Cezar Lupu





E-MAIL: lupucezar@gmail.com, cel47@pitt.edu

Blog Stats

  • 82,341 hits

An introduction to the Riemann zeta function, multiple zeta function, multiple zeta values (MZV or Euler-Zagier sums) and some of their special evaluations

  1. The Riemann zeta function and values is defined, for s\in\mathbb{C} as

       \displaystyle\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}, Res>1.

 Moreover, it has an integral representation in terms of Euler’s gamma function, \displaystyle\zeta(s)=\frac{1}{\Gamma(s)}\int_0^{\infty}\frac{x^{s-1}}{e^x-1}dx. It can actually be extended to a meromorphic function on the whole complex plane with a simple pole at s=1.

 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 \zeta(s) lie on the line Res=\frac{1}{2}) with tremendous consequences in number theory and beyond!

  Let us recall what happens when we evaluate \zeta(s) at integers. First, let us start with Euler’s result from 1734 which asserts that

\displaystyle \zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}.

Later, in 1740, the same Euler proved the following generalization:

\displaystyle \zeta(2k)=\sum_{n=1}^{\infty}\frac{1}{n^{2k}}=(-1)^{k+1}\frac{2^{2k-1}B_{2k}}{(2k)!}\cdot\pi^{2k},

where B_{k} are the Bernoulli numbers and they are given by the Taylor series expansion \displaystyle\frac{z}{e^z-1}=\sum_{k=0}^{\infty}\frac{B_{k}}{k!}\cdot z^k, |z|<2\pi. More about Euler and the zeta values can be found in this paper.

 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:

\displaystyle \pi\cot(\pi x)=\frac{1}{x}+\sum_{n\geq 1}\frac{2x}{x^2-n^2}.

Expanding the quotient inside the sum sign as a geometric series and interchanging the order of summation, we obtain the following identity:

\displaystyle \pi\cot(\pi x)=\frac{1}{x}-2\sum_{k\geq 1}\zeta(2k)x^{2k-1}.


  •  Euler’s formula implies the following equality of subrings of \mathbb{R}:

            \displaystyle\mathbb{Q}[\zeta(2), \zeta(4), \ldots]=\mathbb{Q}[\pi^2].

  •   Thanks to the functional equation

            \displaystyle \pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)=\pi^{-\frac{1-s}{2}}\Gamma\left(\frac{1-s}{2}\right)\zeta(1-s),

one can deduce the values of \zeta(s) at negative integers: \zeta(-k)=-\frac{B_{k+1}}{k+1} for all k\geq 1. In particular, \zeta(-2k)=0 for all k\geq 1; we call these values the trivial zeros of the function \zeta. Also, we have \zeta(0)=-\frac{1}{2} and \zeta(-1)=-\frac{1}{12}.

Question: What can we say about \zeta(s) when s is odd?

Unfortunately, not too much is known, We cannot even find a closed formula for \zeta(2n+1) in terms of \pi. This led to the following:

CONJECTURE. (Transcedence conjecture). The numbers \pi, \zeta(3), \zeta(5), \ldots are algebraically independent, that is, for each k\geq 0 and each nonzero polynomial P\in\mathbb{Z}[x_{0}, x_{1}, \ldots, x_{k}], we have

\displaystyle P(\pi, \zeta(3), \zeta(5), \ldots, \zeta(2k+1))\neq 0.  



SEAM’17-The 33rd SouthEastern Analysis Meeting, March 17-19, 2017

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:

     Plenary Speakers                 Participants                Abstract of Talks                Schedule

header-logo UTK

This slideshow requires JavaScript.

Some important integral inequalities in mathematical analysis and beyond

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) h:[a, b]\to\mathbb{R}, we have

\displaystyle\int_a^bh^2(x)dx\geq 0.

For any real number \lambda, consider h(x)=f(x)-\lambda g(x), h\in C([a, b]). After computations, this us give us the celebrated Cauchy-Schwarz’s inequality,


Moreover, the Cauchy-Schwarz’s inequality can be obtained by integrating over [a, b]\times [a, b] the symmetrizing functions u\mapsto f(x)g(y) and v\mapsto f(y)g(x) together with the elementary inequality 2uv\leq u^2+v^2 and Fubini’s theorem.

