In mathematics, the Riemann–Siegel theta function is defined in terms of the gamma function as for real values of t. Here the argument is chosen in such a way that a continuous function is obtained and holds, i.e., in the same way that the principal branch of the log-gamma function is defined. It has an asymptotic expansion which is not convergent, but whose first few terms give a good approximation for . Its Taylor-series at 0 which converges for is where denotes the polygamma function of order . The Riemann–Siegel theta function is of interest in studying the Riemann zeta function, since it can rotate the Riemann zeta function such that it becomes the totally real valued Z function on the critical line . Curve discussion The Riemann–Siegel theta function is an odd real analytic function for real values of with three roots at and . It is an increasing function for , and has local extrema at , with value . It has a single inflection point at with , which is the minimum of its derivative. Theta as a function of a complex variable We have an infinite series expression for the log-gamma function where γ is Euler's constant. Substituting for z and taking the imaginary part termwise gives the following series for θ(t) For values with imaginary part between −1 and 1, the arctangent function is holomorphic, and it is easily seen that the series converges uniformly on compact sets in the region with imaginary part between −1/2 and 1/2, leading to a holomorphic function on this domain. It follows that the Z function is also holomorphic in this region, which is the critical strip. We may use the identities to obtain the closed-form expression which extends our original definition to a holomorphic function of t. Since the principal branch of log Γ has a single branch cut along the negative real axis, θ(t) in this definition inherits branch cuts along the imaginary axis above i/2 and below −i/2. Gram points The Riemann zeta function on the critical line can be written If is a real number, then the Z function returns real values. Hence the zeta function on the critical line will be real when . Positive real values of where this occurs are called Gram points, after J. P. Gram, and can of course also be described as the points where is an integer. A