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Hardy–Littlewood Zeta-function Conjectures
In mathematics, the Hardy–Littlewood zeta-function conjectures, named after Godfrey Harold Hardy and John Edensor Littlewood, are two conjectures concerning the distances between zeros and the density of zeros of the Riemann zeta function. Conjectures In 1914, Godfrey Harold Hardy proved that the Riemann zeta function \zeta\bigl(\tfrac+it\bigr) has infinitely many real zeros. Let N(T) be the total number of real zeros, N_0(T) be the total number of zeros of odd order of the function \zeta\bigl(\tfrac+it\bigr), lying on the interval (0,T]. Hardy and Littlewood claimed two conjectures. These conjectures – on the distance between real zeros of \zeta\bigl(\tfrac+it\bigr) and on the density of zeros of \zeta\bigl(\tfrac+it\bigr) on intervals (T,T+H] for sufficiently great T > 0, H = T^ and with as less as possible value of a > 0, where \varepsilon > 0 is an arbitrarily small number – open two new directions in the investigation of the Riemann zeta function. 1. For any \varep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Godfrey Harold Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of population genetics. G. H. Hardy is usually known by those outside the field of mathematics for his 1940 essay ''A Mathematician's Apology'', often considered one of the best insights into the mind of a working mathematician written for the layperson. Starting in 1914, Hardy was the mentor of the Indian mathematician Srinivasa Ramanujan, a relationship that has become celebrated.THE MAN WHO KNEW INFINITY: A Life of the Genius Ramanujan . Retrieved 2 December 2010. Hardy almost immediately rec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Edensor Littlewood
John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations, and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanujan and Mary Cartwright. Biography Littlewood was born on 9 June 1885 in Rochester, Kent, the eldest son of Edward Thornton Littlewood and Sylvia Maud (née Ackland). In 1892, his father accepted the headmastership of a school in Wynberg, Cape Town, in South Africa, taking his family there. Littlewood returned to Britain in 1900 to attend St Paul's School in London, studying under Francis Sowerby Macaulay, an influential algebraic geometer. In 1903, Littlewood entered the University of Cambridge, studying in Trinity College. He spent his first two years preparing for the Tripos examinations which qualify undergraduates for a bachelor's degree where he emerged in 1905 as Senior Wrangler bracketed with James Mercer (Mercer had already ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > 1 and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atle Selberg
Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded the Fields Medal in 1950 and an honorary Abel Prize in 2002. Early years Selberg was born in Langesund, Norway, the son of teacher Anna Kristina Selberg and mathematician Ole Michael Ludvigsen Selberg. Two of his three brothers, Sigmund and Henrik, were also mathematicians. His other brother, Arne, was a professor of engineering. While he was still at school he was influenced by the work of Srinivasa Ramanujan and he found an exact analytical formula for the partition function as suggested by the works of Ramanujan; however, this result was first published by Hans Rademacher. During the war he fought against the German invasion of Norway, and was imprisoned several times. He studied at the University of Oslo and completed his PhD in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Selberg's Zeta Function Conjecture
In mathematics, the Selberg conjecture, named after Atle Selberg, is a theorem about the density of zeros of the Riemann zeta function ζ(1/2 + ''it''). It is known that the function has infinitely many zeroes on this line in the complex plane: the point at issue is how densely they are clustered. Results on this can be formulated in terms of ''N''(''T''), the function counting zeroes on the line for which the value of ''t'' satisfies 0 ≤ ''t'' ≤ ''T''. Background In 1942 Atle Selberg investigated the problem of the Hardy–Littlewood conjecture 2; and he proved that for any :\varepsilon > 0 there exist :T_0 = T_0(\varepsilon) > 0 and :c = c(\varepsilon) > 0, such that for :T \geq T_0 and :H=T^ the inequality :N(T+H)-N(T) \geq cH\log T holds true. In his turn, Selberg stated a conjecture relating to shorter intervals, namely that it is possible to decrease the value of the exponent ''a'' = 0.5 in :H=T^. Proof of the conjecture In 1984 Anatolii Kar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjectures
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
