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On the Behavior of Power Series with Completely Additive Coefficients

Petrushov O.

Bulletin of the Polish Academy of Sciences. Mathematics

2015

Vol. 63, no. 3

217--225

Consider the power series A(z)=∑∞n=1 α (n) zn, where α(n) is a completely additive function satisfying the condition α(p) = o(lnp) for prime numbers p. Denote by e(l/q) the root of unity e2πil/q. We give effective omega-estimates for A(e(l/pk)r) when r→1−. From them we deduce that if such a series has non-singular points on the unit circle, then it is a zero function.

Composition of arithmetical functions with generalization of perfect and related numbers

Shukla D. P., Yadav S.

Commentationes Mathematicae

2012

Vol. 52, [Z] 2

153--170

In this paper we have studied the deficient and abundent numbers connected with the composition of φ,φ*, σ,σ* and ψ arithmetical functions , where φ is the Euler totient, φ* is the unitary totient, σ is the sum of divisors, σ* is the unitary sum of divisors and ip is the Dedekind function. In 1988, J. Sandor conjectured that ψ(φ(m))≥m, for all odd m and proved that this conjecture is equivalent to ψ(φ(m))≥m/2 for all m. Here we have studied this equivalent conjecture. Further, a necessary and sufficient conditions of primitivity for unitary r-deficient numbers and unitary totient r-deficient numbers have been obtained . Finally, we have discussed the generalization of perfect numbers for an arithmetical function Eα.

On Unitary Analogue of f_g-perfect numbers and Ψ_s-perfect numbers

Shukla D. P., Pandey S.

Commentationes Mathematicae

2011

Vol. 51, [Z] 2

141-153

In this paper unitary analogue of f_g-Perfect numbers and some properties of Dedekind's function and all the Ψ-perfect numbers have been discussed.