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### Vijayaraghavan, T.

- An Analogue of Laurent's Theorem for a Simply Connected Region

#### Authors

**Affiliations**

1 The Institute of Science, Bombay, IN

2 Ramanujan Institute of Mathematics (Karaikudi), Madras, IN

#### Source

The Journal of the Indian Mathematical Society, Vol 16 (1952), Pagination: 25-30#### Abstract

Our purpose is to establish the following theorem.

Theorem I. Given open disks D_{1} and D_{2} in the complexplane such that D = D_{1} ∩ D_{2} 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 f_{1} and f_{2} such that f_{1} is holomorphic and reciprocable on D_{1} and f_{2} is holomorphic and reciprocable on D_{2}.

- On a Problem in Elementary Number Theory

#### Authors

^{1}

**Affiliations**

1 Ramanujan Institute of Mathematics, Madras, IN

#### Source

The Journal of the Indian Mathematical Society, Vol 15 (1951), Pagination: 51-56#### 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 e_{1}, e_{2}, e_{3},..., 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).

- On Two Problems Relating to Liner Connected Topological Spaces

#### Authors

^{1}

**Affiliations**

1 Andhra University, IN

#### Source

The Journal of the Indian Mathematical Society, Vol 11 (1947), Pagination: 28-30#### 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?.

- On the Largest Prime Divisors of Numbers

#### Authors

#### Source

The Journal of the Indian Mathematical Society, Vol 11 (1947), Pagination: 31-37#### 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→∞?.

- A Power Series that Converges and Diverges at Everywhere Dense Sets of Points on its Circle of Convergence

#### Authors

^{1}

**Affiliations**

1 Andhra University, IN