certain absolute constants. The argument of the Riemann zeta function The function is called the argument of the Riemann zeta function. Here is the increment of an arbitrary continuous branch of along the broken line joining the points , and . There are some theorems on properties of the function . Among those results are the mean value theorems for and its first integral on intervals of the real line, and also the theorem claiming that every interval for contains at least points where the function changes sign. Earlier similar results were obtained by Atle Selberg for the case Representations Dirichlet series An extension of the area of convergence can be obtained by rearranging the original series. The series converges for , while converges even for . In this way, the area of convergence can be extended to for any negative integer . Mellin-type integrals The Mellin transform of a function is defined as in the region where the integral is defined. There are various expressions for the zeta function as Mellin transform-like integrals. If the real part of is greater than one, we have and , where denotes the gamma function. By modifying the contour, Riemann showed that for all (where denotes the Hankel contour). We can also find expressions which relate to prime numbers and the prime number theorem. If is the prime-counting function, then for values with . A similar Mellin transform involves the Riemann function , which counts prime powers with a weight of , so that Now These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's prime-counting function is easier to work with, and can be recovered from it by Möbius inversion. Theta functions The Riemann zeta function can be given by a Mellin transform in terms of Jacobi's theta function However, this integral only converges if the real part of is greater than 1, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all except 0 and 1: Laurent series The Riemann zeta function is meromorphic with a single pole of order one at . It can therefore be expanded as a Laurent series about ; the series development is then The constants here are called the Stieltjes constants and can be defined by the limit The constant term is the Euler–Mascheroni constant. Integral