An introduction to the Riemann zeta function, multiple zeta function, multiple zeta values (MZV or Euler-Zagier sums) and some of their special evaluations
- The Riemann zeta function and values is defined, for as
Although, it is undoubtebly the most important function in mathematics, the Riemann zeta function still keeps many misteries. The most important and impenetrable of them is the Riemann hypothesis (all non-trivial zeroes of the lie on the line ) with tremendous consequences in number theory and beyond!
Let us recall what happens when we evaluate at integers. First, let us start with Euler’s result from 1734 which asserts that
Later, in 1740, the same Euler proved the following generalization:
Let us remark that the key ingredient in the classical proof of Euler’s formula is the following cotangent identity which is also due to Euler:
Expanding the quotient inside the sum sign as a geometric series and interchanging the order of summation, we obtain the following identity:
- Euler’s formula implies the following equality of subrings of :
- Thanks to the functional equation
one can deduce the values of at negative integers: for all . In particular, for all ; we call these values the trivial zeros of the function . Also, we have and .
Question: What can we say about when is odd?
Unfortunately, not too much is known, We cannot even find a closed formula for in terms of . This led to the following:
CONJECTURE. (Transcedence conjecture). The numbers are algebraically independent, that is, for each and each nonzero polynomial , we have
The 33rd Southeastern Analysis Meeting (SEAM’17) took place at the University of Tennessee, Knoxville, TN between 17-19 March. More than 60 mathematicians affiliated with universities from USA, Canada and Sweden participated at the meeting.
The conference was organized by Stefan Richter and Carl Sundberg of University of Tennessee-Knoxville and it included 3 contributed sections. Although, the contributed sections involved more operator theory, it featured many other fields such as PDEs, number theory, combinatorics and probability. My talk is part of the contributed sections, and it will take place on Saturday March 18, 2017, 10:20 a.m.-10:40 a.m. in Room 524. More details about the conference including the abstract of the talks are given below:
Integral inequalities are often a very important tool in mathematical analysis, number theory, partial differential equations, differential geometry, probability, statistics, etc.
The most basic integral inequality is given by the following: Given a continuous function (Riemann integrable is sufficient) , we have
For any real number , consider , . After computations, this us give us the celebrated Cauchy-Schwarz’s inequality,
Moreover, the Cauchy-Schwarz’s inequality can be obtained by integrating over the symmetrizing functions and together with the elementary inequality and Fubini’s theorem.
A generalization of Cauchy-Schwarz’s inequality was given by Rogers (1888) and Holder (1889),
where such that . This follows from Young’s inequality, for all applied for and and an integration afterwards. Here is the -norm and it is defined as . An self-extension of Holder’s inequality reads as follows:
where such that . This last inequality follows easily from the first Holder inequality applied for and with exponents . By an easy induction, the above inequality can be generalized as follows:
where and . Applications of Holder’s inequality are the following inequalities due to Minkovki:
All these inequalities presented above are discussed in details here. Some of these inequalities such as Cauchy-Schwarz, Holder and Minkowski (with their proofs) are also presented in the video below.
Recent and upcoming conferences, workshops, seminar talks, summer schools and other related mathematical activities, 2017
- Number Theory Seminar (November 7, University of California Irvine)
- The 2017 AMS Fall Western Sectional Meeting (November 4-5, University of California at Riverside, Riverside, CA)
Program Abstract book Presenters
- Analysis, Geometry, and Topology Seminar (October 31, University of Pittsburgh-Department of Mathematics, Pittsburgh, PA)
Talk: Clausen function and a dilogarithmic integral arising in quantum field theory (abstract, notes)
- “Nicolae Popescu” Number Theory Seminar (September 27, Simon Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania)
- The Undergraduate Mathematics Seminar (September 12, University of Pittsburgh-Department of Mathematics, Pittsburgh, PA)
- Algebra, Combinatorics and Geometry Graduate Student Research Seminar (September 7, University of Pittsburgh-Department of Mathematics, Pittsburgh, PA)
Talk: The Riemann zeta function. Analytic continuation and functional equation. (abstract, notes, link)
- The 2017 Mathematical Olympiad Summer Program (June 6-July 1, Carnegie Mellon University- Department of Mathematics & Mathematical Association of America)
Instructor: Topics in algebra and number theory (algebraic integers, advanced analytic techniques in number theory), Euclidian geometry and beyond (geometric & trigonometric inequalities), elementary real analysis (sequences and series of real numbers, applications of derivatives and integrals)
Seminar talk: Euler’s formula for Apery’s constant
- The 3rd Annual Graduate Student Conference in Algebra, Geometry and Topology (June 3-5, Temple University-Department of Mathematics, Philadelphia, PA)
Talk: Analytic aspects in the evaluation of some multiple zeta values and multiple Hurwitz zeta values (abstract, slides)
- 7th Upstate New York Number Theory Conference (May 6-7, Binghamton University (SUNY)-Department of Mathematics, Binghamton, NY)
- Algebra, Combinatorics and Geometry Seminar (April 6, University of Pittsburgh-Department of Mathematics, Pittsburgh, PA)
- The 33rd Southeastern Analysis Meeting-SEAM’17 (March 17-19, University of Tennessee Knoxville-Department of Mathematics)
- Analysis on Metric Spaces Conference (March 10-11, University of Pittsburgh-Department of Mathematics, Pittsburgh, PA)
- The Undergraduate Mathematics Seminar (February 22, University of Pittsburgh-Department of Mathematics, Pittsburgh, PA)
- Algebra, Combinatorics and Geometry Graduate Student Research Seminar (February 2, University of Pittsburgh-Department of Mathematics, Pittsburgh, PA)
- Algebra, Combinatorics and Geometry Graduate Student Research Seminar (January 26, University of Pittsburgh-Department of Mathematics, Pittsburgh, PA)
- The 2017 Joint Mathematics Meeting of the AMS & MAA (January 4-7, Marriott Marquis & Hyatt Regency, Atlanta, GA)
This Spring, I shall be the recitation instructor for the Advanced Calculus II (MATH 1540-graduate version). This course is designed to prepare the 1st and 2nd year graduate students for the Preliminary examination in real analysis. This exam is offered twice a year (May and August) by the department. My teaching page is the following: https://lupucezar.wordpress.com/teaching/.
