Tirukkannapuram Vijayaraghavan

Last updated

Thirukannapuram Vijayaraghavan
Born(1902-11-30)30 November 1902
Died20 April 1955(1955-04-20) (aged 52)

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.

Related Research Articles

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".

In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes.

<span class="mw-page-title-main">Gamma function</span> Extension of the factorial function

In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n,

<span class="mw-page-title-main">Natural logarithm</span> Logarithm to the base of the mathematical constant e

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, logex, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann.

In mathematics, Catalan's constantG, is defined by

<span class="mw-page-title-main">Srinivasa Ramanujan</span> Indian mathematician (1887–1920)

Srinivasa Ramanujan was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable.

<span class="mw-page-title-main">Euler's totient function</span> Number of integers coprime to and not exceeding n

In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as or , and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ kn for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n.

<span class="mw-page-title-main">Euler's constant</span> Relates logarithm and harmonic series

Euler's constant is a mathematical constant usually denoted by the lowercase Greek letter gamma.

<span class="mw-page-title-main">Partition function (number theory)</span> The number of partitions of an integer

In number theory, the partition functionp(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4.

<span class="mw-page-title-main">Divisor function</span> Arithmetic function related to the divisors of an integer

In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.

In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1, all of whose Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in 1919 within the context of diophantine approximation. They became widely known after the publication of Charles Pisot's dissertation in 1938. They also occur in the uniqueness problem for Fourier series. Tirukkannapuram Vijayaraghavan and Raphael Salem continued their study in the 1940s. Salem numbers are a closely related set of numbers.

In mathematics and the field of number theory, the Landau–Ramanujan constant is the positive real number b that occurs in a theorem proved by Edmund Landau in 1908, stating that for large , the number of positive integers below that are the sum of two square numbers behaves asymptotically as

In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field.

In mathematics, plurisubharmonic functions form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions. However, unlike subharmonic functions plurisubharmonic functions can be defined in full generality on complex analytic spaces.

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 ).

<span class="mw-page-title-main">Riemann hypothesis</span> Conjecture on zeros of the zeta function

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.

In mathematics, specifically in number theory, the extremal orders of an arithmetic function are best possible bounds of the given arithmetic function. Specifically, if f(n) is an arithmetic function and m(n) is a non-decreasing function that is ultimately positive and

<span class="mw-page-title-main">Grunsky matrix</span>

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.


  1. M.S. Raghunathan, Artless innocents and ivory-tower sophisticates: Some personalities on the Indian mathematical scene.
  2. 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.