A generalization of Cauchy-Schwarz’s inequality was given by Rogers (1888) and Holder (1889),


where p, q>1 such that \frac{1}{p}+\frac{1}{q}=1. This follows from Young’s inequality, \displaystyle \frac{a^p}{p}+\frac{b^q}{q}\geq ab for all a, b\geq 0 applied for a=\frac{|f(x)|}{||f||_{p}} and b=\frac{|g(x)|}{||g||_{q}} and an integration afterwards. Here ||f||_{p} is the p-norm and it is defined as \displaystyle ||f||_{p}:=\left(\int_a^b|f(x)|^pdx\right)^{1/p} .  An self-extension of Holder’s inequality reads as follows:

\displaystyle\left(\int_a^b|f(x)g(x)|dx\right)^{1/r}\leq\left(\int_a^b|f(x)|^pdx\right)^{1/p}\left(\int_a^b|g(x)|^qdx\right)^{1/q}-\textbf{(extended Holder}),

where p, q, r>0 such that \displaystyle \frac{1}{p}+\frac{1}{q}=\frac{1}{r}. This last inequality follows easily from the first Holder inequality applied for x\mapsto |f(x)|^r and x\mapsto |g(x)|^r with exponents \displaystyle p_{1}=\frac{p}{r}, q_{1}=\frac{q}{r}. By an easy induction, the above inequality can be generalized as follows:

\displaystyle\left(\int_a^b |f_{1}(x)f_{2}(x)\ldots f_{k}(x)|^rdx\right)^{1/r}\leq \left(\int_a^b|f_{1}(x)|^{p_{1}}\right)^{1/p_{1}}\left(\int_a^b|f_{2}(x)|^{p_{2}}\right)^{1/p_{2}}\ldots \left(\int_a^b|f_{k}(x)|^{p_{k}}\right)^{1/p_{k}}-\textbf{(generalized Holder}),

where \displaystyle p_{i}, r\geq 1 and \displaystyle\sum_{i=1}^k\frac{1}{p_{i}}=\frac{1}{r}. Applications of Holder’s inequality are the following inequalities due to Minkovki:

\displaystyle\left(\int_a^b|f(x)+g(x)|^pdx\right)^{1/p}\leq \left(\int_a^b|f(x)|^pdx\right)^{1/p}+\left(\int_a^b|g(x)|^pdx\right)^{1/p}-\textbf{(Minkowski}),


\displaystyle\left(\int_a^b\left|\int_c^d f(x, y)dy\right|^pdx\right)^{1/p}\leq \int_c^d\left(\int_a^b|f(x, y)|^pdx\right)^{1/p}dy-\textbf{(generalized Minkowski})

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

        Talk: Analytic aspects in the evaluation of multiple zeta values and multiple Hurwitz zeta values (abstract)

Program                                              Abstract book                                    Presenters

Talk: How to discover and train talented undergraduate students for the Putnam competition (abstract, slides)

           Talk: Clausen function and a dilogarithmic integral arising in quantum field theory (abstract, notes)

          Talk: Analytic aspects in the evaluation of the evaluation of the multiple zeta values and multiple Hurwitz zeta values (abstract)

           Talk: An invitation to the Putnam Seminar (abstract, poster)

           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 \zeta(3)

         Detailed Schedule                                 Talk Abstracts

          Talk: Analytic aspects in the evaluation of some multiple zeta values and multiple Hurwitz zeta values (abstract, slides)

Detailed Schedule                                 Talk Abstracts

         Talk: Analytic aspects in the evaluation of some multiple zeta values and multiple Hurwitz zeta values (abstract, slides)

           Talk: Multiple zeta values and multiple Hurwitz zeta values. Analytic and combinatorial aspects (abstract, notes)

          Schedule                                  Abstract of Talks                                       Participants

           Talk: Analytic aspects in the evaluation of some multiple zeta values (abstract, slides)

          Talk: How to discover and train talented undergraduate students for the Putnam competition (abstract, slides)

          Talk: The Riemann and Hurwitz zeta functions, Apery’s constant and new rational series representations involving \zeta(2k) (abstract, slides, preprint)

           Talk: A simple proof of Euler’s formula for \zeta(2k) and some rational series representations involving \zeta(2k) (abstract, notes, paper)

 Program                                  Abstract book                                       Presenters

            Talk: The Riemann zeta function for integer values and evaluation of some multiple zeta values (abstract, slides, video)

Information on the Advanced Calculus II-MATH 1540 (graduate), Spring 2017

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:
  •  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:
  • 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 AMS and MAA

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.

