Local zeta function: Difference between revisions

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Revision as of 03:31, 26 March 2018

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In [[number theory]], the '''local zeta function''' ''<math>Z''(''V'',&nbsp;''s'') of ''V''</math> (sometimes called the '''congruent zeta function''') is defined as

:<math>Z(V, s) = \exp\left(\sum_{m = 1}^\infty \frac{N_m}{m} (q^{-s})^m\right)</math>

where ''N''<submath>''m''N_m</submath> is the number of points of ''<math>V''</math> defined over the degree ''<math>m''</math> extension{{explain|date=September 2017}} '''F'''<submath>''q''^{''\mathbf{F}_q^m''}</submath> of '''F'''<submath>''q''\mathbf{F}_q^m</submath>, and ''<math>V''</math> is a [[non-singular]] ''<math>n''</math>-dimensional [[projective algebraic variety]] over the field '''F'''<submath>''q''\mathbf{F}_q^m</submath> with ''<math>q''</math> elements. By the variable transformation <math>u=q^{-s}</math>, then it is defined by

:<math>

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</math>

as the [[formal power series]] of the variable ''<math>u''</math>.

Equivalently, sometimesthe itlocal zeta function sometimes is defined as follows:

:<math>

(1)\ \ \mathit{Z} (V,0) = 1 \,

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<!--In [[number theory]], a '''local zeta-function'''

:''<math>Z''(''-t'')</math>

is a function whose [[logarithmic derivative]] is a [[generating function]]