The Journal of the Indian Mathematical Society

P. Masani ¹, T. Vijayaraghavan ²

Affiliations
1 The Institute of Science, Bombay, IN
2 Ramanujan Institute of Mathematics (Karaikudi), Madras, IN

Abstract

Our purpose is to establish the following theorem.

Theorem I. Given open disks D₁ and D₂ in the complexplane such that D = D₁ ∩ D₂ is non-void, and a function f on D with values in a (non-commutative) Banach algebra over the complex field such that f is holomorphic and reciprocable on D, then f is the product of two functions f₁ and f₂ such that f₁ is holomorphic and reciprocable on D₁ and f₂ is holomorphic and reciprocable on D₂.

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T. Vijayaraghavan ¹

Affiliations
1 Ramanujan Institute of Mathematics, Madras, IN

Abstract

Let A be any positive integer and let Φ(x, A) denote the number of positive integers which are prime to A and less than or equal to x. The object of this note is to prove a result about the upper and lower bounds of e₁, e₂, e₃,..., where

e_n = e_A,n = Φ(n,A)-n Φ(A)/A, n = 1, 2, 3 , . . .,

and Φ(A) stands for Φ(A, A).

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T. Vijayaraghavan ¹

Affiliations
1 Andhra University, IN

Abstract

In this note I answer two questions of Dr R. Vaidyanathaswamy.

The questions are

(i) Suppose that a connected linearly ordered topological space S has the power of the continuum; does it follow that S has an everywhere dense enumerable subset?

(ii) A point P of a connected topological space S is said to be a cut point if its removal splits S into two (and only two) disjoint connected open topological spaces. If every point P of a connected topological space S is a cut point of S does it imply then that S is a linear space?.

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S. D. Chowla , T. Vijayaraghavan

Abstract

The object of this note is to give an answer to the

QUERY: Let g(m) denote the largest prime divior of m. In what range does g(m) lie for all almost all values of m ≤ x?

More precisely, suppose that h(x) and H(x) aie two functions of x; let N(x) = N(h, H x) denote the number of numbers m ≤ x for which h(x) ≤ g(m) ≤ H(x). For what choices of h(x) and H(x) can we say that N(x)/x→I, as x→∞?.

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T. Vijayaraghavan ¹

Affiliations
1 Andhra University, IN

Abstract

It is known that the behaviour of a power series on the circumference of its circle of convergence can be very complicated.

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T. Vijayaraghavan

Abstract

The object of this note is to discuss the existence of certain configurations of points on the circumference of a circle, which involve in their definitions partly combinatory moments and partly such connected with the theory of sets of points. The interest of this discussion lies perhaps primarily in its connection with some theorems on the existence of certain configurations of singular points on the circle of convergence. We must therefore, before formulating the results, state these connections.

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