On Arithmetical Properties of Lambert Series

rr=l u=l Chowla* has proved that if t is an integer 2 5, then g(x it) is irrational. He also conjectures that for rational 1 x I< J bothJ(x) and g(x) are irrational. In the present note we prove the following THEOREM. Let [ t [ > 1 be an.2 integer. Then both f(I It) and g(I /t) are irm tionak. We only give the details forf(I/t) ; the proof for g(I/t) follows by the method of this note and that of Chowla. Let us first assume that t is positive and that n is large. Put k = [(log n) " " '] and let pl, fir,. .. , be the sequence of consecutive primes greater than (log n)2. Put Ad I 1 a; i; Rck+l?sIt. 2 J From elementary results about the distribution of primes it follows that pi < 2 (log n)2 for k < WSI)-2 * Thus by a simple computation we obtain A < { 2 (log n) }th2 < e(log n)*14, (1) Consider now the following congruences : x =plf-l (mod pi) x+1 = (Pz IW-~ { mod (P2~dt) 4..