On the papers of Ramachandra and Kátai

where the sum over m is over all positive integers and p runs over all primes. For any complex constant z, we can define (L(s, χ)) as exp(z logL(s, χ)). Let S2 consist of the set of all derivatives of L(s, χ) for all L-series and let S3 denote the set of logarithms as defined above for all L-series. Let P1(s) be any finite power product (with complex exponents) of functions of S1. Let P2(s) be any finite power product (with non-negative integral exponents) of functions of S2. Also let P3(s) denote any finite power product (with non-negative integral exponents) of functions of S3. Let cn (n = 1, 2, 3, . . .) be complex numbers which are Oε(n) for every positive constant ε and suppose that F0(s) = ∑∞ n=1 cnn −s is absolutely convergent in Re s > 1/2. Finally, put