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Contributed talks
11th Atelier PARI/GP
(April 16-17th)
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Diophantine approximation with prime variables
Alessandro Gambini
(Università di Parma)
This talk concerns a Diophantine approximation problem with 3 prime variables: we will prove that the inequality
$$
|\lambda_1p_1+\lambda_2p_2+\lambda_3p_3^k-\omega|\le (\max(p_1,p_2,p_3^k))^{\psi(k)+\varepsilon}
$$
where
$$ \psi (k) =
\begin{cases}
(3-2k)/6k, & \text{if $1 < k \le 6/5$} \\
1/12, & \text{if $6/5 < k \le 2$}\\
(3-k)/6k & \text{if $2 < k < 3$}\\
1/24 & \text{if $k=3$}
\end{cases}
$$
has infinitely many solutions in prime variables $p_1$, $p_2$ and $p_3$ for any given real number $\omega$, with $\lambda_1$,
$\lambda_2$ and $\lambda_3$ non-zero real numbers, not all of the same sign and such that $\lambda_1/\lambda_2$ is not rational, and $1 \le k \le 3$ real.
The proof uses a variant of the circle method technique introduced by Davenport and Heilbronn and the Harman technique. This is a joint work with Alessandro Languasco
and Alessandro Zaccagnini.
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Waring's theorem for binary powers
Carlo Sanna
(Università di Torino)
A natural number is a binary $k^{\text th}$ power if its binary representation consists of $k$ consecutive identical blocks. We prove an analogue
of Waring's theorem for sums of binary $k^{\text th}$ powers. More precisely, we show that for each integer $k \ge 2$, there exists a positive integer $W(k)$
such that every sufficiently large multiple of $E_k :=\operatorname{gcd}(2^k - 1, k)$ is the sum of at most $W(k)$ binary $k^{\text th}$ powers.
(The hypothesis of being a multiple of $E_k$ cannot be omitted, since we show that the gcd of the binary $k^{\text th}$ powers is $E_k$). Also,
we explain how our results can be extended to arbitrary integer bases $b > 2$. This is a joint work with Daniel M. Kane and Jeffrey Shallit: https://arxiv.org/abs/1801.04483
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The exact measures of the Sierpinski d-dimensional tetrahedron in connection with a Diophantine nonlinear system
Fabio Caldarola
(Università della Calabria)
The Sierpinski d-dimensional tetrahedron $\Delta^d$ is the generalization of the most known Sierpinski gasket which appears in many fields of mathematics.
Considering the sequences of polytopes $\Delta^d_{n}$ that generate $\Delta^d$, we find closed formulas for the sum $v^{d,k}_n$ of the measures of the k-dimensional
elements of $\Delta^d_n$, deducing the behavior of the sequences $v^{d,k}_n$. It becomes quite clear that traditional analysis does not have the adequate
language and notations to go further, in an easy and manageable way, in the study of the previous sequences and their limit values; contrariwise, by adopting the
new computational system for infinities and innitesimals developed by Y.D. Sergeyev, we achieve precise evaluations for every
$k$-dimensional measure related to each $\Delta^d$, obtaining a set $W$ of values expressed in the new system, which leads us to a
Diophantine problem in terms of classical number theory. To solve it, we work with traditional tools from algebra and mathematical analysis.
In particular, we define two kinds of equivalence relations on $W$ and we get a detailed description of the partition of
various of its subsets together with the exact composition of the corresponding classes of equivalence. Finally, we also show as the unique Sierpinski
tetrahedron for each dimension d, is replaced, if we adopt Sergeyev's framework, by a whole family of infinitely many Sierpinski d-dimensional tetrahedrons.
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Correlations of Ramanujan expansions
Giovanni Coppola
(Università di Salerno)
After the series of papers, coauthored, "Finite Ramanujan expansions and shifted convolution sums of arithmetical functions" and two papers on the arxiv,
about the same studies, we keep up to date our investigations about the "Reef"
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On the Baez-Duarte criterion about the Riemann hypothesis
Goubi Mouloud
(Université Mouloud Maameri de Tizi Ouzou))
This work is a tentative to prove that there exist a Beurling function which satisfies the Baez-Duarte Criterion about the Riemann hypothesis.
Furthermore we connect the distance of the indicatrice function to space generated by any Beurling function in means of the well known digamma function.
