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K. G. Ramanathan ¹

Affiliations
1 Tata Institute of Fundamental Research, Bombay, IN

Abstract

In the previous paper Minakshisundaram has obtained a generalization of the Riemann sphere to spaces of complex symmetric matrices.

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K. G. Ramanathan ¹

Affiliations
1 Tata Institute of Fundamental Research, Bombay, IN

Abstract

In 1924 Hasse proved a fundamental theorem concerning quadratic forms over algebraic number fields, namely that two quadratic forms with coefficients in an algebraic number field K are equivalent if and only if they are so in every completion K_p of K by valuations of K.

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K. G. Ramanathan ¹

Affiliations
1 Tata Institute of Fundamental Research, Bombay, IN

Abstract

E being the unit matrix of order m. The symplectic group Sp(m) is defined to be the set of real matrices M such that

M'JM=J, (1)

M' being the transposed of M.

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K. G. Ramanathan ¹

Affiliations
1 Osmania University, IN

Abstract

The object of this paper is to prove some theorems regarding the product of elements in a finite abelian group.

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K. G. Ramanathan ¹

Affiliations
1 University of Madras, IN

Abstract

Let T(n) be Ramanujan's function defined by

Σ T(n) xⁿ = x [(I-x) (I-x⁾²)...]²⁴.

It was recently proved by me that

T(kn-I) = 0 (mod k),            (I. I)

where k = 3, 4, 6, 8, 12 or 21 and n ≥ I. The object of this paper is to find the residues of T(n) to the moduli 3, 4 and 7.

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K. G. Ramanathan ¹

Affiliations
1 University of Madras, IN

Abstract

Dr. R. Vaidyanathaswamy has proved that every quadratic multiplicative arithmetic function satisfies a certain identity.

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K. G. Ramanathan ¹

Affiliations
1 Institute for Advanced Study, Princeton, NJ 08540, US

Abstract

In this note we prove a general theorem, stated as Theorem 2 below, as an application of the classical Kronecker limit formula and relate it to several problems arising in Ramanujan’s work concerning singular values of certain modular functions.

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K. G. Ramanathan ¹

Affiliations
1 The Institute for Advanced Study, Princeton, N.J.-08540, US

Abstract

Srinivasa Ramanujan was born at Erode, about 250 miles west south west of Madras, the capital city of Madras Presidency, on Thursday, 22 December 1887 at 6 : 00 P.M. as the son of Srinivasa Aiyangar of Kumbakonam, a town in the Thanjavur (Tanjore) district of the Madras Presidency, and his wife Komalattammal of Erode. Ramaujan’s mother Komalattammal was a strong-willed and intelligent woman, daughter of Narayana Aiyangar who was well known in Erode though he was only a modest official in the District Court.

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