generalizing L-functions and modular forms

A Dirichlet L-function looks like this:

L(\chi,s)=\sum_{n=1}^{\infty} \frac{\chi(n)}{n^{s}}

http://en.wikipedia.org/wiki/Dirichlet_L-function

 

And the related modular form looks something like this:

f(\tau)=c_{0}+\sum_{n=1}^{\infty} \chi(n)e^{2\pi i \tau}

What about the following:

G(\chi,\tau,s) = c_{0}+\sum_{n=1}^{\infty} \frac{\chi(n)e^{2\pi i \tau n}}{n^{s}}

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 G(\chi,\tau,s) 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

q-analogizing Glasser’s hypergeometric solution to the quintic

Start with Glasser’s hypergeometric solution to the general quintic:

(assume that you’ve already performed the Tschirnhausen transformation)

F_1(t)  = \,_4F_3\left(\frac{-1}{20},  \frac{3}{20}, \frac{7}{20}, \frac{11}{20} ;\frac{1}{4}, \frac{1}{2}, \frac{3}{4}; \frac{3125t^4}{256}\right)
F_2(t)  = \,_4F_3\left(\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} ;\frac{1}{2}, \frac{3}{4}, \frac{5}{4}; \frac{3125t^4}{256}\right) \\[6pt]
F_3(t)  = \,_4F_3\left(\frac{9}{20}, \frac{13}{20}, \frac{17}{20}, \frac{21}{20} ;\frac{3}{4}, \frac{5}{4}, \frac{3}{2}; \frac{3125t^4}{256}\right)
F_4(t)  = \,_4F_3\left(\frac{7}{10}, \frac{9}{10}, \frac{11}{10}, \frac{13}{10} ;\frac{5}{4}, \frac{3}{2}, \frac{7}{4}; \frac{3125t^4}{256}\right)

(the signs are wrong on the wikipedia page. they have been corrected below)

x_1  = {} tF_2(t)
x_2  =  {} F_1(t)    - \frac{1}{4}tF_2(t)  -  \frac{5}{32}t^2F_3(t)  -  \frac{5}{32}t^3F_4(t)

x_3  =  -F_1(t)  -  \frac{1}{4}tF_2(t)  +  \frac{5}{32}t^2F_3(t)  -  \frac{5}{32}t^3F_4(t)

x_4  =  {} +{\mathrm{i}}F_1(t)  -  \frac{1}{4}tF_2(t)  +  \frac{5}{32}{\mathrm{i}}t^2F_3(t)  +  \frac{5}{32}t^3F_4(t)

x_5  =  -{\mathrm{i}}F_1(t) -  \frac{1}{4}tF_2(t)  +  \frac{5}{32}{\mathrm{i}}t^2F_3(t)  +  \frac{5}{32}t^3F_4(t)

I was reading Gasper’s Basic Hypergeometric Series, and since I’ve been playing around (amongst other things) with the hypergeometric solution to the quintic, and in the first few pages Gasper mentions {}_a\phi_{b} q-hypergeometric functions, the question leapt into my mind: What would happen if we take the q-analogue of the Glasser’s method hypergeometric solution, in other words:

\Phi_1(t,q)  = \,_4\phi_3\left(q^{\frac{-1}{20}},  q^{\frac{3}{20}}, q^{\frac{7}{20}}, q^{\frac{11}{20}} ;q^{\frac{1}{4}}, q^{\frac{1}{2}}, q^{\frac{3}{4}}; q ;\frac{3125t^4}{256}\right)
\Phi_2(t,q)  = \,_4\phi_3\left(q^{\frac{1}{5}}, q^{\frac{2}{5}}, q^{\frac{3}{5}}, q^{\frac{4}{5}} ;q^{\frac{1}{2}}, q^{\frac{3}{4}}, q^{\frac{5}{4}}; q ;  \frac{3125t^4}{256}\right) \\[6pt]
\Phi_3(t,q)  = \,_4\phi_3\left(q^{\frac{9}{20}}, q^{\frac{13}{20}}, q^{\frac{17}{20}}, q^{\frac{21}{20}}; q^{\frac{3}{4}}, q^{\frac{5}{4}}, q^{\frac{3}{2}};q; \frac{3125t^4}{256}\right)
\Phi_4(t,q)  = \,_4\phi_3\left(q^{\frac{7}{10}}, q^{\frac{9}{10}}, q^{\frac{11}{10}}, q^{\frac{13}{10}} ;q^{\frac{5}{4}}, q^{\frac{3}{2}}, q^{\frac{7}{4}};q \frac{3125t^4}{256}\right)

x_1(q)  = {} t\Phi_2(t,q)
x_2(q)  =  {} \Phi_1(t,q)    - \frac{1}{4}t\Phi_2(t,q)  -  \frac{5}{32}t^2\Phi_3(t,q)  -  \frac{5}{32}t^3\Phi_4(t,q)

x_3(q)  =  -\Phi_1(t,q)  -  \frac{1}{4}t\Phi_2(t,q)  +  \frac{5}{32}t^2\Phi_3(t,q)  -  \frac{5}{32}t^3\Phi_4(t,q)

x_4(q)  =  {} +{\mathrm{i}}\Phi_1(t,q)  -  \frac{1}{4}t\Phi_2(t,q)  +  \frac{5}{32}{\mathrm{i}}t^2\Phi_3(t,q)  +  \frac{5}{32}t^3\Phi_4(t,q)

x_5(q)  =  -{\mathrm{i}}\Phi_1(t,q) -  \frac{1}{4}t\Phi_2(t,q)  +  \frac{5}{32}{\mathrm{i}}t^2\Phi_3(t,q)  +  \frac{5}{32}t^3\Phi_4(t,q)

The idea here would be to choose a fixed t, one whose roots were all within the unit disk, and then to make five phase portraits of their q-analogues

Remark: I am tempted to look at cases where q=g^{20n}, n\in\mathbb{N} because all of those fractional powers will vanish

Additionally, suppose we were to get \prod_{n=1}^{5} (x_{n}(q)-t) into Bring form and then solve that with the non-q-analogized version.(matrix identity to be worked out soon)

Root finding algorithms and identities

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?

rebranding.

 

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

Hypergeometric and Theta Functions Solution to the Quintic (in python!)

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 “q^{5/8}(q^{5})^{-1/8}
coefficient, and is painful to write, but is elegant.

The polynomial in question is w(z) = z^5 -z + \frac{4}{5^{5/4}}, 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 \sqrt{2}-1 and thus to nome e^{-\pi\sqrt{2}}, and in the hypergeometric case, the fourth power of this means that the z-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 z 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.