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FUNCTIONS Ci{x,y) AND Si\x\ y)*> TAMOTSU TSUCHIKURAMr. Gennόsuke Hara has introduced the following functions 1 *:where LF 0 (2/, 2y) a ^d U 1 (2f>2ji) are LcmmeΓs fmctions of order zero and one respectively.In this note we shall establish some relations concerning these functions, especially Hara's function γ(j/)* £) FORMULA ( 1). If ftv > -1, then (sd = TMT) ί 1 -Z \?Ji k dπ 0) cos PROOF. We shall prove the first part. (! ε sto "> cos " + '' •* -*) Received May 1, 1946. 2) In the sequel, we adopt the notations in the book: Watson, Theory of BesseJ functions, 1922.

ByTAMOTSU TSUCHIKURA This paper contains three § § on different problems. In the first § we shall concern the problem of J.D.Hill and S.Kakutani, in the second the extension of a theorem of H. Steinhaus, and in the third a property of sυbserios of di* vergent series with function-terms. § 1. On a problem of J. D. Hill and S. Kakutani.Let (r«(x)} be the Rademacher system, JL D. Hill (£23) and S. Kakutani (CO) proposed the problem, whether, for the sequence 0<^pi^p. 2 S>• °° such that p n /fa + -+ p n ) -> 0 {n -• oo), the Riesz meantends to zero almost everywhere as n -> oo.I was told that this problem had been solved negatively by S. Kakutani, but I am not aware of his precise result. On the other hand, using the independency of the system, G. Maruyama recently gave a negative example using the Kolmogorofϊ lemma on the law of the iterated logarithm (£53).We shall give here another negative example (£93) by an elementary lemma of S. Mazur and W. Orlicz LEMMA. Let f(x) be a measurable function of period 1. Then for any sequence of positive numbers n±< n%<• », we have', for almost all x,•) Received Aug. 20, 1949.

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