Dirichlet seriesIn mathematics, a Dirichlet series is any series of the form where s is complex, and is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet.
Dirichlet characterIn analytic number theory and related branches of mathematics, a complex-valued arithmetic function is a Dirichlet character of modulus (where is a positive integer) if for all integers and : that is, is completely multiplicative. (gcd is the greatest common divisor) that is, is periodic with period . The simplest possible character, called the principal character, usually denoted , (see Notation below) exists for all moduli: The German mathematician Peter Gustav Lejeune Dirichlet—for whom the character is named—introduced these functions in his 1837 paper on primes in arithmetic progressions.
Dirichlet convolutionIn mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet. If are two arithmetic functions from the positive integers to the complex numbers, the Dirichlet convolution f ∗ g is a new arithmetic function defined by: where the sum extends over all positive divisors d of n, or equivalently over all distinct pairs (a, b) of positive integers whose product is n.
XenonXenon is a chemical element with the symbol Xe and atomic number 54. It is a dense, colorless, odorless noble gas found in Earth's atmosphere in trace amounts. Although generally unreactive, it can undergo a few chemical reactions such as the formation of xenon hexafluoroplatinate, the first noble gas compound to be synthesized. Xenon is used in flash lamps and arc lamps, and as a general anesthetic. The first excimer laser design used a xenon dimer molecule (Xe2) as the lasing medium, and the earliest laser designs used xenon flash lamps as pumps.
Xenon difluorideXenon difluoride is a powerful fluorinating agent with the chemical formula XeF2, and one of the most stable xenon compounds. Like most covalent inorganic fluorides it is moisture-sensitive. It decomposes on contact with water vapor, but is otherwise stable in storage. Xenon difluoride is a dense, colourless crystalline solid. It has a nauseating odour and low vapor pressure. Xenon difluoride is a linear molecule with an Xe–F bond length of 197.73pm in the vapor stage, and 200 pm in the solid phase.