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In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler .
By the fundamental theorem of arithmetic, the partial product when expanded out gives a sum consisting of those terms n −s where n is a product of primes less than or equal to q. The inequality results from the fact that therefore only integers larger than q can fail to appear in this expanded out partial product.
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 +:
Euler Mathematical Toolbox (or EuMathT; formerly Euler) is a free and open-source numerical software package. It contains a matrix language, a graphical notebook style interface, and a plot window. Euler is designed for higher level math such as calculus, optimization, and statistics.
These are similar to the Riemann zeta function: they have a functional equation, and an infinite product similar to the Euler product but taken over closed geodesics rather than primes. The Selberg trace formula is the analogue for these functions of the explicit formulas in prime number theory.
Euler's continued fraction formula connecting a finite sum of products with a finite continued fraction; Euler product formula for the Riemann zeta function. Euler–Maclaurin formula (Euler's summation formula) relating integrals to sums; Euler–Rodrigues formula describing the rotation of a vector in three dimensions