There are many outstanding articles written in the last 20 years in Number Theory. What do you think what is the most interesting and groundbreaking article in Number Theory?
Naturally this will depend on the specific type of subfield in number theory and even then there are is no clear winner in most cases. Anyway personaly I would pick Harri's and Taylor's "The geometry and cohomology of some simple Shimura varieties" as it is one of the first big steps in the Langlands Program but of course there is no universal answer.
Norbert Tihanyi For me, recently, this one: "The Bernoulli numbers and Unity of mathematics".
https://people.math.harvard.edu/~mazur/papers/slides.Bartlett.pdf
Carlos Hernan Lopez Zapata , The article you mentioned is indeed very interesting. By the way, I am fascinated in prime number theory so my favourite article is from Yitang Zhang. Bounded gaps between primes. This was a huge breakthrough in number theory.
https://annals.math.princeton.edu/wp-content/uploads/Yitang_Zhang.pdf
Norbert
Norbert Tihanyi Thanks for this remarkable article. I am fascinated by the possibility to 'decode' more the Bernoulli numbers. Now, I have completed an article which describes how the Bernoulli numbers B_2k are related to the Turán moments with tremendous importance within the context of the entire function or the Riemann Xi function, especially at s=1/2 +t*i. I have developed a formula that lets calculate the Bernoulli B_2k based on these Turán moments and other formula based on the coefficients of the Jensen polynomials analyzed for the Riemann Xi function as well. The formulas set an important link considered as a consequence of the assumption of the point s=1/2 +t*i and the validity of taking seriously the hyperbolicity of Jensen polynomials for Xi.The main result in my research is an expression for the Euler-Mascheroni constant written by a summation based on the Turán moments, of course, jumping to the Jensen and even Taylor coefficients for the Riemann Xi's series form. There are details evidenced by numerical computation and even a graphic of the Riemann Xi function after using the data analyzed. I have reached also the way to write the even derivatives of the Riemann Xi at s=1/2 by the coefficients mentioned above, so it is a set of numbers that contributes to represent other important numbers like the Bernoulli and the Euler Mascheroni constant. Crossing fingers for the proper dissemination in magazines.
Clearly the Riemann Hypothesis.It combines many areas in number theory beyond analysis and even physics.I consider it the most comprehensive topic that one can take up and explore infinite other areas of learning.
Nikos Mantzakouras The Riemann Hypothesis and Riemann zeta function, fascinating topics. I hope that a new age of findings starts soon. Greetings.
Dear Nikos Mantzakouras and Carlos Hernan Lopez Zapata
I also like the math behind the Riemann Hypothesis. I did my PhD thesis regarding peak values of the Riemann zeta function. It is really interesting.
You can find my thesis here:
https://edit.elte.hu/xmlui/bitstream/handle/10831/46542/Theses_ENG.pdf
Regards,
Norbert
Norbert Tihanyi thank you very much for sharing your thesis. I will read this work.
There is many information around the Riemann zeta function and Riemann Hypothesi, it is so wonderful to be able to read new documents that are not easy to find.
Take care
Dear Norbert Tihany interesting your paper I will read it in detail soon.The number theory is certainly very worthwhile with the analysis in all kinds of work in mathematics. There are many issues unexplored that I believe with a free spirit and will can be solved.Not on blind theories but on the analysis of transcendental equations mainly.I wish the best to all colleagues.
Very interesting question. I have a few about NT, they verse about harmonic numbers, Riemann zera function and Dirichlet series.
It's impossible to rank NT articles, all have their value and importance.
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