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What are people thinking?


Walter Von Moo Moo

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11 minutes ago, hornedlizardman said:

I don't know how the weather is in New Jersey, but in Houston, Texas, it's usually air conditioner weather all year round.  i

Houston runs about 55 degrees, mid day, December to mid March.

But, I know the summer heat waves there.  98 to 105 with intense humidity.  That's air conditioner time.

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So I am currently thinking about dinner and Artin L-functions, and the Artin conjecture in general, which has remained elusive to proof!

So, essentially an Artin L-function is a Dirichlet series with a linear representation p of a Galois group G in a finite dimensional complex vector space V, such that G is the Galois group of the finite extension L/K of number fields, and the Artin L-function L(p,s) is defined by an Euler product. 

There is an Euler factor for every prime ideal in K's ring of integers! The simplest case of this to define is when is unramified in L, where the Frobenius element Frob(p) is a conjugacy class in G. 

Therefore, the characteristic polynomial of p(Frob(p)) is defined. The Euler factor of is defined as:

1/(poly(p(Frob(p)))) = 1/(det[I-tp(Frob(p))]), 

where ploy is the characteristic polynomial, det is the determinant. This is a rational function of t, evaluated at t = s/(N(p)), with s being a complex variable in the usual Riemann zeta function notation. 

If we are dealing with the ramified pI is the inertia group which is a subgroup of G, where the same process is done, but to a subspace with V fixed by I. The Artin L-function L(p, s) is then the infinite product over all prime ideals of these factors. 

Being able to prove this, and other problems and statements related to it are some of the most cutting edge problems in modern mathematics and science. 

So I guess you could say that is what is on my mind today. 

Oh wait! You were baiting people into this thread with a clickbait title haha my bad, I thought you were just asking what I was thinking about! 

Edited by atomicinumatt
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2 hours ago, Walter Von Moo Moo said:

Houston runs about 55 degrees, mid day, December to mid March.

But, I know the summer heat waves there.  98 to 105 with intense humidity.  That's air conditioner time.

It can be cold outside and I'll still sleep with the fan on. Because that's how we roll in Houston.

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