1. This is a directed course which among others, means it is flexible in terms of its contents, in order to accommodate the participants interests, but it is not flexible in terms of its work load.

2. The whole course will run on a Debate-Session dynamics, with minimum, indispensable, short lectures from the Professor.

3. Each week a work sheet will posted in the Course Website for the problems and work of the week.

4. The whole course will run in English for the Debate-Sessions, course documents, papers and slides.

5. Participants are expected to participate actively in at least 80% of the sessions.

6. Each participant is expected to prepare one 30-minute presentation in the semester, the slides must be written in Beamer LaTeX. The slides will posted in the Course Website, a template can be downloaded here.

7. In order to develop leadership skills, each participant is expected to be the chair of the Debate-Session, once in the semester. The responsibility includes preparing the week work sheet and lead the session. As a preparation for the task, each participant will be invited to act as chair in one Debate-Session in the lower level course on Set Theory & Combinatorics.

8. Each participant is expected to turn in a final expository monograph, no longer than three pages, in the vast field of Number Theory i.e. it may not be restricted to Analytic Number Theory, but any other areas related to Number Theory. The paper must be written in English and addressed to a general audience. The format will be the Elsevier article Format, and it will be posted in the Course Website. The Template can be downloaded here. Acceptable topics are.

   • Exposition of one or more, state of the art open problems.

   • Exposition of one or more solutions of corner-stone problems.

   • Cultural-Anthropological and/or Historic-Critical surveys.

   • Survey on Expository Works intended to present iconic questions to general audience and even for improving Mental Conditions, Recreation, Art Creation or other relevant topic.

9. The final grade will be 50% participation in the Debate-Sessions, 25% from the exposition and 25% from the paper.

1. Introduction to Analytic Number Theory, Hildebrand. MAIN REFERENCE.

2. Introduction to Analytic Number Theory, Apostol.

3. A Course in Analytic Number Theory, Overholt.

4. Analytic Number Theory: Exploring the Anatomy of Integers, De Koninck & Luca.

5. Multiplicative Number Theory, Davenport.

6. A Classical Introduction to Modern Number Theory, Ireland.

7. Algebraic Number Theory, Lang.

8. Functions of One Complex Variable, Conway.

9. Complex Analysis, Stein & Shakarchi.

10. On Transcendental Progressions. Leonhard Euler. Translation by Stacy Langton.

OPEN ACCESS JOURNALS

You can access the full content of the following journals online, for free.

1. Integers, Electronic Journal of Combinatorial Number Theory.

2. Journal of Integer Sequences.

3. Notes on Number Theory and Discrete Mathematics.

4. Acta Arithmetica. (Free access to articles older than eightyears)

5. The Fibonacci Quarterly. (Free access to articles older than eight years)

LIMITED ACCESS JOURNALS

You can partially access the content of these journals, through the data basis of the university hosted in SINAB. All the following journals belong to the Springer Journal. You can log in to here.

1. Journal of Number Theory.

2. Research in Number Theory.

3. The Ramanujan Journal.

PAID ACCESS JOURNALS

You can access the following prestigious journals only by paying their fees. However, occasionally the Editor liberates some important articles.

1. International Journal of Number Theory.

2. Journal de Théorie des Nombres de Bordeaux.

1. August the 6th, 2015. Meetings scheduled on Wednesdays 16:00 - 18:00, room 43-326. Problem Set 1 and Problem Set 2 are Available.

2. August the 12th, 2015. First Meeting. Discussion of problems 1.5 and 1.6 Problem Set 2. Reading Assignment: Leonhard Euler's deduction of the Gamma function as a generalization of the factorial concept.

3. August the 19th, 2015. Second Meeting. Discussion of problem 18 (iv) Problem Set 1. There was need for a lower bound for the growth of the counting primes function, check out the updated version (now is the problem 18(v)). Problem Set 3 Available. Brief lecture on the main traits of Analytic Number Theory

   1. The "blending" of the multiplicative integer structure and convergence concepts, in analogy with the fussion of linear algebraic structure with topological distance of functional analysis. The role of the Riemann-Zeta function as a means to this connection.

   2. The concept of "separation of total orders" in analogy with the "separation of variables" from PDE.

   3. The role of Dirichlet products as binary isomorphic operation in mulplicative algebraic structures.

4. August the 26th, 2015. Third Meeting. Discussion of problem 18 (v) Problem Set 1, no solution has been found, it remains an open question. Discussion of Problem 1.13, Problem Set 2 solved.

5. September the 2nd, 2015. Problem Set 4 Available. Discussion of problems 2.2, 2.3, 2.4, 2.5, 2.6, 2.8 Problem Set 2. Comments and debate on

• • Absolute and conditional convergence of the Dirichlet Series.

  • Constrast between power series and Dirichlet series. The power functions versus the exponential functions.

  • Briefs on the analytic structure of the Dirichlet series and its regularization requirements by exponential growth conditions.

  • Challenges to seek out reasons for the remarkable success of Dirichlet series.

  • Combinatorial meaning of the Moebius map.

  • Briefs on integration by parts and duality. Abel's integration by parts to include integer-evaluations in the duality statements inherent to integration by parts.

6. September the 9th, 2015. Problem Set 5 Available. Discussion of problem 1.22 Problem Set 2. Discussion of problems 1.2, 1.3, 1.4, 1.5, 1.6 and 1.8 Problem Set 5.

• • Comments on the asymptotic order of the number of divisors as it is controlled from above by the root of any order and bounded from below by any power ov the logarithm. The exact blow rate asymptotic order remains an open question.

  • Observations on the average order of the log and the relative errors in the Cesaro Average.

  • Brief lecture on the technique of graphic estimation of errors and alignment of errors by translation.

7. September the 16th, 2015. Problem Set 6 Available. Discussion of Problem 1.22 Problem Set 2.; (i), (ii), (iii), (iv) are solved. Part (v) remains an open question.

• • Briefs on Characters and the motivation to introduce them.

  • Comments on the wild nature of the groups of units.

8. September the 23th, 2015. Suspended due to University's Celebration Week. Find the Analytic proof of the Prime Number Theorem by D.J. Newman here (you have access to JSTOR with your user, through SINAB Universidad Nacional de Colombia).

9. September the 30th, 2015. Problem Set 7 Available. Discussion of problems 1.4, (i), (ii), (iii). Problem 1.5 (i), (ii) Problem Set 2.

10. October the 7th, 2015. Problem Set 8 Available. Solution of problems 1.19 Problem Set 5 , Problem 2.9 Problem Set 6. Discussion of problem 2.4 (i), which remains open, Problem Set 6.

11. October the 14th, 2015. Canceled due to lecturer's trip.

12. October the 21st, 2015. Discussion of Problems 1.9, 1.10, 1.11 I(i), (ii), (iii), II(i), (ii) and (iii) Problem Set 4.

13. October the 28th, 2015. Discussion of Problems 2.1, 2.3 Problem 2.4 remains an open question. Problem Set 6.

14. November the 4th, 2015. Suspended due to Lecturer's obligations with the Conference in Mathematical Biology.

15. November the 11th, 2015. Suspended due to lack of quorum.

16. November the 18th, 2015. Suspended due to participants' multiple obligations.

17. November the 25th, 2015. Suspended due to participants' multiple obligations. Final debate session of the semester.