In mathematics, the Riemann–Siegel theta function is defined in terms of the gamma function as
θ ( t ) = arg ( Γ ( 1 4 + i t 2 ) ) − log π 2 t {\displaystyle \theta (t)=\arg \left(\Gamma \left({\frac {1}{4}}+{\frac {it}{2}}\right)\right)-{\frac {\log \pi }{2}}t}
for real values of t. Here the argument is chosen in such a way that a continuous function is obtained and θ ( 0 ) = 0 {\displaystyle \theta (0)=0} holds, i.e., in the same way that the principal branch of the log-gamma function is defined. It has an asymptotic expansion
θ ( t ) ∼ t 2 log t 2 π − t 2 − π 8 + 1 48 t + 7 5760 t 3 + ⋯ {\displaystyle \theta (t)\sim {\frac {t}{2}}\log {\frac {t}{2\pi }}-{\frac {t}{2}}-{\frac {\pi }{8}}+{\frac {1}{48t}}+{\frac {7}{5760t^{3}}}+\cdots }
which is not convergent, but whose first few terms give a good approximation for t ≫ 1 {\displaystyle t\gg 1} . Its Taylor-series at 0 which converges for | t | < 1 / 2 {\displaystyle |t|<1/2} is
θ ( t ) = − t 2 log π + ∑ k = 0 ∞ ( − 1 ) k ψ ( 2 k ) ( 1 4 ) ( 2 k + 1 ) ! ( t 2 ) 2 k + 1 {\displaystyle \theta (t)=-{\frac {t}{2}}\log \pi +\sum _{k=0}^{\infty }{\frac {(-1)^{k}\psi ^{(2k)}\left({\frac {1}{4}}\right)}{(2k+1)!}}\left({\frac {t}{2}}\right)^{2k+1}}
where ψ ( 2 k ) {\displaystyle \psi ^{(2k)}} denotes the polygamma function of order 2 k {\displaystyle 2k} . 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 s = 1 / 2 + i t {\displaystyle s=1/2+it} .
Curve discussion The Riemann–Siegel theta function is an odd real analytic function for real values of t {\displaystyle t} with three roots at 0 {\displaystyle 0} and ± 17.8455995405 … {\displaystyle \pm 17.8455995405\ldots } . It is an increasing function for | t | > 6.29 {\displaystyle |t|>6.29} , and has local extrema at ± 6.289835988 … {\displaystyle \pm 6.289835988\ldots } , with value ∓ 3.530972829 … {\displaystyle \mp 3.530972829\ldots } . It has a single inflection point at t = 0 {\displaystyle t=0} with θ ′ ( 0 ) = − ln π + γ + π / 2 + 3 ln 2 2 = − 2.6860917 … {\displaystyle \theta ^{\prime }(0)=-{\frac {\ln \pi +\gamma +\pi /2+3\ln 2}{2}}=-2.6860917\ldots } , which is the minimum of its derivative.
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