arXiv subjects
Publications and source records attributed to Marc Munsch.
We obtain new lower bounds on the number of smooth squarefree integers up to $x$ in residue classes modulo a prime $p$, relatively large compared to $x$, which in some ranges of $p$ and $x$ improve that of A. Balog and C. Pomerance (1992). We also estimate the smallest squarefull number in almost all residue classes modulo a prime $p$.
In recent years a variant of the resonance method was developed which allowed to obtain improved $Ω$-results for the Riemann zeta function along vertical lines in the critical strip. In the present paper we show how this method can be adapted to prove the existence of large values of $|L(σ, χ)|$ in the range $σ\in (1/2,1]$, and to estimate the proportion of characters for which $|L(σ, χ)|$ is of such a large order. More precisely, for every fixed $σ\in (1/2,1)$ we show that for all sufficiently large $q$ there is a non-principal character $χ$ (mod $q$) such that $\log |L(σ,χ)| \geq C(σ) (\log q)^{1-σ} (\log \log q)^{-σ}$. In the case $σ=1$ we show that there is a non-principal character $χ$ (mod $q$) for which $|L(1,χ)| \geq e^γ\left(\log_2 q + \log_3 q - C \right)$. In both cases, our results essentially match the prediction for the actual order of such extreme values, based on probabilistic models.
For any polynomial $P(x)\in\mathbb{Z}[x],$ we study arithmetic dynamical systems generated by $\displaystyle{F_P(n)=\prod_{k\le n}}P(n)(\text{mod}\ p),$ $n\ge 1.$ We apply this to improve the lower bound on the number of distinct quadratic fields of the form $\mathbb{Q}(\sqrt{F_P(n)})$ in short intervals $M\le n\le M+H$ previously due to Cilleruelo, Luca, Quirós and Shparlinski. As a second application, we estimate the average number of missing values of $F_P(n)(\text{mod}\ p)$ for special families of polynomials, generalizing previous work of Banks, Garaev, Luca, Schinzel, Shparlinski and others.
Let $χ$ range over the $(p-1)/2$ even Dirichlet characters modulo a prime $p$ and denote by $θ(x,χ)$ the associated theta series. The asymptotic behaviour of the second and fourth moments proved by Louboutin and the author implies that there exists at least $ \gg p/ \log p$ characters such that the associated theta function does not vanish at a fixed point. Constructing a suitable mollifier, we improve this result and show that there exists at least $ \gg p/ \sqrt{\log p}$ characters such that $θ(x,χ) \neq 0$ for any $x>0$. We give similar results for odd Dirichlet characters mod $p$.
A. Booker and C. Pomerance (2017) have shown that any residue class modulo a prime $p\ge 11$ can be represented by a positive $p$-smooth square-free integer $s = p^{O(\log p)}$ with all prime factors up to $p$ and conjectured that in fact one can find such $s$ with $s = p^{O(1)}$. Using bounds on double Kloosterman sums due to M. Z. Garaev (2010) we prove this conjecture in a stronger form $s \le p^{3/2 + o(1)}$ and also consider more general versions of this question replacing $p$-smoothness of $s$ by the stronger condition of $p^α$-smoothness. Using bounds on multiplicative character sums and a sieve method, we also show that we can represent all residue classes by a positive square-free integer $s\le p^{2+o(1)}$ which is $p^{1/(4e^{ /2})+o(1)}$-smooth. Additionally, we obtain stronger results for almost all primes $p$.
In the present note, we prove new lower bounds on large values of character sums $Δ(x,q):=\max_{χ\neq χ_0} \vert \sum_{n\leq x} χ(n)\vert$ in certain ranges of $x$. Employing an implementation of the resonance method developed in a work involving the author in order to exhibit large values of $L$- functions, we improve some results of Hough in the range $\log x = o(\sqrt{\log q})$. Our results are expressed using the counting function of $y$- friable integers less than $x$ where we improve the level of smoothness $y$ for short intervals.
In the present paper, we study large values of Dirichlet $L$- functions inside the critical strip. For every $1/2<σ<1$, we show that for $q$ sufficiently large, there exists a non-principal character $χ$ modulo $q$ and a constant $c(σ)>0$ such that $\log \vert L(σ,χ)\vert \gg c(σ)(\log q)^{1-σ}(\log\log q)^{-σ}$. This matches the believed prediction for these values which was previously known only for the Riemann zeta function since Montgomery, or conditionally on GRH for quadratic $L$- functions due to Lamzouri. In a recent work involving the author, a new implementation of the resonance method was presented in order to exhibit large values of the Riemann zeta function on the line $\Re(s)=1$. We show how to adapt the argument to our setting.
We prove that there are arbitrarily large values of $t$ such that $|ζ(1+it)| \geq e^γ (\log_2 t + \log_3 t) + \mathcal{O}(1)$. This essentially matches the prediction for the optimal lower bound in a conjecture of Granville and Soundararajan. Our proof uses a new variant of the "long resonator" method. While earlier implementations of this method crucially relied on a "sparsification" technique to control the mean-square of the resonator function, in the present paper we exploit certain self-similarity properties of a specially designed resonator function.
In this note, we give an upper bound for the number of elements from the interval $[1,p^{1/4e^{1/2}+ε}]$ necessary to generate the finite field $\mathbb{F}_{p}$ with $p$ an odd prime.
We use character sum estimates to give a bound on the least square-full primitive root modulo a prime. Specifically, we show that there is a square-full primitive root mod $p$ less than $p^{2/3 + 3/(4 \sqrt{e})+ ε}$, and we give some conditional bounds.
Assuming the Riemann Hypothesis, Soundararajan showed that $\displaystyle{\int_{0}^{T} \vert ζ(1/2 + it)\vert^{2k} \ll T(\log T)^{k^2 + ε}}$ . His method was used by Chandee to obtain upper bounds for shifted moments of the Riemann Zeta function. Building on these ideas, we obtain, conditionally, upper bounds for shifted moments of Dirichlet $L$- functions which allow us to derive upper bounds for moments of theta functions.
We obtain optimal lower bounds for moments of theta functions. On the other hand, we also get new upper bounds on individual theta values and moments of theta functions on average over primes. The upper bounds are based on bounds of character sums and in particular on a modification of some recent results of M. Z. Garaev.
We prove that the sequence $n!\,(\bmod\,p)$ occupies at least $\sqrt{\frac{3}{2}N}$ residue classes in the short interval $H\le n \le H+N$ and $N\gg p^{\frac{1}{4}}$ improving previously known trivial bound $\sqrt{N}.$ In the other direction, we estimate the average number of residue classes missed by the sequence $n!\,(\bmod\,p)$ for $p\le x.$
We obtain new results about the representation of almost all residues modulo a prime $p$ by a product of a small integer and also an element of small multiplicative subgroup of $({\mathbb Z}/p{\mathbb Z})^*$. These results are based on some ideas, and their modifications, of a recent work of J. Cilleruelo and M. Z. Garaev (2014),