Research Interest

• Analysis (real functions, generalized derivatives, mean-value theorems, convexity, functional and probabilistic inequalities, operator inequalities, Sobolev spaces, fine properties of measurable functions, isoperimetric inequalities)
• Number Theory (special values of L-functions and multiple zeta functions, modular forms, special functions in p-adic setting, Mahler measure of (multivariable) polynomials)
• Special Functions ((generalized) Hypergeometric functions, (quantum & multiple) polylogarithms, orthogonal polynomials, summation formulas)
• Geometry (convex geometry, minimal surfaces, isoperimetric inequalities for curves, symmetric spaces, inequalities for the eigenvalues of Laplacian, Chern-Simmons theory of knot invariants)

Research Description

My main research interests are in the areas of analysis, number theory and special functions. Also, I am interested to their applications in geometry and physics. Most of my research is centered around \textcolor{blue}{\href{https://lupucezar.wordpress.com/2017/03/26/multiple-zeta-values-euler-zagier-sums-and-some-evaluations/}{special values of L-functions and multiple zeta functions}} (\textcolor{blue}{\href{https://www.youtube.com/watch?v=0VaKtD7lpZM}{Youtube video: An introduction to the Riemann zeta and multiple zeta functions and their special values}}) which play an important role at the interface of \textcolor{blue}{analysis, number theory, geometry, and physics} with applications ranging from periods of mixed Tate motives to evaluating Feynman integrals in quantum field theory. Moreover, it seems that these numbers appear in \textcolor{blue}{\href{https://www.quantamagazine.org/strange-numbers-found-in-particle-collisions-20161115/}{particle collisions}}. In my research, I employ methods from real \& complex analysis and special functions. Apart from that, I am also interested in connections with modular forms, special functions in $p$-adic setting, Mahler measure of multivariable polynomials and Chern-Simmons theory of knot invariants and connections between estimates of eigenvalue problems of the Laplacian in certain domains and multiple zeta values.

The objects I study are the Riemann zeta and multiple zeta functions (Euler-Riemann-Zagier zeta function) and their special values. They are defined by the following,
$$\displaystyle\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}, \operatorname{Re}(s)>1,$$
$$\displaystyle\zeta(s_{1}, s_{2}, \ldots, s_{r})=\sum_{1\leq n_{1}1, \sum_{j=1}^r\operatorname{Re}(s_{j})>r, $$
for all $j=1, 2, \ldots, r.$

In contrast with the Riemann zeta function, the multiple zeta functions are still a mystery! At this point, we know that $\zeta(s_{1}, \ldots, s_{r})$ has an analytic continuation to the whole $\mathbb{C}^r$ except certain hyperplanes containing single poles \cite{Zhao}. In this matter, Zhao \cite{Zhao} formulated the following open problems.\

\textbf{Conjecture 1.} \textit{Determine the complete set of trivial (respectively non-trivial) zeros of the multiple zeta functions.}
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\textbf{Conjecture 2.} \textit{Determine the functional equations (if any) of the multiple zeta functions which generalize the classical functional equation of the Riemann zeta function.}

\subsection{Riemann zeta values.} Initially, the Riemann (single) and double zeta values were defined by Euler long time ago starting with his sensational proof of $\zeta(2)=\frac{\pi^2}{6}$. This was followed by a generalization given by the same Euler in 1740,
$$\displaystyle\zeta(2k)=\sum_{n=1}^{\infty}\frac{1}{n^{2k}}=(-1)^{k+1}\cdot\frac{2^{2k-1}B_{2k}}{(2k)!}\cdot\pi^{2k},$$
The above formula has been a subject to numerous recent papers over the last decades. In a very recent paper \cite{Alladi-Defant} another proof was given using Parseval’s identity for the Fourier coefficients of $x^k$. In \cite{Lupu-Lupu} we provide three much more simplified proofs which avoids any use of Fourier series! On the other hand, when it comes to odd zeta values $\displaystyle\zeta(3), \zeta(5), \ldots, \zeta(2n+1)$ very little is known. The current status of the known results are that $\zeta(3)$ is irrational (Apery \cite{Apery}, 1978) and that there infinitely many irrational numbers among odd zeta values (Ball and Rivoal \cite{Ball-Rivoal}, 2002). Moreover, the odd zeta values can be represented as rational zeta series involving $\zeta(2n)$.
One of the main goals would be to prove the following\

