ON DIRICHLET CONVOLUTION METHOD

In this paper we have proved limitation theorem for (D,h(n)) summability methods and have shown that it is best possible.


INTRODUCTION
In his studies on the prime number theorem, Ingham defined a novel summability method called (I) This was generalized by Segal [2] and he defined the notion of (D,h(n)) summability, where h N R denotes a function with h(1) I We define the "Dirichlet inverse" h*(n A series an is said to be (D,h(n)) summable to L ifand onlyif n oo-v adh(v/d) L.
(1 l) v=l Given a series an and a specific h(n), define the function it clearly makes no difference to the existence or value of the limit (l 2) Since D([t])= whether c is through real values or integers.Ingham's method corresponds to the case h(n) Segal [3] proved the limitation theorem for (I) summability.If an is (I) summable, then ' an o(log x) and has shown in the following theorem that his result is best possible TBEOREM A [4] Let e (x) be any positive function decreasing to 0 monotonically but arbitrarily slowly as x oo.Then there exists a series Y a, which is (I) summable and such that E a, : O( 6 (x)logx) as x oo.
Sukla [5] has shown an analogous limitation theorem for (D, h(n)) summability.TREOREM B. If a is (D,h(n)) summable then an O(logx) if (i) H'(r) E h'(n) 0 (1)   and (ii) It is remarked in that paper that the condition (ii) cannot be dropped However if we replace (i) by a slightly stronger condition then we get the result to be true without assuming (ii) In section 4 we show that our revised version of Theorem A is best possible.
(2 2) We will show that (3 1) is a best possible result THEOREM 2. Let E (2:) be any positive function decreasing to 0 monotonically but arbitrarily slowly as 2: oo.Then there exists a series a, which is (D, h(n)) summable and (3 2) holds and l<d< (2 3) does not tend to zero as r oo holds and such that E :/: o( (x) E lh*(n)[) as 2: PROOF OF THEOREM 1.For m _> 0, let roD(m) if m_>l K(m)= 0  SUKLA Since in this last sum + < 1 the inner sum contains at most one term, and so 1 a<]H.(r) (r)] 1 n<_r n_r tends to infinity as r oo since by (2.3) the expression in the bracket does not tend to zero as r oo This completes the proof of Theorem 2 Agnew [6] showed directly that, for r )0 the Ces/u'o and Riesz transforms C(n), P(n) respectively of a given series an are equiconvergent i.e C(n), Re(n) exist for each n and [lib O.
These concepts are applied to arithmetic summation methods (I) and (D, h(n)) for particular values of h(n) by Jukes [7] He has found different conditions under which the equiconvergence of (I) and (D, " 2 ( ' ----2 ) , have been established.The (D, "-), and (I) transform are given by b,, TItEOREM C [7] Tauherian constants M do not exist for comparisons of conservative matrices with non-conservative matrices xarORr , tTa

A2<oo
We have proved (see Kuttner and Sukla  [8]) that TI:IEOREM E. The (D,h(n)) is conservative if and onlyif Ih(n)[ < oo It isto notethat if rt--1 part of the above theorem was proved earlier by Jukes [9] See S. L. Segal, Math.Reviews 86e 11093 (May 1986, p 1864) THEOREM 3. The (D,h(n)) and (I) are not equiconvergent whenever A < oo and E Ih(n)l < oo PROOF.By Theorem C since (I) is not conservative and (D,h(n)) is conservative for Ih(n))] < oo whenever A2 < oo(D, h(n)) and (I) are not equiconvergent From Theorem E also we get that the following theorem of Jukes as corollaries COROLLARY 2. (D,#2(n)/n) and (D, E A(n)/Trg-(n)) transforms are not equiconvergent whenever A2 < oo.

ACKNOWLEDGMENT. improvement of the paper
We are thankfial to the referee for his valuable suggestions for the (2:) O for x < 1 PROOF OF THEOREM 2. Define b, by

1 H
does not hold then b,=n<_r o ( e ( r ) -n _ < r h" (n) known that in order for c,d to transform all sequences tending to 0 into sequences tending to 0, for all r where c is independent of r rl/2<d<_r

COROLLARY 1 .
The methods (D, (-) and (D, x( -n2) are not conservative PROOF.Sinceand are not absolutely convergent So by Theorem 3 the result n=l n=l follows.