EFFECTIVE LOWER BOUND FOR THE VARIANCE OF DISTRIBUTION OF PRIMES IN ARITHMETIC PROGRESSIONS

2008 ◽  
Vol 04 (01) ◽  
pp. 45-56 ◽  
Author(s):  
EMMANUEL KNAFO
Keyword(s):  
Lower Bound ◽  
Arithmetic Progressions ◽  
Distribution Of Primes ◽  
Primes In Arithmetic Progressions ◽  
Siegel Zeros

Through a refinement for the estimation of the effect of Siegel zeros, we show how to avoid the use of Siegel's theorem in order to obtain the first effective lower bound for the variance of distribution of primes in arithmetic progressions.

MULTIPLE PATTERNS OF k-TUPLES OF INTEGERS

2011 ◽  
Vol 07 (07) ◽  
pp. 1761-1779
Author(s):  
PAULO RIBENBOIM
Keyword(s):  
Lower Bound ◽  
Arithmetic Progressions ◽  
Carmichael Numbers ◽  
Pigeon Hole Principle ◽  
Primes In Arithmetic Progressions

The first proposition and its corollary are a transfiguration of Dirichlet's pigeon-hole principle. They are applied to show that a wide variety of sequences display arbitrarily large patterns of sums, differences, higher differences, etc. Among these, we include sequences of primes in arithmetic progressions, of powerful integers, sequences of integers with radical index having a prescribed lower bound, and many others. We also deal with patterns in iterated sequences of primes, patterns of gaps between primes, patterns of values of Euler's φ-function, or their gaps, as well as patterns related to the sequence of Carmichael numbers.

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