- Homework (30% of your final grade!) will be assigned and collected weekly by Dr. DeBlois and you can find it on his webpage here. The homework will be graded and returned the following week. Late homework is accepted only with the instructor’s permission.
- The homework assigned last Spring’16 semester by Dr. Xu and me were the following:
- HW 1-LINEAR MAPS, LIMITS, CONTINUITY AND DIFFERENTIABILITY OF FUNCTIONS OF SEVERAL VARIABLES
- HW 2-THE INVERSE AND IMPLICIT FUNCTION THEOREMS; EXTREMUM PROBLEMS AND LAGRANGE MULTIPLIERS
- HW 3-INTEGRAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES
- HW4-SETS OF MEASURE ZERO AND LEBESGUE INTEGRATION
- I encourage you to solve as many problems as you can from this last year’s homework. All of them have the same caliber as prelim problems from previous years.
- Moreover, I shall assign four WORKSHEETS for this semester as follows:
- LINEAR MAPS, LIMITS, CONTINUITY AND DIFFERENTIABILITY OF FUNCTIONS OF SEVERAL VARIABLES.
- THE INVERSE AND IMPLICIT FUNCTION THEOREMS; EXTREMUM PROBLEMS AND RELATED TOPICS.
- INTEGRAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES.
- LEBESGUE INTEGRATION AND SETS OF MEASURE ZERO.
- My office hours are Tuesday (5-6 PM) & Wednesday (12-2 PM) in the POSVAR LAB and Tuesday (6-7 PM) in the MAC. My office is Thackeray 711.
- I invite you to join our Facebook group. The purpose of this group is to discuss problems from your homework or from previous preliminary exams. Moreover, I shall also post some notes from the recitation as a substitute for review sessions. Maybe, from time to time I shall upload some videos with solved prelim-type problems.
- Last but not least, you should also start working on problems from the famous Berkeley Problems in Mathematics by P. Ney de Souza and J-N. Silva. This book is a must for every graduate student! If you feel discouraged by the difficulty of the problems in the book, please remember to take into account Polya’s advice on how to approach a problem. Other resources can be found on my teaching page (look for Advanced Calculus-undergraduate & graduate 2014, 2015, 2016).
The 2017 Joint Mathematics Meeting of the American Mathematical Society and Mathematical Association of America is the largest mathematics meeting in the world. More than 6000 mathematicians and math enthusiasts met in the beautiful city of Atlanta, GA between January 4-7. The locations were Marriott Marquis and Hyatt Regency. This is the 100th annual winter meeting of MAA and the 123rd annual meeting of AMS. My talk is part of the AMS Contributed Paper Session on Number Theory, III , Friday January 6, 2017, 1:00 p.m.-4:55 p.m at the International 1, International Level, Marriott Marquis.
The 77th annual William Lowell Putnam Mathematical Competition took place on December 5th in 705 Thackeray Hall. Eight Pitt students had the mission of solving the challenging Putnam problem sets. As usual, there were two sessions of 6 problems each. The official Pitt Team members were Alex Mang, Andrew Tindall and Jack Hafer. Other participating students were: Matthew Gerstbrein, Terry Tan, Andrew Klang, Tianke Li and Haoming Yan. Below one can find this year’s Putnam problems. Congratulations to all participants!
Problem A1. Find the smallest positive integer such that for every polynomial with integer coefficients and for every integer the integer
(the -th derivative of at ) is divisible by
Problem A2. Given a positive integer , let be the largest integer such that
Problem A3. Suppose that is a function from to such that
for real . (As usual, means and ).
Problem A4. Consider a rectangular region, where and are integers such that The region is to be tiled using tiles of the two types shown:
(The dotted lines divide the tiles into squares.) The tiles may be rotated and reflected, as long as their sides are parallel to the sides of the rectangular region. They must all fit within the region, and they must cover it completely without overlapping.
What is the minimum number of tiles required to tile the region?
Problem A5. Suppose that is a finite group generated by the two elements and , where the order of is odd. Show that every element of can be written in the form
with and . (Here is the number of elements of .)
Problem A6. Find the smallest constant such that every real polynomial of degree that has a root in the interval ,
Problem B1. Let be the sequence such that and for ,
(as usual, the function is the natural logarithm). Show that the infinite series
converges and find its sum.
Problem B2. Define a positive integer to be squarish if either is itself a perfect square of the distance from to the nearest perfect square is a perfect square. For example, is squarish, because the nearest perfect square to is and is a perfect square. (of the positive integers between and , only and are not squarish.) For a positive integer , let be the number of squarish between and inclusive. Find positive constants and such that
or show that no such constants exist.
Problem B3. Suppose that is a finite set of points in the plane such that the area of the triangle is at most whenever and are in . Show that there exists a triangle of area that (together with its interior) covers the set .
Problem B4. Let be a matrix, with entries choasen indepedently at random. Every entry is chosen to be or , each with probability . Find the expected value of (as a function of ), where is the transpose of .
Problem B5. Find all functions from the interval to with the following property:
if and , then .
Problem B6. Evaluate