Prime number theoremCharacterization of how many integers are prime without a subscript base should be interpreted as a natural logarithm, also commonly written as ln(x){\displaystyle \ln(x)} or loge(x){\displaystyle \log _{e}(x)}.In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of prime numbers among the positive integers.
Riemann hypothesisUnsolved problem in mathematicsDo all non-trivial zeros of the Riemann zeta function have a real part equal to one half?More unsolved problems in mathematicsIn 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 part1/2. Many consider it to be the most important unsolved problem in pure mathematics.
Riemann zeta functionIn mathematics, the Riemann zeta function, or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ {\displaystyle \zeta } (zeta), is a function of a complex variable defined as ζ ( s ) = ∑ n = 1 ∞ 1 n s = 1 1 s + 1 2 s + 1 3 s + ⋯ {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots } for R e ( s ) > 1 {\displaystyle \mathrm {Re} (s)>1} , and its analytic continuation elsewhere.
Prime numberNumber divisible only by 1 and itselfComposite numbers can be arranged into rectangles but prime numbers cannot.A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself.
On-Line Encyclopedia of Integer SequencesThe On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009, and is its chairman. OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited.
ConjectureIn mathematics, a conjecture is a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjecturesProofFormal mathematics is based on provable truth.Möbius functionThe Möbius function μ ( n ) {\displaystyle \mu (n)} is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula.
Real numberIn mathematics, a real number is a number that can be used to measure a continuous one-dimensionalquantity such as a length, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.
Function (mathematics)Association of one output to each inputIn mathematics, a function from a setX to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time.
Adrien-Marie LegendreAdrien-Marie Legendre (; French: [adʁiɛ̃ maʁi ləʒɑ̃dʁ]; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. He is also known for his contributions to the method of least squares, and was the first to officially publish on it, though Carl Friedrich Gauss had discovered it before him.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.PiThe number π ( ; spelled out as pi) is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter. It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.*As an Amazon Associate I earn from qualifying purchases.