A Dirichlet L-function looks like this:
And the related modular form looks something like this:
What about the following:
Mathworld, in the above page, mentions that: “Dirichlet L-series can be written as sums of Lerch transcendents with z a power of e^(2pii/k).”
But an explicit google search for the following does not appear to result in much useful things:
“lerch transcendent” “modular forms” Hecke. I may generate a video showing the interpolation of for a given Dirichlet character from L-function to modular form shortly.
But, lo, I spoke too soon: THE LERCH ZETA FUNCTION I. ZETA INTEGRALS
What can the algebraic topology of complex projective varieties tell us about analytic combinatorics and vice versa, what can analytic combinatorics tell us about the algebraic topology of complex projective varieties?
For the quintic, the hypergeometric and theta methods must necessarily find the same results, and this must also be the case for many different sorts of root finding methods, Newton’s, Householders, Halley’s, Steffenson’s, etc. So why not construct new identities for the roots between all these different root finding methods?
Suppose I have a transcendental function , and I wish
to construct a new function from all Padé approximants to it, like thus:
Why do this? Well, the idea here is to cancel out the common and make something new.
And now time for a little rebranding. All of my written math journals of late have been generally written in <a href="Apica CD40SN journals. All of them are named after butterfly species, and the butterfly depicted on the cover image of each is a from a picture of a living butterfly. This online journal-thingy is no different, so therefore this one is Orange-Banded Shoemaker
On github, in my math diary (which is unorganized at the moment), there is code which implements (essentially, it’s only a special case, but is easily generalized0 both the hypergeometric (Glasser) and the theta functions solution of the quintic. The Glasser method quoted on the wikipedia page about the Bring Radical contains sign errors but is essentially correct. The theta functions solution on the mathworld page about the quintic also appears to be essentially correct, although the orthography on that page contains some spurious mistakes — such as the “”
coefficient, and is painful to write, but is elegant.
The polynomial in question is , and this one was chosen in particular because the constant interacts nontrivially with both of these solutions: For the theta function solution it corresponds to elliptic modulus and thus to nome , and in the hypergeometric case, the fourth power of this means that the -slot in the hypergeometric functions is simply 1 — which is a little bit of a problem for mpmath, and it’s noted in the documentation that there might be problems when is around one. I am looking forward to reimplementing both of these in arb
owen@orrery ~/atal » ./hypg.verify.py
(0.66880437318301406386 - 0.006218779022452095584j)
(0.66879191050396671825 + 0.0062116365382047596402j)
(-1.1038422688861613827 + 8.7204858053738321163e-19j)
(-0.11681917053334134398 + 1.03445357140566252j)
(-0.11681917053334134475 - 1.0344535714056625207j)
owen@orrery ~/atal » ./theta.verify.py
(-1.103842268886161584 + 8.6656680999461563393e-17j)
(0.66874030497642225453 + 4.459681438480688112e-9j)
(-0.11681917053334137041 - 1.0344535714056627406j)
(-0.11681917053334137041 + 1.0344535714056627406j)
(0.66874030497642225453 - 4.459681438480688112e-9j)
More generally different root finding methods must give the same results, and I suspect there are nifty things to find by equating the Umemura/theta functions approach with the Glasser/differential resolvents approach. I did ask, and will update relationship between solution of quintic in terms of 4F3 hypergeometric function and theta functions on m.se, but given how scant public knowledge of these methods are and the errors in their transcriptions, I rather doubt there has been detailed study so far.