    Abstract book                                     Program                                      Presenters

The 2016 William Lowell Putnam Competition Exam at the University of Pittsburgh

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 j such that for every polynomial p(x) with integer coefficients and for every integer k, the integer

\displaystyle p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}

(the j-th derivative of p(x) at k) is divisible by 2016.

Problem A2. Given a positive integer n, let M(n) be the largest integer m such that

\displaystyle \binom{m}{n-1}>\binom{m-1}{n}.



Problem A3. Suppose that f is a function from \mathbb{R} to \mathbb{R} such that

\displaystyle f(x)+f\left(1-\frac{1}{x}\right)=\arctan x

for real x\neq 0. (As usual, y=\arctan x means -\frac{\pi}{2}<y<\frac{\pi}{2} and \tan y=x).


\displaystyle \int_0^1f(x)dx.

Problem A4. Consider a (2m-1)\times(2n-1) rectangular region, where m and n are integers such that m,n\ge 4. The region is to be tiled using tiles of the two types shown:


(The dotted lines divide the tiles into 1\times 1 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 G is a finite group generated by the two elements g and h, where the order of g is odd. Show that every element of G can be written in the form

\displaystyle g^{m_{1}}h^{n_{1}} g^{m_{2}}h^{n_{2}}\ldots g^{m_{r}}h^{n_{r}}

with 1\leq r\leq |G| and m_{1}, n_{1}, m_{2}, n_{2}, \ldots m_{r}, n_{r}\in \{1, -1\}. (Here |G| is the number of elements of G.)

Problem A6. Find the smallest constant C such that every real polynomial P(x) of degree 3 that has a root in the interval [0, 1],

\displaystyle \int_0^1|P(x)|dx\leq C\cdot\max_{x\in [0,1]}|P(x)|.


Problem B1. Let x_{0}, x_{1}, \ldots, x_{n}, \ldots be the sequence such that x_{0}=1 and for n\geq 0,

\displaystyle x_{n+1}=ln(e^{x_{n}}-x_{n}).

(as usual, the function ln is the natural logarithm). Show that the infinite series

\displaystyle x_{0}+x_{1}+x_{2}+\ldots

converges and find its sum.

Problem B2. Define a positive integer n to be squarish if either n is itself a perfect square of the distance from n to the nearest perfect square is a perfect square. For example, 2016 is squarish, because the nearest perfect square to 2016 is 45^2=2025 and 2025-2016=9 is a perfect square. (of the positive integers between 1 and 10, only 6 and 7 are not squarish.) For a positive integer N, let S(N) be the number of squarish between 1 and N inclusive. Find positive constants \alpha and \beta such that


or show that no such constants exist.

Problem B3. Suppose that S is a finite set of points in the plane such that the area of the triangle \Delta ABC is at most 1 whenever A, B and C are in S. Show that there exists a triangle of area 4 that (together with its interior) covers the set S.

Problem B4. Let A be a 2n\times 2n matrix, with entries choasen indepedently at random. Every entry is chosen to be 0 or 1, each with probability \frac{1}{2}. Find the expected value of \det(A-A^{t}) (as a function of n), where A^{t} is the transpose of A.

Problem B5. Find all functions f from the interval (1, \infty) to (1, \infty) with the following property:

if x, y\in (1, \infty) and x^2\leq y\leq x^3, then (f(x))^2\leq f(y)\leq (f(x)^3.

Problem B6. Evaluate

\displaystyle \sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\cdot\sum_{n=0}^{\infty}\frac{1}{k2^n+1}.


This slideshow requires JavaScript.