For all , , the integral relation (cf. Abel–Plana formula) holds true, which may be used for a numerical evaluation of the zeta function. Rising factorial Another series development using the rising factorial valid for the entire complex plane is This can be used recursively to extend the Dirichlet series definition to all complex numbers. The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the Gauss–Kuzmin–Wirsing operator acting on ; that context gives rise to a series expansion in terms of the falling factorial. Hadamard product On the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansion where the product is over the non-trivial zeros of and the letter again denotes the Euler–Mascheroni constant. A simpler infinite product expansion is This form clearly displays the simple pole at , the trivial zeros at −2, −4, ... due to the gamma function term in the denominator, and the non-trivial zeros at . (To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i.e. the factors for a pair of zeros of the form and should be combined.) Globally convergent series A globally convergent series for the zeta function, valid for all complex numbers except for some integer , was conjectured by Konrad Knopp in 1926 and proven by Helmut Hasse in 1930 (cf. Euler summation): The series appeared in an appendix to Hasse's paper, and was published for the second time by Jonathan Sondow in 1994. Hasse also proved the globally converging series in the same publication. Research by Iaroslav Blagouchine has found that a similar, equivalent series was published by Joseph Ser in 1926. Peter Borwein has developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series suitable for high precision numerical calculations. Series representation at positive integers via the primorial Here is the primorial sequence and is Jordan's totient function. Series representation by the incomplete poly-Bernoulli numbers The function can be represented, for , by the infinite series where , is the th branch of the Lambert -function, and is an incomplete poly-Bernoulli number. The Mellin transform of the Engel map The function is iterated to find the coefficients appearing in Engel expansions. The Mellin transform of the map is related to the Riemann zeta function by the formula Thue-Morse sequence Certain