scholarly journals Nonvanishing for cubic L-functions

2021 ◽  
Vol 9 ◽  
Author(s):  
Chantal David ◽  
Alexandra Florea ◽  
Matilde Lalin
Keyword(s):  
Critical Point ◽  
Number Field ◽  
Upper Bound ◽  
Riemann Hypothesis ◽  
Implied Constant ◽  
Field Case ◽  
Second Moment ◽  
Positive Proportion ◽  
First Moment

Abstract We prove that there is a positive proportion of L-functions associated to cubic characters over $\mathbb F_q[T]$ that do not vanish at the critical point $s=1/2$ . This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic L-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester and Radziwiłł, which in turn develops further ideas from the work of Soundararajan, Harper and Radziwiłł. We work in the non-Kummer setting when $q\equiv 2 \,(\mathrm {mod}\,3)$ , but our results could be translated into the Kummer setting when $q\equiv 1\,(\mathrm {mod}\,3)$ as well as into the number-field case (assuming the generalised Riemann hypothesis). Our positive proportion of nonvanishing is explicit, but extremely small, due to the fact that the implied constant in the upper bound for the mollified second moment is very large.

scholarly journals ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

2020 ◽  
pp. 1-10
Author(s):  
CLEMENS FUCHS ◽  
SEBASTIAN HEINTZE
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Linear Recurrences ◽  
Field Case ◽  
Power Sum ◽  
Linear Recurrence Sequence

Abstract Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

scholarly journals Laver sequences for extendible and super-almost-huge cardinals

1999 ◽  
Vol 64 (3) ◽  
pp. 963-983 ◽  
Author(s):  
Paul Corazza
Keyword(s):  
Critical Point ◽  
Upper Bound ◽  
Large Cardinal ◽  
Strong Cardinal ◽  
Regular Class ◽  
Strong Cardinals ◽  
Extendible Cardinals

AbstractVersions of Laver sequences are known to exist for supercompact and strong cardinals. Assuming very strong axioms of infinity, Laver sequences can be constructed for virtually any globally defined large cardinal not weaker than a strong cardinal; indeed, under strong hypotheses. Laver sequences can be constructed for virtually any regular class of embeddings. We show here that if there is a regular class of embeddings with critical point κ, and there is an inaccessible above κ, then it is consistent for there to be a regular class that admits no Laver sequence. We also show that extendible cardinals are Laver-generating, i.e., that assuming only that κ is extendible, there is an extendible Laver sequence at κ. We use the method of proof to answer a question about Laver-closure of extendible cardinals at inaccessibles. Finally, we consider Laver sequences for super-almost-huge cardinals. Assuming slightly more than super-almost-hugeness, we show that there are super-almost-huge Laver sequences, improving the previously known upper bound for such Laver sequences. We also describe conditions under which the canonical construction of a Laver sequence fails for super-almost-huge cardinals.

scholarly journals A REMARK ON PRIMALITY TESTING AND DECIMAL EXPANSIONS

2011 ◽  
Vol 91 (3) ◽  
pp. 405-413 ◽  
Author(s):  
TERENCE TAO
Keyword(s):  
Upper Bound ◽  
Primality Testing ◽  
Positive Proportion ◽  
Fixed Base ◽  
Covering Set ◽  
Selberg Sieve

AbstractWe show that for any fixed base a, a positive proportion of primes become composite after any one of their digits in the base a expansion is altered; the case where a=2 has already been established by Cohen and Selfridge [‘Not every number is the sum or difference of two prime powers’, Math. Comput.29 (1975), 79–81] and Sun [‘On integers not of the form ±pa±qb’, Proc. Amer. Math. Soc.128 (2000), 997–1002], using some covering congruence ideas of Erdős. Our method is slightly different, using a partially covering set of congruences followed by an application of the Selberg sieve upper bound. As a consequence, it is not always possible to test whether a number is prime from its base a expansion without reading all of its digits. We also present some slight generalisations of these results.

