​The Riemann Zeta Function

I. The Riemann Hypothesis - What and Why?

The Riemann hypothesis was conjectured by B. Riemann in 1859 in his monumental monograph [1]. Today, the conjecture is considered one of the most notorious and, by many, one of the most important questions in math. See [2]. Like many good problems the Riemann hypothesis is straight-forward to state: ​The Riemann zeta function ζ(z) admits non-trivial zeros only along the critical line Re(z)=0.5. You can also see a nice accessible video description: 



But - why is the Riemann hypothesis actually so central and notorious? One can give a few kinds of answers:
(a) The Riemann hypothesis is important (relations to prime distribution, etc...), (b) The Riemann hypothesis is hard (computing zeta is hard, zeta has chaotic features...) (c) The Riemann hypothesis is mythological (open for over a century, tackled by some of the greatest mathematicians...)

The aim of this "blog" is to mainly discuss various aspects of (b) - with two parallel resources in mind - (1) the vast remarkable literature on zeta (spreading over more than a century, until current times) (2) modern computers - which allow us to compute and visualize zeta in ways which were not commonly avilable in the past.

II. The ζ-monotonicity conjecture.





























In the 1960's Robert Spira conducted various empirical studies of the Riemann zeta function, the results of which are documented in [7]. The studies were done in Fortran code runing on an IBM 7040 machine (see picture).  

In a later work [7], Spira introduced and further studied the following remarkable montonicity conjecture: 

The ζ-monotonicity conjecture : For any y>8 the function ∣ζ(x+yi)∣ is strictly-decreasing in the half-line x<0.5.

Clearly, monotonicity implies the Riemann hypothesis. In fact, the two conjectures have been shown to be equivalent, see [4,7] for ξ-monotonicity and [3] for ζ-monotonicity, as well. The following figure illustrates the ζ-monotonicty property, showing ∣ζ(0.5+yi)∣<∣ζ(0.05+yi)∣ in the domain 0














And, similarly, in the bigger domain 0
In a recent preprint [5] we have studied various questions related to the  ξ-monotonicity conjecture. Our aim in [5] is to explain why the monotonicity property, viewed empiricaly above, holds. ​ 
References

[1]  B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie, November 1859. 

See link: 
http://www.claymath.org/sites/default/files/zeta.pdf

​[2] E. Bombieri, The Riemann Hypothesis – official problem description, 2000.

See link:
http://www.claymath.org/sites/default/files/official_problem_description.pdf


[3]  J. Sondow and C. Dumitrescu, A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis 

See link:
https://link.springer.com/article/10.1007/s10998-010-1037-3

​[4] Matiyasevich-Saidak-Zvengrowski, Horizontal monotonicity of the modulus of the Riemann zeta-function, and related functions 

See link: 
https://www.researchgate.net/publication/224951648_Horizontal_Monotonicity_of_the_Modulus_of_the_Riemann_Zeta_Function_andRelated_Functions​


​[5] Y. Jerby An experimental study of the monotonicity property of the Riemann zeta function, 2017, Arxiv:1707.01754.

See link: 
https://arxiv.org/abs/1707.01754

[6] R. Spira Check Values, Zeros and Fortran Programs for the Riemann Zeta Function and
its First Three Derivatives. Report No. 1, University Computation Center, University of
Tennessee, Knoxville, Tennessee.

See link:

[7] R. Spira Zeros of ζ’(s) and the Riemann hypothesis. llinois J. Math. 17 (1973), no. 1, 147--152.

See link:
https://projecteuclid.org/euclid.ijm/1256052045

III. Voronin's universality theorem and Garunkstis effective results.
​(in progress)

An early "vlog" can be found below