Efficient Prime Counting and the Chebyshev Primes

@article{Planat2013EfficientPC,
  title={Efficient Prime Counting and the Chebyshev Primes},
  author={Michel Planat and Patrick Sol'e},
  journal={arXiv: Number Theory},
  year={2013},
  volume={2013},
  pages={1-11}
}
The function where is the logarithm integral and the number of primes up to is well known to be positive up to the (very large) Skewes' number. Likewise, according to Robin's work, the functions and , where and are Chebyshev summatory functions, are positive if and only if Riemann hypothesis (RH) holds. One introduces the jump function at primes and one investigates , , and . In particular, , and for . Besides, for any odd , an infinite set of the so-called Chebyshev primes. In the context of… Expand

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References

SHOWING 1-10 OF 19 REFERENCES
Multiplicative Number Theory
From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The… Expand
An Introduction to the Theory of Numbers
THIS book must be welcomed most warmly into X the select class of Oxford books on pure mathematics which have reached a second edition. It obviously appeals to a large class of mathematical readers.… Expand
An Introduction to the Theory of Numbers
This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford,… Expand
Stream Ciphers and Number Theory
Contents Preface to the Revised Edition Preface to the First Edition. 1 Introduction. 2 Stream Ciphers. 3 Primes, Primitive Roots and Sequences. 4 Cyclotomy and Cryptographic Functions. 5 Special… Expand
The On-Line Encyclopedia of Integer Sequences
  • N. Sloane
  • Mathematics, Computer Science
  • Electron. J. Comb.
  • 1994
TLDR
The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used. Expand
Riemann's Zeta Function
Preface This document grew out of lecture notes for a course taught in Cambridge during Lent 2014 by Adam Harper on the theory of the Riemann zeta function. There are likely to be errors, which are… Expand
THE DISTRIBUTION OF PRIME NUMBERS
This is an expanded account of three lectures on the distribution of prime numbers given at the Montreal NATO school on equidistribution.
β€œA and B”:
Direct fabrication of large micropatterned single crystals. p1205 21 Feb 2003. (news): Academy plucks best biophysicists from a sea of mediocrity. p994 14 Feb 2003.
Sur la diffΓ©rence Li(ΞΈ(x)) βˆ’ Ο€(x),
  • Annales de la Faculté des Sciences de Toulouse, Série 5,
  • 1984
...
1
2
...