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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.
Both sides of the Euler product formula converge for Re(s) > 1. The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s = 1, diverges, Euler's formula (which becomes Π p p / p − 1 ) implies that there are infinitely many primes. [5]
Download as PDF; Printable version; ... Pages in category "Infinite products" ... Proof of the Euler product formula for the Riemann zeta function; Q.
Download as PDF; Printable version; ... (Viète's formula, the first published infinite product in mathematics) ... This is a special case of the Euler product.
Let K be an algebraic number field.Its Dedekind zeta function is first defined for complex numbers s with real part Re(s) > 1 by the Dirichlet series = (/ ())where I ranges through the non-zero ideals of the ring of integers O K of K and N K/Q (I) denotes the absolute norm of I (which is equal to both the index [O K : I] of I in O K or equivalently the cardinality of quotient ring O K / I).
Q-series generalize Euler's function, which is closely related to the Dedekind eta function, and occurs in the study of modular forms. The modulus of the Euler function (see there for picture) shows the fractal modular group symmetry and occurs in the study of the interior of the Mandelbrot set .
The formula shows that the L-function of χ is equal to the L-function of the primitive character which induces χ, multiplied by only a finite number of factors. [ 6 ] As a special case, the L -function of the principal character χ 0 {\displaystyle \chi _{0}} modulo q can be expressed in terms of the Riemann zeta function : [ 7 ] [ 8 ]