Thirukannapuram Vijayaraghavan | |
---|---|

Born | 30 November 1902 |

Died | 20 April 1955 52) | (aged

Occupation | mathematician |

**Tirukkannapuram Vijayaraghavan** (Tamil : திருக்கண்ணபுரம் விஜயராகவன்; 30 November 1902 – 20 April 1955) was an Indian mathematician from the Madras region. He worked with G. H. Hardy when he went to Oxford in mid-1920s on Pisot–Vijayaraghavan numbers. He was a fellow of Indian Academy of Sciences elected in the year 1934.

Vijayaraghavan was well versed in Sanskrit and Tamil. He was a close friend of André Weil. He served with him in Aligarh Muslim University. He later moved to the University of Dhaka in protest at Weil's firing from AMU.^{ [1] }

Vijayaraghavan proved a special case of Herschfeld's theorem on nested radicals:^{ [2] } For

converges if and only if

where denotes the limit superior.

In number theory, an **arithmetic**, **arithmetical**, or **number-theoretic function** is for most authors any function *f*(*n*) whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of *n*".

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In mathematics, the **Hardy–Ramanujan theorem**, proved by G. H. Hardy and Srinivasa Ramanujan (1917), states that the normal order of the number ω(*n*) of distinct prime factors of a number *n* is log(log ).

In mathematics, the **Riemann hypothesis** is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.

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In complex analysis and geometric function theory, the **Grunsky matrices**, or **Grunsky operators**, are infinite matrices introduced in 1939 by Helmut Grunsky. The matrices correspond to either a single holomorphic function on the unit disk or a pair of holomorphic functions on the unit disk and its complement. The **Grunsky inequalities** express boundedness properties of these matrices, which in general are contraction operators or in important special cases unitary operators. As Grunsky showed, these inequalities hold if and only if the holomorphic function is univalent. The inequalities are equivalent to the inequalities of Goluzin, discovered in 1947. Roughly speaking, the Grunsky inequalities give information on the coefficients of the logarithm of a univalent function; later generalizations by Milin, starting from the Lebedev–Milin inequality, succeeded in exponentiating the inequalities to obtain inequalities for the coefficients of the univalent function itself. The Grunsky matrix and its associated inequalities were originally formulated in a more general setting of univalent functions between a region bounded by finitely many sufficiently smooth Jordan curves and its complement: the results of Grunsky, Goluzin and Milin generalize to that case.

- ↑ M.S. Raghunathan, Artless innocents and ivory-tower sophisticates: Some personalities on the Indian mathematical scene.
- ↑ Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: American Mathematical Society (2000), p. 348.

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