arXiv subjects

Nicolas Robles

Publications and source records attributed to Nicolas Robles.

29 records · Page 2Linked to original sources

arXiv2019

More than five-twelfths of the zeros of $ζ$ are on the critical line

The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form $(μ\star Λ_1^{\star k_1} \star Λ_2^{\star k_2} \star \cdots \star Λ_d^{\star k_d})$ is computed unconditionally by means of the autocorrelation of ratios of $ζ$ techniques from Conrey, Farmer, Keating, Rubinstein and Snaith (2005), Conrey, Farmer and Zirnbauer (2008) as well as Conrey and Snaith (2007). This in turn allows us to describe the combinatorial process behind the mollification of \[ ζ(s) + λ_1 \frac{ζ'(s)}{\log T} + λ_2 \frac{ζ''(s)}{\log^2 T} + \cdots + λ_d \frac{ζ^{(d)}(s)}{\log^d T}, \] where $ζ^{(k)}$ stands for the $k$th derivative of the Riemann zeta-function and $\{λ_k\}_{k=1}^d$ are real numbers. Improving on recent results on long mollifiers and sums of Kloosterman sums due to Pratt and Robles (2017), as an application, we increase the current lower bound of critical zeros of the Riemann zeta-function to slightly over five-twelfths.

math.NT↗

arXiv2018

Breaking the $\frac{1}{2}$-barrier for the twisted second moment of Dirichlet $L$-functions

We study the second moment of Dirichlet $L$-functions to a large prime modulus $q$ twisted by the square of an arbitrary Dirichlet polynomial. We break the $\frac{1}{2}$-barrier in this problem, and obtain an asymptotic formula provided that the length of the Dirichlet polynomial is less than $q^{51/101} = q^{1/2 +1/202}$. As an application, we obtain an upper bound of the correct order of magnitude for the third moment of Dirichlet $L$-functions. We give further results when the coefficients of the Dirichlet polynomial are more specialized.

math.NT↗

arXiv2018

Random permutations with logarithmic cycle weights

We consider random permutations on $\Sn$ with logarithmic growing cycles weights and study asymptotic behavior as the length $n$ tends to infinity. We show that the cycle count process converges to a vector of independent Poisson variables and also compute the total variation distance between both processes. Next, we prove a central limit theorem for the total number of cycles. Furthermore we establish a shape theorem and a functional central limit theorem for the Young diagrams associated to random permutations under this measure. We prove these results using tools from complex analysis and combinatorics. In particular we have to apply the method of singularity analysis to generating functions of the form $\exp\left( (-\log(1-z))^{k+1} \right)$ with $k\geq 1$, which have not yet been studied in the literature.

math.PR↗

arXiv2018

Perturbed moments and a longer mollifier for critical zeros of $ζ$

Let $A(s)$ be a general Dirichlet polynomial and $Φ$ be a smooth function supported in $[1,2]$ with mild bounds on its derivatives. New main terms for the integral $I(α,β)=\int_{\mathbb{R}} ζ(\frac{1}{2}+α+it)ζ(\frac{1}{2}+β+it)|A(\frac{1}{2}+it)|^2 Φ(\frac{t}{T})dt$ are given. For the error term, we show that the length of the Feng mollifier can be increased from $θ< \frac{17}{33}$ to $θ< \frac{6}{11}$ by decomposing the error into Type I and Type II sums and then studying the resulting sums of Kloosterman sums. As an application, we slightly increase the proportion of zeros of $ζ(s)$ on the critical line.

math.NT↗

arXiv2018

Polynomial partition asymptotics

Let $f \in \mathbb{Z}[y]$ be a polynomial such that $f(\mathbb{N}) \subseteq \mathbb{N}$, and let $p_{\mathcal{A}_{f}}(n)$ denote number of partitions of $n$ whose parts lie in the set $\mathcal{A}_f:=\{f(n):n \in \mathbb{N}\}$. Under hypotheses on the roots of $f-f(0)$, we use the Hardy--Littlewood circle method, a polylogarithm identity, and the Matsumoto--Weng zeta function to derive asymptotic formulae for $p_{\mathcal{A}_f}(n)$ as $n$ tends to infinity. This generalises asymptotic formulae for the number of partitions into perfect $d$th powers, established by Vaughan for $d=2$, and Gafni for the case $d \geq 2$, in 2015 and 2016 respectively.

math.NT↗

arXiv2017

On mean values of mollifiers and L-functions associated to primitive cusp forms

We study the second moment of the L-function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers. This family is a natural extension of the mollifers considered by Conrey and by Bui, Conrey and Young. As an application, we improve the current lower bound on critical zeros of holomorphic primitive cusp forms.

math.NT↗

arXiv2016

On a mollifier of the perturbed Riemann zeta-function

The mollification $ζ(s) + ζ'(s)$ put forward by Feng is computed by analytic methods coming from the techniques of the ratios conjectures of $L$-functions. The current situation regarding the percentage of non-trivial zeros of the Riemann zeta-function on the critical line is then clarified.

math.NT↗

arXiv2015

Moments of averages of generalized Ramanujan sums

Let $β$ be a positive integer. A generalization of the Ramanujan sum due to Cohen is given by \begin{align} c_{q,β}(n) := \sum\limits_{{{(h,{q^β})}_β} = 1} {e^{2πinh/{q^β}}}, \nonumber \end{align} where $h$ ranges over the the non-negative integers less than $q^β$ such that $h$ and $q^β$ have no common $β$-th power divisors other than $1$. The distribution of the average value of the Ramanujan sum is a subject of extensive research. In this paper, we study the distribution of the average value of $c_{q,β}(n)$ by computing the $k$-th moments of the average value of $c_{q,β}(n)$. In particular we have provided the first and second moments with improved error terms. We give more accurate results for the main terms than our predecessors. We also provide an asymptotic result for an extension of a divisor problem and for an extension of Ramanujan's formula.

math.NT↗

arXiv2015

Koshliakov kernel and identities involving the Riemann zeta function

Some integral identities involving the Riemann zeta function and functions reciprocal in a kernel involving the Bessel functions $J_{z}(x), Y_{z}(x)$ and $K_{z}(x)$ are studied. Interesting special cases of these identities are derived, one of which is connected to a well-known transformation due to Ramanujan, and Guinand.

math.NT↗

arXiv2015

Explicit formulas of a generalized Ramanujan sum

Explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized.

math.NT↗

arXiv2013

Zeta functions on tori using contour integration

A new, seemingly useful presentation of zeta functions on complex tori is derived by using contour integration. It is shown to agree with the one obtained by using the Chowla-Selberg series formula, for which an alternative proof is thereby given. In addition, a new proof of the functional determinant on the torus results, which does not use the Kronecker first limit formula nor the functional equation of the non-holomorphic Eisenstein series. As a bonus, several identities involving the Dedekind eta function are obtained as well.

math-ph↗