ON A LATTICE GENERALISATION OF THE LOGARITHM AND A DEFORMATION OF THE DEDEKIND ETA FUNCTION
2020 ◽
Vol 102
(1)
◽
pp. 118-125
Keyword(s):
We consider a deformation $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ of the Dedekind eta function depending on two $d$-dimensional simple lattices $(L,\unicode[STIX]{x1D6EC})$ and two parameters $(m,t)\in (0,\infty )$, initially proposed by Terry Gannon. We show that the minimisers of the lattice theta function are the maximisers of $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ in the space of lattices with fixed density. The proof is based on the study of a lattice generalisation of the logarithm, called the lattice logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterised by a variational problem over a class of one-dimensional lattice logarithms.
Keyword(s):
1997 ◽
Vol 52
(2)
◽
pp. 327-340
◽
Keyword(s):
2016 ◽
Vol 50
(1)
◽
pp. 015301
◽
Keyword(s):
1995 ◽
2007 ◽
Vol 21
(02n03)
◽
pp. 139-154
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Keyword(s):