A NEW UPPER BOUND FOR RANDOM (2 + p)-SAT BY FLIPPING TWO VARIABLES

2013 ◽  
Vol 24 (06) ◽  
pp. 899-912 ◽  
Author(s):  
GUANGYAN ZHOU ◽  
ZONGSHENG GAO
Keyword(s):  
Phase Transition ◽  
Computational Complexity ◽  
Upper Bound ◽  
Moment Method ◽  
Transition Point ◽  
Phase Transition Point ◽  
Rigorous Results ◽  
Unit Clause ◽  
First Moment ◽  
Local Maximality

The random (2 + p)-SAT model has been proposed [18] to study the possible relation between the “order” of phase transitions and computational complexity. It was also claimed that there exists pc > 0, such that for p < pc the random (2 + p)-SAT instance behaves like 2-SAT. Later, Achlioptas et al. [3] obtained the first rigorous results that 0.4 ≤ pc ≤ 0.695, the methods they use are the first moment method and the simple Unit-Clause algorithm. In this paper, we try to optimize the local maximality condition of the truth assignments when implementing the first moment method. We prove that the phase transition point of clauses-to-variables ratio r (dependent on p) can be improved. Moreover, we show that the upper bound of pc can be reduced to 0.6846. This fact implies that, for a constant λ < 1, a random (2 + p)-SAT formula with λn 2-clauses and 2.17n 3-clauses is almost surely unsatisfiable.

Cohomology ofS-arithmetic subgroups in the number field case

1994 ◽  
Vol 116 (1) ◽  
pp. 75-93 ◽  
Author(s):  
Don Blasius ◽  
Jens Franke ◽  
Fritz Grunewald
Keyword(s):  
Number Field ◽  
Field Case

scholarly journals MARKETING RESEARCH PROCESSES. A PERSPECTIVE OF THE FUTURE FROM A QUALITATIVE VIEW

2020 ◽  
pp. 111-130
Author(s):  
Luz Alexandra Montoya-Restrepo ◽  
Iván Alonso Montoya-Restrepo ◽  
Sandra Rojas-Berrio
Keyword(s):  
Social Networking ◽  
Quantitative Research ◽  
Market Research ◽  
Marketing Research ◽  
Future Research ◽  
Scientific Journals ◽  
Second Moment ◽  
The Future ◽  
First Moment

The objective of this document is to resume the increasing importance of qualitative market research, which has shown growth not just in scientific journals but also has larger numbers compared to quantitative research. The first step was reflecting on research in marketing and the possibilities it offers to get to know consumers and purchasing habits. The methodology applied is divided in two moments: the first moment is based on bibliometrics, which reviews trends in publications, and in the second moment, different scenarios from marketing research are proposed based on experts’ opinions and prospective methods (Smic Prob-Expert method). The conclusion is that in the future, research will become a discipline that is largely associated to sensory and neurological studies, operated with social networking strategies and oriented to the description of specific phenomena, all of which will lead to a new scientific anthropo-marketing.

The Kernel of and the 4-Rank of K2(O)

1992 ◽  
Vol 35 (3) ◽  
pp. 295-302 ◽  
Author(s):  
Ruth I. Berger
Keyword(s):  
Number Field ◽  
Upper Bound ◽  
Class Group ◽  
Number Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Ring Of Integers

AbstractAn upper bound is given for the order of the kernel of the map on Sideal class groups that is induced by For some special types of number fields F the connection between the size of the above kernel for and the units and norms in are examined. Let K2(O) denote the Milnor K-group of the ring of integers of a number field. In some cases a formula by Conner, Hurrelbrink and Kolster is extended to show how closely the 4-rank of is related to the 4-rank of the S-ideal class group of

A note about a partial no-go theorem for quantum PCP

2011 ◽  
Vol 11 (11&12) ◽  
pp. 1019-1027
Author(s):  
Itai Itai Arad
Keyword(s):  
Critical Point ◽  
Upper Bound ◽  
Open Problem ◽  
Preliminary Step ◽  
Important Open Problem

This is not a disproof of the quantum PCP conjecture! In this note we use perturbation on the commuting Hamiltonian problem on a graph, based on results by Bravyi and Vyalyi, to provide a very partial no-go theorem for quantum PCP. Specifically, we derive an upper bound on how large the promise gap can be for the quantum PCP still to hold, as a function of the non-commuteness of the system. As the system becomes more and more commuting, the maximal promise gap shrinks. We view these results as possibly a preliminary step towards disproving the quantum PCP conjecture posed in \cite{ref:Aha09}. A different way to view these results is actually as indications that a critical point exists, beyond which quantum PCP indeed holds; in any case, we hope that these results will lead to progress on this important open problem.

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