\textbf{Transcendence conjecture.} \textit{The numbers $\displaystyle \pi, \zeta(3), \zeta(5), \ldots, \zeta(2n+1)$ are algebraically independent over $\mathbb{Q}$.}

\subsection{Multiple zeta values.} In 1992, M. E. Hoffman \cite{Hoffman} and independently D. Zagier \cite{Zagier} generalized Euler’s single and double zeta values to multiple zeta values which sometimes are called Euler-Zagier sums. For $k_{1}, k_{2}, \ldots, k_{r-1}\geq 1$ and $k_{r}\geq 2$, we define
$$\displaystyle\zeta(k_{1}, k_{2}, \ldots, k_{r})=\sum_{1\leq n_{1}<n_{2}<\ldots<n_{r}}\frac{1}{n_{1}^{k_{1}}n_{2}^{k_{2}}\ldots n_{r}^{k_{r}}},$$
where we fix the weight $k=k_{1}+k_{2}+\ldots+k_{r}$ and the depth (length) $r$. For example, there are $2^{13}$ such numbers in weight $15$, but they form a vector space over $\mathbb{Q}$ of dimension at most $28$. The main goal is to understand $\mathbb{Q}$-linear relations among multiple zeta values and this will turn out to be equivalent with understanding polynomial relations in $\pi$ and odd zeta values. Also, M. Kontsevich observed that all multiple zeta values are periods in the sense of the definition \cite{Kontsevich-Zagier}.
Let us denote by $\mathcal{Z}$ the $\mathbb{Q}$-vector space spanned by all multiple zeta values. It is not hard to see that $\mathcal{Z}$ has the structure of an algebra.
The first question to ask is what is a basis for $\mathcal{Z}$ as a $\mathbb{Q}$-vector space? Also if we consider the $\mathbb{Q}$-vector space $\mathcal{Z}_{k}$ of all multiple zeta values of weight $k$ we can ask what is its dimension over $\mathbb{Q}$? In other words, we have the following\

\textbf{Dimension conjecture (Zagier).} \textit{$\dim_{\mathbb{Q}}\mathcal{Z}{k}=d{k}$, where $d_{k}$ satisfies $d_{k}=d_{k-2}+d_{k-3}$, $d_{0}=1$, $d_{1}=0$, $d_{2}=1$.}
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A very important fact is that Zagier’s dimension conjecture implies the transcendence conjecture! In connection with Zagier’s conjecture there is another result which was conjectured by M. Hoffman \cite{Hoffman1} in 1997 and solved by F. Brown \cite{Brown} in 2012.
\bt
Every multiple zeta value of weight $k$ can be expressed as a $\mathbb{Q}$-linear combination of multiple zeta values of the same weight involving $2$’s and $3$’s.
\et
In other words, every multiple zeta value of weight $k$ is a $\mathbb{Q}$-linear combination of $\zeta(k_{1}, k_{2}, \ldots, k_{r})$, where $k_{i}\in {2, 3}$ and $\sum_{i=1}^{r}k_{i}=k$. The arguments used by Brown in proving Theorem 1.1 are purely motivic and it used motivic multiple zeta values. This theorem gives us the upper bound $\dim_{\mathbb{Q}}\mathcal{Z}{k}\leq d{k}$. In fact, Brown reduces the problem to show that the multiple zeta values involving Hoffman elements,
$\displaystyle H(a, b)=\zeta(\underbrace{2, 2, \ldots, 2}{\text{$a$}}, 3, \underbrace{2, 2, \ldots, 2}{\text{$b$}} )$ can be expressed as a $\mathbb{Q}$-linear combination of products $\pi^{2m}\zeta(2n+1)$, with $m+n=a+b+1$. In another direction, F. Brown also proved \cite{Brown} that all periods of mixed Tate motives over $\mathbb{Z}$ are $\mathbb{Q}[(2\pi i)^{\pm 1}]$-linear combinations of multiple zeta values.

Last but not least, regarding these linear combinations among MZV’s there is the following
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\textbf{Folklore conjecture.} \textit{Regularized double shuffle relations are enough to characterize all $\mathbb{Q}$-linear relations among multiple zeta values.}