man n = Res s= +a k ζ( s a j ) xs s + +O( xc+ɛ T )+O(xc T ɛ ). Taking T = x d for some sufficiently large d>0we obtain (2) and theorem is proved. Case A. Letb =...= b n =,a j >, 0 a a j < /2 forj =,...,n.if n m((/2) + a a j ) m...m n x m a...m an n = Res s=+a j ζ(s a j ) xs s + O(xa +(/2)+ɛ ) for every ɛ>0. For example, let a > anda j = a j,(j =2,...,n). be real numbers. 2n For 4 n 0, from Theorem 8.4. [2], the hypothesis () is verified and the above formula holds. Case B. Leta =...= a n = a> y0<b... b n.if ( +a + a) < ( b 2 b 2 + a)+,j =,...,n 2 and, besides m b...m bn n m(( /b )( + a) a) 2 (m...m n ) a = Res s= +a ζ( s a) xs s +O(x b (a+(/2)+ɛ ). In particular if a =and0<b < 4b 3 and if the condition about the moments is satisfied, the order of the O-term is x 3 +ɛ for every ɛ>0.

5 Generalized divisor problem 63 References [] A. Ivić, On the number of finite non isomorphic abelian groups in short intervals. Math. Nachr. 0(98), [2] A. Ivić, The Riemann zeta function. John Wiley and sons. New York 985. [3] A. Ivić, The general divisor problem. Journal of number theory, 27 (987), [4] E. Krätzel, Lattice Points. Kluwer Academic Publishers. Dordrech Boston London [5] E. Krätzel, Teilerproblem in drei dimensionen. Math. Nachr. 42 (969), [6] E. Krätzel, Divisor problems and powerful numbers. Math. Nachr. 4 (983) [7] E. Landau, Au sujet d une certaine expression asymptotique. L intermédiaire des math. Vol. 20 (93), p.55. [8] H.E. Richert, Uber die anzahl abelscher gruppen gegebener ordnung I. Math. Zeit. 56 (952), [9] P.G.Schmidt. Zur Anzahl abelscher gruppen gegebener ordnung I. J.Reine Angew. Math. 229 (968), [0] E.C.Titchmarsh, The theory of the Riemann zeta function. Clarendon Press, Oxford (95). [] M. Vogts, Teilerproblem in drei dimensionen. Math. Nachr. 0(98), [2] M. Vogts, Many dimensional generalized divisor problems. Math. Nachr. 24 (985) C.Calderón and M.J. Zárate Universidad del País Vasco Departamento de Matemáticas E Bilbao- España MTPCAGAC@lg.ehu.es