Large oscillations of the argument of the Riemann zeta‐function
Bulletin of the London Mathematical Society, Volume 53, Issue 6, Page 1776-1785, December 2021., 2021 Abstract Let S(t) denote the argument of the Riemann zeta‐function, defined as S(t)=1πImlogζ(1/2+it).Assuming the Riemann hypothesis, we prove that S(t)=Ω±logtlogloglogtloglogt.This improves the classical Ω‐results of Montgomery (Theorem 2; Comment. Math. Helv. 52 (1977) 511–518) and matches with the Ω‐result obtained by Bondarenko and Seip (Theorem 2;
Andrés Chirre, Kamalakshya Mahatab
wiley +1 more source
Linear correlations of multiplicative functions
Proceedings of the London Mathematical Society, Volume 121, Issue 2, Page 372-425, August 2020., 2020 Abstract We prove a Green–Tao type theorem for multiplicative functions.
Lilian Matthiesen
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Counting Fixed Points, Two-Cycles, and Collisions of the Discrete Exponential Function using p-adic Methods [PDF]
, 2011 Brizolis asked for which primes p greater than 3 does there exist a pair (g, h) such that h is a fixed point of the discrete exponential map with base g, or equivalently h is a fixed point of the discrete logarithm with base g.
Bourbaki +9 more
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The shifted convolution of generalized divisor functions [PDF]
, 2016 We prove an asymptotic formula for the shifted convolution of the divisor functions $d_k(n)$ and $d(n)$ with $k \geq 4$, which is uniform in the shift parameter and which has a power-saving error term, improving results obtained previously by Fouvry and ...
Topacogullari, Berke
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The least common multiple of a sequence of products of linear polynomials
, 2011 Let $f(x)$ be the product of several linear polynomials with integer coefficients. In this paper, we obtain the estimate: $\log {\rm lcm}(f(1), ..., f(n))\sim An$ as $n\rightarrow\infty $, where $A$ is a constant depending on $f$.Comment: To appear in ...
A. Selberg +12 more
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Inertia, positive definiteness and $\ell_p$ norm of GCD and LCM matrices and their unitary analogs [PDF]
, 2017 Let $S=\{x_1,x_2,\dots,x_n\}$ be a set of distinct positive integers, and let $f$ be an arithmetical function. The GCD matrix $(S)_f$ on $S$ associated with $f$ is defined as the $n\times n$ matrix having $f$ evaluated at the greatest common divisor of ...
Haukkanen, Pentti, Tóth, László
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On Diamond's $L^1$ criterion for asymptotic density of Beurling generalized integers [PDF]
, 2019 We give a short proof of the $L^{1}$ criterion for Beurling generalized integers to have a positive asymptotic density. We actually prove the existence of density under a weaker hypothesis.
Debruyne, Gregory, Vindas, Jasson
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Asymptotic behavior of the least common multiple of consecutive arithmetic progression terms
, 2013 Let $l$ and $m$ be two integers with $l>m\ge 0$, and let $a$ and $b$ be integers with $a\ge 1$ and $a+b\ge 1$.
B. Farhi +10 more
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On the $k$-free values of the polynomial $xy^k+C$ [PDF]
, 2015 Consider the polynomial $f(x,y)=xy^k+C$ for $k\geq 2$ and any nonzero integer constant $C$. We derive an asymptotic formula for the $k$-free values of $f(x,y)$ when $x, y\leq H$.
Lapkova, Kostadinka
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Arithmetic functions at consecutive shifted primes [PDF]
, 2014 For each of the functions $f \in \{\phi, \sigma, \omega, \tau\}$ and every natural number $k$, we show that there are infinitely many solutions to the inequalities $f(p_n-1) < f(p_{n+1}-1) < \dots < f(p_{n+k}-1)$, and similarly for $f(p_n-1) > f(p_{n+1 ...
Pollack, Paul, Thompson, Lola
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