What are some fundamental symmetries of Riemann Zeta (Xi) function are important to Riemann Hypothesis ?

(Beside the obvious symmetry of Riemann Xi function, s <--> 1-s reflection)

IMHO, at the end of the day, when dust settles, maybe we will find that it is some fundamental symmetries that causes Riemann Hypothesis to be true.

What are those symmetries ?


First of all, it is important to understand what we mean by symmetry. I will take 'symmetry' to be invariance under a one-to-one self-map $\phi$ of $\mathbb{C}$. Strict global symmetry then is the functional equation $$\xi(z)=\xi(\phi(z)).$$ Using the conformal characterization of biholomorphic functions on $\mathbb{C}$ I will let you show that $\phi$ has to be conformal on $\mathbb{C}$. The only conformal 1-1 maps on $\mathbb{C}$ are $\phi(z) = az+b$: complex rescalings (rotations and dilations) and translations. Using the location of the zeros of $\xi$ one can see that the only symmetry is precisely $\phi(z) =1-z$.

Taking our cure from the functional equation itself, however, we may ask the vague question of a 'sort-of' symmetry: not invariance, but a nice transformation under some simple function: $$\xi(z) = m(z)\xi(\phi(z)) $$ for some manageable multiplier $m$. The problem with this is that if you pick your favorite transformation $\phi$, you can define $$m(z) = \frac{\xi(z)}{\xi(\phi(z))} $$ so you are running in circles. In fact, all useful transformations we know ultimately lead back to the functional equation so there is no symmetry or quasi-symmetry that we know of truly independent of the functional equation.

However, this does not mean $\zeta$ is not related to symmetrical objects. On the contrary: there is a device, called the Mellin transform, which converts modular forms to $L$-functions, and $\zeta(z)$ is the regularized Mellin transform of a famous modular form, the Jacobi theta function. As a modular form, this function has a vast and complex group of symmetries. These symmetries were in fact instrumental in proving the functional equation in the first place, as well as providing the Fourier analytic input necessary for estimates on zeta in strips. From this point of view, not only are symmetries crucial to the study of zeta, they lie at its core and they are certainly not a new idea.

However, translating between the symmetries of modular forms and regularity of the corresponding $L$-function is no trivial task. Both the Mellin device and the required regularization are highly transendental analytic operations that prevent the symmetries from manifesting clearly on the target.

Upshot: The zeta function is linked to functions with vast symmetries but in a very subtle way. Other $L$-functions, coming in families, have even tighter links to symmetry groups and this connection lies at the forefront of the theory of automorphic $L$-functions.

I cannot close this note without mentioning another link between the Riemann zeta function and symmetry, one that begins with the Riemann Hypothesis. You see, apart from the horizontal location of zeros in the critical strip, there is the issue of the vertical distribution. It turns out that zeros of the Riemann zeta function are not randomly distributed in the vertical direction, but obey the Montgomery-Odlyzko Law, a distribution which matches precisely that of a highly symmetric random matrix ensemble, the Gaussian Unitary Ensemble.

This mysterious observation actually clicks (in retrospect) with another structure in number theory: the zeta function associated to a projective variety. There, too, there exist statements similar to the Riemann hypothesis, bundled up in the so-called Weil conjectures, with one big difference: they have been proven. The case of curves was already proven by Weil, and in all these investigations a key role was the manifestation of that zeta function as a kind of generalized determinant of a linear operator on linear spaces associated with the variety. The horizontal distribution was then a manifestation of a symmetry (self-adjointness) of the corresponding operator.

Furthermore, in that algebraic setting this manifestation was related to the membership of the zeta function in an entire family of zeta functions, bound together by the symmetries of certain compact Lie groups; the vertical distribution of the zeros in the entire family is controlled by the nature of the group of symmetries. This is still, I believe, cutting edge research, and the complete exposition of these results can be found in Sarnak and Katz, Random Matrices, Frobenius Eigenvalues, and Monodromy.

It is expected by some experts that both of these intuitions somehow carry over to the usual zeta fucnction, and although nobody has managed to realize this expectation, most researchers make investigations inspired by those results in the function field case and the connection with physical ensembles.

The upshot of this second part: apart from symmetries of zeta itself or other functions the zeta is related to, there are seeming manifestations of symmetry either "from the outside", from the space of functions where zeta lies, and from statistical rather than functional connections to symmetric objects.

  • $\begingroup$ what about the Euler product ? in short RH is conjectured for every $L$ function with a functional equation AND an Euler product en.wikipedia.org/wiki/Selberg_class and is known to fail for all others Dirichlet series $\endgroup$ – reuns Aug 23 '16 at 0:20

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