The evaluation of two-dimensional lattice sums via Ramanujan's theta functions

2014 ◽  
Vol 165 (1) ◽  
pp. 57-89
Author(s):  
Ping Xu
Keyword(s):  
Theta Functions ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
Lattice Sums

Exact values of some two-dimensional lattice sums

1975 ◽  
Vol 8 (6) ◽  
pp. 874-881 ◽  
Author(s):  
I J Zucker ◽  
M M Robertson
Keyword(s):  
Two Dimensional ◽  
Dimensional Lattice ◽  
Lattice Sums

On the Riemann property of angular lattice sums and the one-dimensional limit of two-dimensional lattice sums

2008 ◽  
Vol 464 (2100) ◽  
pp. 3327-3352 ◽  
Author(s):  
Ross C McPhedran ◽  
I.J Zucker ◽  
Lindsay C Botten ◽  
Nicolae-Alexandru P Nicorovici
Keyword(s):  
Critical Line ◽  
Quadratic Function ◽  
Quadratic Functions ◽  
Square Lattice ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Lattice Sums ◽  
The One ◽  
Complex Powers

We consider a general class of two-dimensional lattice sums consisting of complex powers s of inverse quadratic functions. We consider two cases, one where the quadratic function is negative definite and another more restricted case where it is positive definite. In the former, we use a representation due to H. Kober, and consider the limit u →∞, where the lattice becomes ever more elongated along one period direction (the one-dimensional limit). In the latter, we use an explicit evaluation of the sum due to Zucker and Robertson. In either case, we show that the one-dimensional limit of the sum is given in terms of ζ (2 s ) if Re( s )>1/2 and either ζ (2 s −1) or ζ (2−2 s ) if Re( s )<1/2. In either case, this leads to a Riemann property of these sums in the one-dimensional limit: their zeros must lie on the critical line Re( s )=1/2. We also comment on a class of sums that involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle. We show that certain of these sums can have their zeros on the critical line but not in a neighbourhood of it; others are identically zero on it, while still others have no zeros on it.

scholarly journals Dirichlet L -series with real and complex characters and their application to solving double sums

2008 ◽  
Vol 464 (2094) ◽  
pp. 1405-1422 ◽  
Author(s):  
I.J Zucker ◽  
R.C McPhedran
Keyword(s):  
Closed Form ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
Lattice Sums ◽  
Complex Characters

A description of the properties of L -series with complex characters is given. By using these, together with the more familiar L -series with real characters, it is shown how certain two-dimensional lattice sums, which previously could not be put into closed form, may now be expressed in this way.

scholarly journals DISCRETE AND CONTINUUM APPROACHES TO THREE-DIMENSIONAL QUANTUM GRAVITY

1991 ◽  
Vol 06 (39) ◽  
pp. 3591-3600 ◽  
Author(s):  
HIROSI OOGURI ◽  
NAOKI SASAKURA
Keyword(s):  
Gauge Theory ◽  
Moduli Space ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Analogue Model ◽  
Gauge Invariant ◽  
Invariant Functions ◽  
Dimensional Lattice ◽  
Chern Simons ◽  
Dimensional Surface

It is shown that, in the three-dimensional lattice gravity defined by Ponzano and Regge, the space of physical states is isomorphic to the space of gauge-invariant functions on the moduli space of flat SU(2) connections over a two-dimensional surface, which gives physical states in the ISO(3) Chern–Simons gauge theory. To prove this, we employ the q-analogue of this model defined by Turaev and Viro as a regularization to sum over states. A recent work by Turaev suggests that the q-analogue model itself may be related to an Euclidean gravity with a cosmological constant proportional to 1/k2, where q=e2πi/(k+2).

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