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zeta_functions [2015/02/07 23:42] nikolaj |
zeta_functions [2015/02/07 23:47] nikolaj |
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| (maybe with parts/states/aspects $s_0, s_1, s_2, s_3, s_4,\dots$) | (maybe with parts/states/aspects $s_0, s_1, s_2, s_3, s_4,\dots$) | ||
| - | |||
| - | $\dfrac{1}{S}$ ... flipped encoding, switches low and far behaviour, represents the weight of $X$. | ||
| == traceless part == | == traceless part == | ||
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| or similar ... here $1$ is the neutral/constant thing in the theory and $T$ resp. $t$ is what's really interesting about $S$. | or similar ... here $1$ is the neutral/constant thing in the theory and $T$ resp. $t$ is what's really interesting about $S$. | ||
| - | E.g. you have a scattering matrix and $1$ is the free propagation and $Y$ is the interaction. | + | E.g. you have a scattering matrix and $1$ is the free propagation and $T$ is the interaction. |
| == Q == | == Q == | ||
| - | $Q(S):=\dfrac{1}{1-t}=\sum_{n=0}^\infty t^n$ | + | $\dfrac{1}{S}$ ... flipped encoding, switches low and far behavior, represents the weight of $X$. |
| - | ... flipped linear encoding of $X$ with emphasis on the interesting part. | + | $Q(t):=\dfrac{1}{1-t}=\sum_{n=0}^\infty t^n$ |
| - | It starts out as $Q(S)=1+t+{\marhcal O}(t^2) \approx 1-T$, but it diverges once $t$ reaches $1$. | + | It starts out as $Q(t)=1+t+{\marhcal O}(t^2) \approx 1-T$, but it diverges once $t$ reaches $1$. |
| - | (And I observe $S\,Q(S)=2\,Q(S)-1$.) | + | (And I observe $S\,Q(t)=2\,Q(t)-1$.) |
| == log == | == log == | ||
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| We want to understand this, in a broad sense, as tamed version of the original: $\log(T) < T$. | We want to understand this, in a broad sense, as tamed version of the original: $\log(T) < T$. | ||
| - | But, for $Q(S)$ interpreted in a field, its proper singularity isn't tamed by $\log$: | + | But, for $Q(t)$ interpreted in a field, its proper singularity isn't tamed by $\log$: |
| - | $\log(Q(S))=\log\left(\dfrac{1}{1-t}\right)=-\log(1-t)=\sum_{n=0}^\infty\frac{1}{n}t^n$ | + | $\log(Q(t))=\log\left(\dfrac{1}{1-t}\right)=-\log(1-t)=\sum_{n=0}^\infty\frac{1}{n}t^n$ |
| still diverges at $\lim{t\to 1}$. | still diverges at $\lim{t\to 1}$. | ||
| == zeta == | == zeta == | ||
| - | $\zeta(S)$ ... Some gluing together of data of $S$. | + | $\zeta_S$ ... Some gluing together of data of $S$. |
| Sometimes zetas are somewhat obscured using $\exp$'s chained with $\log$'s, in the spirit of above. | Sometimes zetas are somewhat obscured using $\exp$'s chained with $\log$'s, in the spirit of above. | ||
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| == Riemann zeta == | == Riemann zeta == | ||
| - | For primes $p$, set $S_p=1-p^{-z}$ and define | + | For primes $p$, set $t_z=p^{-z}$ and define |
| - | $\zeta_\text{Riemann}(z):=\prod Q(S_p)=\prod_\text{primes p}\frac{1}{1-p^{-z}}$. | + | $\zeta_\text{Riemann}(z):=\prod Q(t_z)=\prod_\text{primes p}\frac{1}{1-p^{-z}}$. |
| == Polylog == | == Polylog == | ||
