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The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows: , = (( (!))) for integer j and k. This shows that the Dirichlet function is a Baire class 2 function.
The Dirichlet L-function L(s, χ) = 1 − 3 −s + 5 −s − 7 −s + ⋅⋅⋅ (sometimes given the special name Dirichlet beta function), with trivial zeros at the negative odd integers. Let χ be a primitive character modulo q, with q > 1. There are no zeros of L(s, χ) with Re(s) > 1. For Re(s) < 0, there are zeros at certain negative ...
All known examples are automorphic L-functions, and the reciprocals of F p (s) are polynomials in p −s of bounded degree. [4] The best results on the structure of the Selberg class are due to Kaczorowski and Perelli, who show that the Dirichlet L-functions (including the Riemann zeta-function) are the only examples with degree less than 2. [5]
It is named after Carl Johannes Thomae, but has many other names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function (not to be confused with the integer ruler function), [2] the Riemann function, or the Stars over Babylon (John Horton Conway's name). [3] Thomae ...
One example of such a function is the indicator function of the rational numbers, also known as the Dirichlet function. This function is denoted as 1 Q {\displaystyle \mathbf {1} _{\mathbb {Q} }} and has domain and codomain both equal to the real numbers .
The set of arithmetic functions forms a commutative ring, the Dirichlet ring, under pointwise addition, where f + g is defined by (f + g)(n) = f(n) + g(n), and Dirichlet convolution. The multiplicative identity is the unit function ε defined by ε ( n ) = 1 if n = 1 and ε ( n ) = 0 if n > 1 .
The method also has theoretical applications: for example, Peter Gustav Lejeune Dirichlet introduced the technique in 1849 to obtain the estimate [1] [2] = + + (), where γ is the Euler–Mascheroni constant.
The most famous example of a Dirichlet series is = =,whose analytic continuation to (apart from a simple pole at =) is the Riemann zeta function.. Provided that f is real-valued at all natural numbers n, the respective real and imaginary parts of the Dirichlet series F have known formulas where we write +: