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The Encyclopedia of Integer Sequences
"... This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs ..."
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This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other
Is P versus NP formally independent
 Bulletin of the European Association for Theoretical Computer Science
, 2003
"... I have moved back to the University of Chicago and so has the web page for this column. See above for new URL and contact informaion. This issue Scott Aaronson writes quite an interesting (and opinionated) column on whether the P = NP question is independent of the usual axiom systems. Enjoy! ..."
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I have moved back to the University of Chicago and so has the web page for this column. See above for new URL and contact informaion. This issue Scott Aaronson writes quite an interesting (and opinionated) column on whether the P = NP question is independent of the usual axiom systems. Enjoy!
On Robin’s criterion for the Riemann Hypothesis
, 2006
"... Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality σ(n): = ∑ dn d < eγn log log n is satisfied for n ≥ 5041, where γ denotes the Euler(Mascheroni) constant. We show by elementary methods that if n ≥ 37 does not satisfy Robin’s criterion it must be e ..."
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Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality σ(n): = ∑ dn d < eγn log log n is satisfied for n ≥ 5041, where γ denotes the Euler(Mascheroni) constant. We show by elementary methods that if n ≥ 37 does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that n must be divisible by a fifth power> 1. As consequence we obtain that RH holds true iff every natural number divisible by a fifth power> 1 satisfies Robin’s inequality.
Criteria equivalent to the Riemann Hypothesis
, 2008
"... We give a brief overview of a few criteria equivalent to the Riemann Hypothesis. Next we concentrate on the Riesz and BáezDuarte criteria. We proof that they are equivalent and we provide some computer data to support them. It is not compressed to six pages version of the talk delivered by M.W. ..."
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We give a brief overview of a few criteria equivalent to the Riemann Hypothesis. Next we concentrate on the Riesz and BáezDuarte criteria. We proof that they are equivalent and we provide some computer data to support them. It is not compressed to six pages version of the talk delivered by M.W.
Eight Hateful Sequences
, 2008
"... Sequences (the OEIS) [12] contains 140000 sequences. Here are eight of them, suggested by the theme of the Eighth Gathering: they are all infinite, and all ’ateful in one way or another. I hope you like ’em! Each one is connected with an interesting unsolved problem. Since this is a 15minute talk, ..."
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Sequences (the OEIS) [12] contains 140000 sequences. Here are eight of them, suggested by the theme of the Eighth Gathering: they are all infinite, and all ’ateful in one way or another. I hope you like ’em! Each one is connected with an interesting unsolved problem. Since this is a 15minute talk, I can’t give many details—see the entries in the OEIS for more information, and for links to related sequences. 1. Hateful or Beastly Numbers The most hateful sequence of all! These are the numbers that contain the string 666 in their decimal expansion:
Superabundant Numbers and the Riemann Hypothesis
"... The function σ(n) = ∑ dn d is the sum of divisors function, so for example σ(12) = 28. In 1913 Gronwall proved that lim sup n→∞ σ(n) e γ n log log n where γ ≈ 0.57721 is Euler’s constant (see [4, Theorem 323]). This says that the maximal size of σ(n) is roughly e γ n log log n. The following theo ..."
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The function σ(n) = ∑ dn d is the sum of divisors function, so for example σ(12) = 28. In 1913 Gronwall proved that lim sup n→∞ σ(n) e γ n log log n where γ ≈ 0.57721 is Euler’s constant (see [4, Theorem 323]). This says that the maximal size of σ(n) is roughly e γ n log log n. The following theorem of Robin [7, Theorem 2] gives a more refined version of this upper bound. Theorem 1 For n ≥ 3 we have σ(n) e γ n log log n
GCD of p  1, q  1 for Random p, q
, 2003
"... Introduction In this note we study the following problem: How big can the greatest common divisor of p 1 be, where p, q are randomly chosen primes in the set {1, . . . , N}? Apart from being of independent interest, this problem arises in security when one wants to use an l (= 1024) bit RSA ..."
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Introduction In this note we study the following problem: How big can the greatest common divisor of p 1 be, where p, q are randomly chosen primes in the set {1, . . . , N}? Apart from being of independent interest, this problem arises in security when one wants to use an l (= 1024) bit RSA crypto coprocessor to do 2l bit cryptography [3]. One can answer this question quickly if one is allowed asymptotic results. But in practice one has N = 2 , so asymptotic results do not make much sense. It was observed that with probability at least than 0.99 ([3]), the g.c.d. is less than 32 bit. In this note we prove exactly this! To do so we combine some nontrivial exact results from analytic number theory. 2 Number Theoretic Preliminaries We use the following notation: #(n) := 1#i#n,(i,n)=1 1, #(n) := 1#d#n, dn d, #(n) denotes the number of primes less than or equal to n, and #(n; r, k) := 1#p#n, p#k(mod r) 1, where p are primes. The following are known facts about arithmetic
ROBIN’S THEOREM, PRIMES, AND A NEW ELEMENTARY REFORMULATION OF THE RIEMANN HYPOTHESIS
"... For n> 1, let G(n) = σ(n) n log log n, where σ(n) is the sum of the divisors of n. We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G(N) ≥ max (G(N/p), G(aN)) , for all prime factors p of N and each positive integer a. The proof uses Robin’s an ..."
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For n> 1, let G(n) = σ(n) n log log n, where σ(n) is the sum of the divisors of n. We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G(N) ≥ max (G(N/p), G(aN)) , for all prime factors p of N and each positive integer a. The proof uses Robin’s and Gronwall’s theorems on G(n). An alternate proof of one step depends on two properties of superabundant numbers proved using Alaoglu and Erdős’s results. 1.
A proof for the Riemann hypothesis
, 2008
"... The Riemann zeta function ζ(s) is defined by ζ(s) = ∑∞ n=1 1 ns for ℜ(s)> 1 and can be extended to a regular function on the whole complex plane deleting its unique pole at s = 1. The Riemann hypothesis is a conjecture made by Riemann in 1859 asserting that all nontrivial zeros for ζ(s) lie on the ..."
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The Riemann zeta function ζ(s) is defined by ζ(s) = ∑∞ n=1 1 ns for ℜ(s)> 1 and can be extended to a regular function on the whole complex plane deleting its unique pole at s = 1. The Riemann hypothesis is a conjecture made by Riemann in 1859 asserting that all nontrivial zeros for ζ(s) lie on the line ℜ(s) = 1 2, which is equivalent to the prime number theorem in the form of π(x)−Li(x) = O(x 1 2 +ǫ) for any positive ǫ, where π(x) = ∑ p≤x 1 with the sum runs through the set of primes is the prime counting function and Li(x) = ∫ x 1 2 log v dv is Gauss ’ logarithmic integral function. In this article, it gives a proof for the density hypothesis and so that settles the long time due justification for the Riemann hypothesis from the equivalence of the density hypothesis and the Riemann hypothesis proved recently in [12], which in turn gives a prime number theorem stated as above.