Ein Beitrag zur analytischen Zahlentheorie

Mertens,

doi:10.1515/9783112389843-002

Cited by 52 publications

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“…≤ log(p m ) + 3 by Mertens's first theorem [4], and because the sum of the second series is less than 0.76. By [6], we have p m < m log m + m log log m if m ≥ 6; thus the rough bound p m ≤ m 2 holds.…”

Section: Preparations For the Proof Of Proposition 12mentioning

confidence: 90%

Upper bounds on the heights of polynomials and rational fractions from their values

Kieffer

2021

Preprint

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Let F be a univariate polynomial or rational fraction of degree d defined over a number field. We give bounds from above on the absolute logarithmic Weil height of F in terms of the heights of its values at small integers: we review well-known bounds obtained from interpolation algorithms given values at d+1 (resp. 2d + 1) points, and obtain tighter results when considering a larger number of evaluation points.

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Section: Preparations For the Proof Of Proposition 12mentioning

confidence: 90%

Upper bounds on the heights of polynomials and rational fractions from their values

Kieffer

2021

Preprint

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“…But p∈P √ n 1 p ∼ loglog(n), by Euler's result on sum of reciprocal of primes (later more rigorously obtained by Mertens in [15]). This identity gives us our result for k = 2.…”

Section: Directly Impliesmentioning

confidence: 92%

Optimal Presentations of Mathematical Objects

Dogra¹

2018

Preprint

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We discuss the optimal presentations of mathematical objects under well defined symbol libraries. We shall examine what light our chosen symbol libraries and syntax shed upon the objects they represent. A major part of this work will focus on discrete sets, particularly the natural numbers, with results that describe the presentation of the natural numbers under specific symbol libraries and what those presentations may reveal about the properties of the natural numbers themselves. We shall present bounds and constraints on the length and shape of presentations, connect already existing problems in other fields of mathematics to questions relevant to these presentations and otherwise illuminate why such a study can produce exciting results.

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“…In 1874, Mertens [15] proved the following estimate for truncated Euler-product of ζ(s), which is also known as Mertens' third theorem given by p≤x…”

Section: Mertens' Theorem For the Class Gmentioning

confidence: 99%

Large values of $L$-functions on $1$-line

Dixit¹,

Mahatab²

2019

Preprint

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In this paper, we study lower bounds of a general family of L-functions on the 1-line. More precisely, we show that for any F (s) in this family, there exist arbitrary large t such that, where m is the order of the pole of F (s) at s = 1. This is a generalization of the same result of Aistleitner, Munsch and the second author for the Riemann zeta-function. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg L-functions of the type L(s, f ×f ) on the 1-line.2010 Mathematics Subject Classification. 11M41.

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Ein Beitrag zur analytischen Zahlentheorie

Cited by 52 publications

References 0 publications

Upper bounds on the heights of polynomials and rational fractions from their values

Upper bounds on the heights of polynomials and rational fractions from their values

Optimal Presentations of Mathematical Objects

Large values of $L$-functions on $1$-line

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