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.

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 MODULAR EXPONENTIATION

2014 ◽  
pp. 49-55
Author(s):  
Sorin Iftene
Keyword(s):  
Number Theory ◽  
Modular Exponentiation ◽  
Fundamental Operation ◽  
Primality Testing ◽  
Computational Number Theory ◽  
Fixed Base

Exponentiation is a fundamental operation in computational number theory. Primality testing and cryptography are important working fields in which the exponentiation is heavily used. In this paper we survey the most popular methods for modular exponentiation: basic techniques, fixed-exponent techniques, fixed-base techniques, and techniques based on modulus particularities. Some aspects related to parallelism are also discussed.

scholarly journals TWISTS OF AUTOMORPHIC L-FUNCTIONS AT THE CENTRAL POINT

2013 ◽  
Vol 09 (04) ◽  
pp. 1015-1053
Author(s):  
H. M. BUI
Keyword(s):  
Upper Bound ◽  
Central Point ◽  
Critical Strip ◽  
Primitive Character ◽  
Central Values ◽  
Positive Proportion ◽  
Primitive Forms ◽  
Derivatives Of

We study the nonvanishing of twists of automorphic L-functions at the center of the critical strip. Given a primitive character χ modulo D satisfying some technical conditions, we prove that the twisted L-functions L(f.χ, s) do not vanish at s = ½ for a positive proportion of primitive forms of weight 2 and level q, for large prime q. We also investigate the central values of high derivatives of L(f.χ, s), and from that derive an upper bound for the average analytic rank of the studied L-functions.

Representation of information in SEM by equal signal lines

1978 ◽  
Vol 36 (1) ◽  
pp. 144-145
Author(s):  
E. Rau ◽  
N. Karelin ◽  
V. Dukov ◽  
M. Kolomeytsev ◽  
S. Gavrikov ◽  
...  
Keyword(s):  
Electronic System ◽  
Space Distribution ◽  
Specimen Surface ◽  
Universal Method ◽  
Base Level ◽  
Signal Value ◽  
Information Signal ◽  
Fixed Base ◽  
Scanning Line ◽  
Amplitude Analyzer

There are different methods and devices for the increase of the videosignal information in SEM. For example, with the help of special pure electronic [1] and opto-electronic [2] systems equipotential areas on the specimen surface in SEM were obtained. This report generalizes quantitative universal method for space distribution representation of research specimen parameter by contour equal signal lines. The method is based on principle of comparison of information signal value with the fixed levels.Transformation image system for obtaining equal signal lines maps was developed in two versions:1)In pure electronic system [3] it is necessary to compare signal U (see Fig.1-a), which gives potential distribution on specimen surface along each scanning line with fixed base level signals εifor obtaining quantitative equipotential information on solid state surface. The amplitude analyzer-comparator gives flare sport videopulses at any fixed coordinate and any instant time when initial signal U is equal to one of the base level signals ε.

A solution to the rear-vision problem in a fixed-base driving simulator

2004 ◽  
Author(s):  
Guihua Yang ◽  
Farnaz Baniahmad ◽  
Beverly K. Jaeger ◽  
Ronald R. Mourant
Keyword(s):  
Driving Simulator ◽  
Fixed Base

A Critical Analysis of Selected Public and Private Mutual Fund Schemes in India

2019 ◽  
Vol 118 (8) ◽  
pp. 28-34
Author(s):  
Dr. V. Murali Krishna ◽  
Dr T. Hima Bindu ◽  
Dr. Ravikumar Gunakala
Keyword(s):  
Mutual Funds ◽  
Capital Market ◽  
Mutual Fund ◽  
Portfolio Diversification ◽  
Financial Intermediaries ◽  
Public And Private ◽  
Fund Managers ◽  
Fixed Base ◽  
Banking Institutions ◽  
Different Levels

Mutual Fund Industry is one of the emerged dominant financial intermediaries in Indian Capital Market. The main objective of investing in a mutual fund is to diversify risk. Though the mutual fund invests in diversified portfolio, the fund managers take different levels of risk in order to achieve the schemes objectives. Mutual funds allow portfolio diversification and relative risk management through collection of funds from the savers/investors, the same investing in equity and debt stocks. This type of invested funds is managed by professional experts called as fund managers Funds are categorized as income should fixed base in India are a kind of mutual fund which makes investment in debt securities that have been issued to the corporate, banking institutions and to government in general

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