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The minimal euclidean algorithm on the Gaussian integers
Hester Graves
(Institute for Defense Analyses)
In 1949, Motzkin introduced a new tool to study Euclidean domains. As a side effect, it described the minimal Euclidean algorithm for agiven domain.
He showed that the minimal Euclidean algorithm onthe integers is $\phi_Z (x) = \log_2 |x|$. Lenstra gave an elegant proof of the minimal Euclidean algorithm on Z[i].
This talk will give anew, elementary proof and an alternate description of said function that allows for fast computation.
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Explicit formula for the average of Goldbach and prime tuples representations
Marco Cantarini
(Università di Parma)
We present the explicit formula, similar to the classical explicit formula for $\psi\left(x\right)$, for the average of the Goldbach representations
$$r_{G}\left(n\right)=\underset{{\scriptstyle m_{1}+m_{2}=n}}{\sum_{m_{1},m_{2}\leq n}}\Lambda\left(m_{1}\right)\Lambda\left(m_{2}\right),\,n\in\mathbb{N}$$
and for the prime tuples representation
$$r_{PT}\left(N,h\right)=\sum_{n=0}^{N}\Lambda\left(n\right)\Lambda\left(n+h\right),\,h,N\in\mathbb{N}$$
where $\Lambda\left(n\right)$ is the Von Mangoldt function.
In the case of $r_{G}\left(n\right)$ we show also a truncated version of the formula.
Furthermore we analyze the possibility to extend the technique to the Cesàro average of $r_{G}\left(n\right)$, that is,
$$\frac{1}{\Gamma\left(k+1\right)}\sum_{n\leq N}r_{G}\left(n\right)\left(N-n\right)^{k},\,k \in \mathbb{R},\,k>0.$$
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Mean values of multiplicative functions over the function fields
Oleksiy Klurman
(KTH, Royal Institute of Technology, Stockholm))
Understanding mean values and correlations of multiplicative functions over number fields plays key role in analytic number theory.
Motivated by the recent work of Granville, Harper and Soundararajan we discuss mean values of multiplicative functions over
the function fields \mathbb{F}_q[x]. In particular, we prove stronger function field analogs of several classical results due to Wirsing, Halasz, Hall,
Tenenbaum explaining some surprising features that are not present in
the number field setting. Our main result describes spectrum of multiplicative functions over the function fields.
This is based on a joint work with K. Soundararajan and C. Pohoata.
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Correlation of Multiplicative Functions
Pranendu Darba
(Institute of Mathematical Sciences, Chennai)
In this talk, after recalling some earlier results I will present some recent progress on asymptotic formula for the following correlation:
$$M_{x}(g_{1}, g_{2}, g_3)=\frac{1}{x}\sum_{n\le x}g_{1}(F_1(n))g_{2}(F_2(n))g_{3} (F_3(n)),$$ where
$F_1(x), F_2(x)$ and $F_3(x)$ are polynomials with integer coefficients and $g_1,g_2,g_3$ are multiplicative functions with modulus less than or equal to $1.$
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Some inequalities associated with the multiple Gamma functions and the Zeta functions
Praveen Agarwal
(Anand International College of Engineering, Jaipur)
In the middle of $1980$, the multiple Gamma functions $\Gamma_n$ have been revived according to the study of determinants of Laplacians and, recently,
found an applicationin giving closed-form evaluations of a class of series involving zeta functions. A class of mathematical constants are naturally
connected with the theory of multipleGamma functions. Here we introduce certain interesting inequalities associated with the multiple
Gamma functions, their related constants, and the zeta functions.
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Multiple zeta values at the non-positive integers
Sadaoui Boualem
(Khemis Miliana Univezrsity)
In this talk, we relate the special value at a non positive integer $\underline{\textbf{s}}=(s_{1}, ..., s_{n})= -\underline{\textbf{N}}= (-N_{1},..., -N_{n})$
obtained by meromorphic continuation of the multiple zeta function
\begin{equation*} Z(\underline{s})= \sum_{\underline{m} \in \mathbb{N}^{*n}}{\prod_{i=1}^{n}{\frac{1}{(m_{1}+\dots +m_{i})^{s_{i}}}}}\end{equation*}
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Non Wieferich primes and Euclidean algorithm
Subramani Muthukrishnan
(Harish Chandra Research Institute, Allahabad)
We prove results on non Wieferich primes in number fields and also it's connection to Euclidean number fields.
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