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created a table-for-inclusion and included it into the relevant entries:
zeta-functions and eta-functions and theta-functions and L-functions – table
Presently it looks like this:
The main statement of this analogy is discussed for instance on the first pages of
I have added some paragraphs at eta invariant, accordingly.
Doesn’t Richardson effectively say that Dirichlet L-function should be in the bottom row? The Riemann zeta function but twisted by some character. Wouldn’t the missing entry be Hecke L-function?
Ah, thanks, yes, that sounds right. Have edited accordingly now.
Another thing:
the Selberg zeta function of a Riemann surface is hopefully identical to the zeta function of an elliptic differential operator for the Laplace operator of the surface. Where would this be discussed?
Probably need to beef up Selberg trace formula:
When Γ is the fundamental group of a Riemann surface, the Selberg trace formula describes the spectrum of differential operators such as the Laplacian in terms of geometric data involving the lengths of geodesics on the Riemann surface. In this case the Selberg trace formula is formally similar to the explicit formulas relating the zeros of the Riemann zeta function to prime numbers, with the zeta zeros corresponding to eigenvalues of the Laplacian, and the primes corresponding to geodesics. Motivated by the analogy, Selberg introduced the Selberg zeta function of a Riemann surface, whose analytic properties are encoded by the Selberg trace formula. (wiki)
Loads, as ever, at Watkin’s site.
Created a stub for Arthur-Selberg trace formula.
Thanks! I have uploaded those lecture notes by Bump here and briefly pointed to them from the main text.
One place I see where the Selberg-type trace formula is discussed together with explicit discussion of the zeta-function of an elliptic operator is
The zeta function itself is (2.13) there.
I have expanded the table by a new row “physics/2dCFT” highlighting the fact that
the theta function is the partition function as a function of choice of polarization and of background gauge fields;
the zeta function is the analyticallyc continued Feynman propagator, its special value is the regularized Feynman propagator;
the eta function is the analyticallyc continued Dirac propagator, its special value is the regularized Dirac propagator;
edited the bottom-right entries in the table
But this needs attention, I need to read up on this. I suppose it starts out as:
over $\mathbb{Q}$ for every Artin L-function is equal to some Dirichlet L-function;
over a general number field, every Artin L-function is equal to some Hecke L-function “of finite order” (is this right?!)
…
What, if anything, is the analogous statement now for function fields?
I have split in the table the column which previously contained both eta functions and L-functions in two.
I had originally taken that idea to have a single column for them from Richardson, first page, which asserts that the eta function is to the Dirichlet L-function as the zeta function of an elliptic operator is to the Riemann zeta… but this is not quite accurate it seems to me.
Rather, the point is that L-functions are twisted zeta functions, specifically a zeta function is the Mellin transform of $\theta(0,-)$ while an L-function is a Mellin transform of $\theta(\mathbf{z},-)$ for general $\mathbf{z}$ (e.g. Kudla77, Stopple 95).
The point of eta-functions is somehow different. They appear in twisted and untwisted form just as the zeta functions, but in some way they divide the parameter $s$ by two. This is clear in the operator-analysis row of the table. I am not sure under which name this secretly appears in the number-theory rows. Only that it makes the special values be at a value of $s$ which looks like $\Re(s) = 1/2$ from the point of view of the zeta function. Which of course reminds one of something…
There are also, I vaguely recall, special values that happen at 3/2, for some L-functions (and possibly other odd numerators as well)
made first row of the table more accurate by pointing to 1-loop vacuum amplitude and to vacuum energy
For the entry to the left of Goss zeta function, perhaps Goss’s A formal mellin transform in the arithmetic of function fields helps.
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