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Register a new userA merry-go-round with the circle map, primes and pseudoprimes
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We show that the use of the main characteristics of the circle map leads naturally to establish a few statements on primes and pseudoprimes. In this way a Fermats theorem on primes and some interesting properties of pseudoprimes are obtained.
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This is an English translation of the Latin original De summa seriei ex numeris primis formatae ${1/3}-{1/5}+{1/7}+{1/11}-{1/13}-{1/17}+{1/19}+{1/23}-{1/29}+{1/31}-$ etc. ubi numeri primi formae $4n-1$ habent signum positivum formae autem $4n+1$ signum negativum (1775). E596 in the Enestrom index. Let $chi$ be the nontrivial character modulo 4. Euler wants to know what $sum_p chi(p)/p$ is, either an exact expression or an approximation. He looks for analogies to the harmonic series and the series of reciprocals of the primes. Another reason he is interested in this is that if this series has a finite value (which is does, the best approximation Euler gets is 0.3349816 in section 27) then there are infinitely many primes congruent to 1 mod 4 and infinitely many primes congruent to 3 mod 4. In section 15 Euler gives the Euler product for the L(chi,1). As a modern mathematical appendix appendix, I have written a proof following Davenport that the series $sum_p frac{chi(p)}{p}$ converges. This involves applications of summation by parts, and uses Chebyshevs estimate for the second Chebyshev function (summing the von Mangoldt function).
In 1967 the Dutch mathemetician F.J.M. Barning described an infinite, planar, ternary tree*. Seven years later, A. Hall independently discovered the same tree. Both used the method of uni-modular matrices to transform one triple to another. A number of rediscoveries have occurred more recently. In this article we announce the discovery of an entirely different ternary tree, and show how it relates to the one found by Barning and Hall.
Dirichlets proof of infinitely many primes in arithmetic progressions was published in 1837, introduced L-series for the first time, and it is said to have started rigorous analytic number theory. Dirichlet uses Eulers earlier work on the zeta function and the distribution of primes. He first proves a simpler case before going to full generality. The paper was translated from German by R. Stephan and given a reference section.
This paper is an exposition and review of the research related to the Riemann Hypothesis starting from the work of Riemann and ending with a description of the work of G. Spencer-Brown.
Throughout more than two millennia many formulas have been obtained, some of them beautiful, to calculate the number pi. Among them, we can find series, infinite products, expansions as continued fractions and expansions using radicals. Some expressions which are (amazingly) related to pi have been evaluated. In addition, a continual battle has been waged just to break the records computing digits of this number; records have been set using rapidly converging series, ultra fast algorithms and really surprising ones, calculating isolated digits. The development of powerful computers has played a fundamental role in these achievements of calculus.
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