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...

\zeta ({\tfrac {1}{2}}+it)=Z(t)e^{-i\theta (t)}} where Hardy's Z function and the Riemann–Siegel theta function θ are uniquely defined by this and the...

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined...

of the Riemann–Siegel theta function and the Riemann zeta function by Z ( t ) = e i θ ( t ) ζ ( 1 2 + i t ) . {\displaystyle Z(t)=e^{i\theta (t)}\zeta...

Riemann hypothesis Riemann hypothesis for curves over finite fields Riemann theta function Riemann Xi function Riemann zeta function Riemann–Siegel formula...

and solitons. Theta functions are parametrized by points in a tube domain inside a complex Lagrangian Grassmannian, namely the Siegel upper half space...

Siegel found while reading through Riemann's unpublished papers. by Siegel: Transcendental numbers, 1949 Analytic functions of several complex variables, Stevens...

Ramanujan theta function, f ( a , b ) {\displaystyle f(a,b)} Neville theta functions Riemann–Siegel theta function, θ ( t ) {\displaystyle \theta (t)} The...

higher dimensions, see Siegel (1955). Meromorphic functions can be given fairly explicitly, in terms of Riemann theta functions and the Abel–Jacobi map...

complex tori and theta functions. Using the Riemann theta function, necessary and sufficient conditions on a lattice were written down by Riemann for a lattice...

for example the Stirling formula applied in Im(z) = t of the Riemann–Siegel theta function on the straight line ⁠1/4⁠ + it. Thomas Bayes showed, in a letter...

Meromorphic function in one-variable complex function were studied in a compact (closed) Riemann surface, because since the Riemann-Roch theorem (Riemann's inequality)...

and elliptic functions. Examples of such functions are very classical - the Jacobi theta functions and the Fourier coefficients of Siegel modular forms...

Explicit formula (L-function) Riemann–Siegel formula (particular approximate functional equation) "§25.15 Dirichlet -functions on NIST". Weisstein, Eric...

701\ k_{\mathrm {B} }T,} where ζ {\displaystyle \zeta } is the Riemann zeta function. In the limit of low frequencies (i.e. long wavelengths), Planck's...

iz}} so that |q| < 1. The theta functions agree with the standard classical formulas for the Jacobi-Riemann theta functions: Θ n , m ( τ , z ) = ∑ k ∈...

{\text{Im}}(B){\text{ is positive definite}}\right\}.} The Riemann θ {\displaystyle \theta } function on C g {\displaystyle \mathbf {C} ^{g}} corresponding...

smooth manifolds Riemann–Siegel formula Riemann–Siegel theta function Riemann–Silberstein vector Riemann–Stieltjes integral Riemann–von Mangoldt formula...

exponential function. An alternative way of expressing the lemniscate functions as a ratio of entire functions involves the theta functions (see Lemniscate...

coming. The Siegel modular forms, for which G is a symplectic group, arose naturally from considering moduli spaces and theta functions. The post-war...

{\displaystyle {\mathcal {Z}}:=\left\{\left(\cos ^{2}\theta ,{\tfrac {1}{2}}\sin 2\theta \right)\mid 0<\theta <\pi \right\}} . ⁠ Z {\displaystyle {\mathcal {Z}}}...

complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann equations. The second...

Rapid Eye Movement Sleep: Regulation and Function. Cambridge University Press. ISBN 978-1-139-50378-5. Lai YY, Siegel JM (1999). "Muscle Atonia in REM Sleep"...

form of the generalized Riemann hypothesis. It is known that the conjecture fails at the endpoint θ = 1 {\displaystyle \theta =1} . In 1986, Bombieri...

in dimension > 1, introducing the Riemann bilinear relations and Riemann theta function. 1865 Carl Johannes Thomae, Theorie der ultraelliptischen Funktionen...

^{+}\left|f(re^{i\theta })\right|d\theta .\,} Finally, define the Nevanlinna characteristic by (cf. Jensen's formula for meromorphic functions) T ( r , f )...

particular case of a Bose–Einstein integral, the polylogarithm, or the Riemann zeta function ζ ( s ) {\displaystyle \zeta (s)} . The value of the integral is...

{\displaystyle f_{\theta }(x)=x^{2}+\lambda x} which has an indifferent fixed point with multiplier λ = e 2 π θ i {\displaystyle \lambda =e^{2\pi \theta i}} at the...

of modular forms, examples such as Hilbert modular forms, Siegel modular forms, and theta-series had been developed. The conjectures have evolved since...

mathematical meanings Function field of an algebraic variety Birational geometry Motive (algebraic geometry) Analytic variety Zariski–Riemann space Semi-algebraic...