Infinite products

Infinite products are an extension of the concept of finite products, where instead of multiplying a finite number of terms together, an infinite sequence of terms is multiplied. The general form of an infinite product is: \[ P = \prod_{n=1}^{\infty} a_n \] where \( a_n \) are the terms in the sequence.

The Hafner–Sarnak–McCurley constant, often denoted as \( C \), is a mathematical constant that arises in number theory, specifically in the context of the distribution of prime numbers, particularly in relation to the number of primes in certain arithmetic sequences. More specifically, it relates to the asymptotic density of prime gaps and primes in certain modular classes.

The Kepler–Bouwkamp constant, denoted as \( K \), is a mathematical constant that appears in the context of the geometrical relationships between regular polygons and circles, particularly in relation to the packing of spheres and the computation of certain areas and volumes in geometry. It can be expressed in terms of elliptic integrals and has a numerical value of approximately: \[ K \approx 0.

Khinchin's constant is a mathematical constant that appears in the context of the theory of continued fractions. Named after the Russian mathematician Aleksandr Khinchin, it is typically denoted by the symbol \( K \) and is approximately equal to \( 2.685452 \). Khinchin's constant arises in the study of the properties of the continued fraction representations of real numbers.

The constant associated with Somos' quadratic recurrence, often denoted as \( c \), is given by the formula: \[ c = \frac{1 + \sqrt{5}}{2} \] This is known as the golden ratio, commonly denoted by \( \phi \). In the context of the recurrence, the sequence defines terms using the previous terms in relation to this constant.

Stephens' constant, often denoted by \( \sigma \), is a physical constant that arises in the context of quantum mechanics and statistical physics. It is specifically associated with the calculation of the density of states for quantum particles in a certain system. However, it is not a universally recognized term like Planck's constant or Boltzmann constant. In many contexts, the term might refer to properties or constants in specific studies related to statistical distributions.

Viète's formulas provide relationships between the coefficients of a polynomial and sums and products of its roots.