is the statement that the positivity of a certain function is equivalent to the Riemann hypothesis. Related is , a statement that the positivity of a certain sequence of numbers is equivalent to the Riemann hypothesis.
The Riemann hypothesis can be generalized by replacing the Riemann zeta-function by the formally similar, but much more general, global . In this broader setting, one expects the non-trivial zeros of the global L-functions to have real part 1/2. It is these conjectures, rather than the classical Riemann hypothesis only for the single Riemann zeta-function, which accounts for the true importance of the Riemann hypothesis in mathematics.
of all time."His book studiedmultivariate polynomials and is consideredthe best mathematics in ancient China and describes methods notrediscovered for centuries; for exampleZhu anticipated the Sylvester matrix method for solving simultaneouspolynomial equations.
(Some of his designs, including the viola organista, hisparachute, and a large single-span bridge, were finally built fivecenturies later.)He developed the mechanical theory of the arch;made advances in anatomy, botany, and other fields of science;he was first to conceive of plate tectonics.
(In a famous leap of over-confidence he claimed he couldcontrol the Nile River; when the Caliph ordered him to do so,he then had to feign madness!)Alhazen has been called the "Father of Modern Optics,"the "Founder of Experimental Psychology" (mainly for his work withoptical illusions), and,because he emphasized hypotheses and experiments,"The First Scientist."In number theory, Alhazen worked with perfect numbers, Mersenne primes,and the Chinese Remainder Theorem.
For example, the Law of the Pendulum, based on Galileo's incorrectbelief that the tautochrone was the circle, conflictedwith his own observations.)Despite his extreme importance to mathematical physics,Galileo doesn't usually appear onlists of greatest .
(For example, some of Euclid's more difficult theorems are easyanalytic consequences of Archimedes' Lemma of Centroids.)His achievements are particularly impressive given thelack of good mathematical notation in his day.
Unlike Diophantus' work, which dealt in specific examples,Al-Khowârizmi was the first algebra text to present general methods;he is often called the "Father of Algebra."(Diophantus did, however, use superior "syncopated" notation.)The word is borrowed from Al-Khowârizmi's name,and is taken from the name of his book.
For example, the area of any right triangle is equal to thesum of the areas of the two lunes formed when semi-circlesare drawn on each of the three edges of the triangle.
I won't try to summarize Leibniz' contributions to philosophyand diverse other fields including biology; as justthree examples: he predicted the Earth's molten core,introduced the notion of subconscious mind,and built the first calculator that could do multiplication.
If proven correct, this would allow to better describe how the are placed among whole numbers. The Riemann hypothesis is so important, and so difficult to prove, that the Clay Mathematics Institute has offered $1,000,000 to the first person to prove it.
He also invented the concept of generating functions; for example,letting p(n) denote the number of partitions of n, Euler found the lovely equation: Σn (n) xn= 1 / Πk (1 - xk)
The denominator of the right side hereexpands to a series whose exponents all have the (3m2+m)/2"pentagonal number" form; Euler found an ingenious proof of this.
The hypothesis is named after Bernhard Riemann. It is about a special , the . This function inputs and outputs values. The inputs that give the output zero are called of the zeta function. Many zeros have been found. The "obvious" ones to find are the negative even integers. This follows from Riemann's functional equation. More have been computed and have part 1/2. The hypothesis states all the undiscovered zeros must have real part 1/2.
The Riemann hypothesis is a mathematical . Many people think that finding a of the is one of the hardest and most important unsolved problems of .
He was noted for his strong belief in determinism, famously replying toNapoleon's question about God with: "I have no need of that hypothesis."Laplace viewed mathematics as just a tool for developing hisphysical theories.