On the average order of a class of arithmetical functions,II

We consider a large class of arithmetical functions generated by Dirichlet series satisfying a very general functional equation with gamma factors. In our previous paper we obtained a “one-sided” Ω result, but here a “two-sided” Ω result is obtained in most cases. Unfortunately, the method fails in the classical circle and Dirichlet divisor problems.     

Asymptotic results for a class of arithmetical functions

In this paper, we obtain asymptotic formulae for the summatory functions of a class of arithmetical functions. These extend, and in certain cases refine, earlier results due toS. S. Pillai andH. G. Kopetzky.     

On the order of a class of meromorphic functions

This paper proves the following result: Letf(z) be a meromorphic function in thez-plane with a deficient value, and δ(θ _k )(k=1,2, ...,q;0≤θ ₁<θ₂<...<θ _q<θ _q+1=θ ₁+2π) beq rays (1≤q<∞) starting at the origin, and letn≥3 be an integer such that for any given positive numberε,0<ε<π/2, $$\overline {\mathop {\lim }\limits_{r \to \infty } } \frac{{\log ^ + n\left\{ { \cup _{k = 1}^q \Omega \left( {\theta _k + \varepsilon ,\theta _{k + 1} - \varepsilon ,r} \right),f\prime f^n = 1} \right\}}}{{\log r}} \leqslant v< \infty ,$$ whereΝ is a constant independent ofε. IfΜ<∞, then we have $$\lambda \leqslant \frac{\pi }{\omega } + v,$$ whereΜ andλ denote the lower order and order off(z), respectively,Ω=minθ _k+1 ?θ _k ;1≤k≤q, andn(E, f=a) is the number of zeros off(z)?a inE with multiple zeros being counted with their multiplicities.     

On random arithmetical functions. I

Let A _Q be the group of complex unit roots of an integer order \( Q \geqslant 2 \). Let \( {\xi_p}\left( {p \in \mathcal{P}} \right) \) be independent random variables distributed uniformly on the set A _Q , where \( \mathcal{P} \)is the set of primes. Let f be a completely multiplicative function defined on \( \mathcal{P} \) by f(p) = ξ _p . We investigate the summatory function of f(n) and the density of those n for which f(n + j) = κ _j (j = 0, …, t), where κ _j ∈ A _Q .     

On the multivalence and order of starlikeness of a class of meromorphic functions

Basically this paper deals with the determination of the radius of starlikeness and radius of univalence of the class of meromorphic functions of the formg(z)=A/z−Φ(z) forA>0, and where Φ(z) is an analytic function defined in the unit disk whose modulus does not exceed unity. We estimate the radius ofp-valence of functions having the fromh(z)=Φ(z)+a/z ^p fora>1,p≧1, and also estimate the radius of starlikeness of certain Blaschke products which is also given as a function of the minimum modulus function. We discuss the question of sharpness of the results and mention some open problems.     

On arithmetical functions related to the Ramanujan sum

Let m and n be positive integers, and μ the M"bius function. And let S _f(m,n) be the function defined by , where f is an arithmetical function. We show that this function has many properties like the Ramanujan sum. Firstly we study the partial summation formula involving S _f(m,n) and taking f=μ, we obtain the Dirichlet series with the coefficients Sμ(m,n) and Sμ(m,n)d(m). Moreover we show a certain property which is analogous to the orthogonality relation of the Ramanujan sums. This revised version was published online in June 2006 with corrections to the Cover Date.     

On arithmetical level of the class of superhigh sets

We determine the proper arithmetical level of the class of superhigh sets.     

Prime-independent arithmetical functions

Summary This paper studies arithmetical functions that are prime-independent and multiplicative. Firstly, it establishes necessary and sufficient conditions for such functions to possess simple formulae relating them to the zeta function. Then it investigates asymptotic average-values and moments of such functions. The results apply to functions of ideals in algebraic numbers fields, or of isomorphism classes in certain categories, as well as to functions of positive integers. Entrata in Redazione il 10 ottobre 1972.