Project 3 (Approach)

 On the Riemann Hypothesis: an Approach Using Algebraic Geometry and Non-standard Analysis (preliminary version)


                          Ovidiu Pasarescu, september 2017


[This text represents an extended abstract of 7 talks which I had in the seminar of Algebraic Geometry from the "Simion Stoilow" Institute of Mathematics of the Romanian Academy during Spring 2018, and of a 50 minutes invited talk which I had to the International Conference on Geometry, Ploiesti, Romania, September 2017]


1.Extended Abstract


 As well-known, the old and unsolved Riemann Hypothesis says that all the nontrivial zeros of the so-called Riemann zeta function (the meromorphic continuation to C of the generalized harmonic series, which is holomorphic on C\{1}, and has a simple pole with residue 1 at z = 1) lie on the line defined by Re(z) = 1/2. An analogue of this Problem, obtained by replacing the Riemann zeta function with Weil zeta functions for smooth projective curves over finite fields of positive characteristic is known to be true, as proved by A. Weil (and by P. Deligne for arbitrary dimensions, greater than 1), result which has been extended more recently from smooth to irreducible projective curves with (isolated) singularities. One of the ideas used in the literature in order to arrive to a solution of the Riemann Hypothesis was to try to imitate somehow the proof of Weil, considering the Riemann zeta function as a Weyl zeta function for some kind of compactification of the scheme Spec(Z) (Z=the ring of integers), arriving in such a way to look for a ”field of characteristic 1” (or a ”field with 1 element”), named F1, such that to get on Z a structure of F1-algebra. Such a field, certainly, does not exist, but there are some kinds of geometric theories (considered as being) over it (constructed by C. Soule', A. Connes, A. Deitmar, Y. Manin, among others), trying to encode the properties of the projective curves used by A. Weil. In this talk, I will consider an enlargement of the usual Von Neumann Univers U (having the apartenence as the primary relation symbol, the axiomatic ZFC, and the standard language in a bivalent logic), namely *U=non-standard univers (having, roughly speaking, the same primary binary relation symbol, one new unary relation symbol S=standard, and 3 more axioms, namely I=Idealization, S=Standardization, T=Transfer, the system of axioms becoming the so-called IST=Internal Set Theory). This is (a version, introduced by E. Nelson, of) the well-known extension to nonstandard analysis (introduced by A. Robinson). We prove that (RH) (=Riemann Hypothesis) is equivalent with its extension to *U, denoted by *(RH). We consider the last one as being associated (in the Weil sense) to some compactification of the internal standard sub-scheme *(Spec(Z)) of the scheme Spec(*Z)(*Z might be considered an ultrapower of Z), extending in such a way the idea from the standard case. Then, I will find an effective field which plays the role of F1, in the univers *U. However, in order to put this F1 to work, I need a further axiom for set theory, namely GCH(=Generalized Continuum Hypothesis). Due to the work of K.Godel, it is known that, given a Von Neumann Universe U_0, with ZF axioms only, it is possible to built a subunivers U_1 of U_0 satisfying the axioms GCH and C (note that ZF +GCH implies the Axiom of Choice C) also (the Godel Constructible Universe). This last axioms are kept in the nonstandard enlargement *U_1 of U_1. Now, the univers *U_1 has as axioms IST+GCH. These transformations of some initial univers U_0, where we consider the (RH)_0 (with ZF only), do not represent a problem, because the truth value of (RH), if any, is independent from nonstandard extensions (see before) and from passing to the Godel Constructible Universe (as a concequence of the Shoenfield’s Absolutness Theorem), that is (RH)_0 (in U_0) and *(RH)_1 (in *U_1) are both either true, or false (or indecidable!). In this talk I will do several steps towards a proof of the fact that *(RH)_1 (so (RH)_0 also) is true.

Note. In this abstract it is possible to replace the Von Neumann Univers with a Grothendieck Universe, if the work is done in elementary Boolean topoi.


2. Contents


  Sections 2-8 are, with the exception of their ordering, not original - in the sense that they are either well-known (sections 2,3,4,8), or known to the experts and partially unpublished (sections 5,6,7).

 1. Introduction.

 2. Riemann zeta function and Riemann Hypothesis (RH).

 3. Weil zeta functions for integral projective curves.

 4. The non-standard Universe *U; internal and external sets in *U.

A set X from *U is called external if there is at least one sentence P true on U for the trace of X in U but false for X in *U (when transferred to *P) and it is called internal otherwise. For example, an ultraproduct of finite fields with characteristics going to infinity does not satisfiy the transfer in *U of the sentence claiming that it has characteristic zero, which is true for such a field when considered in U (see 9. also). One say that such a field has positive internal characteristic, but, in the same time, exterior characteristic equal to zero.

5. Non-standard Riemann zeta function and non-standard Riemann Hypothesis *(RH).

6. Elements of internal and external algebraic geometry (in *U).

7. Internal Weil zeta functions for internal integral projective curves.

8. Shoenfield’s absolutness Theorem, Godel constructible sub-Universe U_1, and *(RH)_1 (in *U_1).

9. Steps towards a proof of Riemann Hypothesis.

We consider an infinite increasing sequence of prime rational integers (so an internal hyper-prime hyper-rational integer from *Z) generating an infinite sequence of cyclic finite fields having these characteristics, and let a F1 (see the abstract) be the ultraproduct of them. This field can be seen both as an external (comming from U_1, and considered in *U_1) and an internal (in *U_1) field, being an ultraproduct of finite fields, so a hyper-finite field (having, in the same time, positive internal hiper-finite characteristic, and external characteristic equal to zero - see section 4; this happens because in the external case the sums and the products in this field are finite, while in the internal case they are hyper-finite). We compare the two projective algebraic geometries of hiper-finite primes over F1 obtained in both situations (which are external, respectively internal in *U_1). In the first case we get external algebraic integral projective curves covering *(Spec(Z)) - see the abstract - (having as zeta function associated truncations of the non-standard Riemann zeta function, converging to it), and in the second case we get internal integral projective curves (having as zeta function associated internal Weil zeta functions). As known from A. Weil, (RH) is true for Weil zeta functions, so, by the transfer principle (T from the IST axiomatics, see the abstract), is also true for internal Weil zeta functions. I am doing several steps in understaning the previous interplay, trying to transfer the information from positive characteristic to characteristic zero.


Aknowledgement. I want to thank to Professor Alexandru Buium for the general discussions which we had on the subject represented by the Riemann Hypothesis, and related topics, both at the University of New Mexico at Albuquerque and at the ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest.


Address of the author. ”Simion Stoilow” Institute of Mathematics of the Romanian Academy(IMAR), Bucharest, Romania. e-mail:  or .