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COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ
"... ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each ..."
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ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n c solutions, for some c> 0. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of φ at p, and on a result of HeathBrown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.
On values taken by the largest prime factor of shifted primes
 J. Aust. Math. Soc
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The Lucas–Pratt primality tree
 Math. Comp
"... Abstract. In 1876, E. Lucas showed that a quick proof of primality for a prime p could be attained through the prime factorization of p − 1 and a primitive root for p. V. Pratt’s proof that PRIMES is in NP, done via Lucas’s theorem, showed that a certificate of primality for a prime p could be obtai ..."
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Abstract. In 1876, E. Lucas showed that a quick proof of primality for a prime p could be attained through the prime factorization of p − 1 and a primitive root for p. V. Pratt’s proof that PRIMES is in NP, done via Lucas’s theorem, showed that a certificate of primality for a prime p could be obtained in O(log 2 p) modular multiplications with integers at most p. We show that for all constants C ∈ R, the number of modular multiplications necessary to obtain this certificate is greater than C log p for a set of primes p with relative asymptotic density 1. 1.
Some normal numbers generated by arithmetic functions, submitted; preprint available online as arXiv:1309.7386
"... Abstract. Let g ≥ 2. A real number is said to be gnormal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let ϕ denote Euler’s totient function, let σ be the sumofdivisors function, and let λ be Carmichael’s lambdafunction. We show that if f ..."
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Abstract. Let g ≥ 2. A real number is said to be gnormal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let ϕ denote Euler’s totient function, let σ be the sumofdivisors function, and let λ be Carmichael’s lambdafunction. We show that if f is any function formed by composing ϕ, σ, or λ, then the number 0.f(1)f(2)f(3)... obtained by concatenating the base g digits of successive fvalues is gnormal. We also prove the same result if the inputs 1, 2, 3,... are replaced with the primes 2, 3, 5,.... The proof is an adaptation of a method introduced by Copeland and Erdős in 1946 to prove the 10normality of 0.235711131719.... 1.
PRIME CHAINS AND PRATT TREES
, 2009
"... We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains with pk � x ..."
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We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains with pk � x (k variable), and P (x; p), the number of chains with p1 = p and pk � px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with pk = p, which is also the height of the Pratt tree for p. We show H(p) � c log log p and H(p) � (log p) 1−c′ for almost all p, with c, c ′ explicit positive constants. We can take, for any ε> 0, c = e − ε assuming the ElliottHalberstam conjecture. A stochastic model of the Pratt tree is introduced and analyzed. The model suggests that for most p � x, H(p) stays very close to e log log x.
SIEVING VERY THIN SETS OF PRIMES, AND PRATT TREES WITH MISSING PRIMES
"... ABSTRACT. Suppose P is a set of primes, such that for every p ∈ P, every prime factor of p − 1 is also in P. We apply a new sieve method to show that either P contains all of the primes or the counting function of P is O(x 1−c) for some c> 0, where c depends only on the smallest prime not in P. O ..."
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ABSTRACT. Suppose P is a set of primes, such that for every p ∈ P, every prime factor of p − 1 is also in P. We apply a new sieve method to show that either P contains all of the primes or the counting function of P is O(x 1−c) for some c> 0, where c depends only on the smallest prime not in P. Our proof makes use of results connected with Artin’s primitive root conjecture. 1
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"... Spectral factorization of biinfinite multiindex block Toeplitz matrices. (English summary) Special issue on structured and infinite systems of linear equations. Linear Algebra Appl. 343/344 (2002), 355–380. Block lowerdiagonalupper LDU and Cholesky LL ∗ factorizations are studied for weighted Wi ..."
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Spectral factorization of biinfinite multiindex block Toeplitz matrices. (English summary) Special issue on structured and infinite systems of linear equations. Linear Algebra Appl. 343/344 (2002), 355–380. Block lowerdiagonalupper LDU and Cholesky LL ∗ factorizations are studied for weighted Wiener classes of biinfinite block Toeplitz matrices A = (Ai−j) i,j∈Zd, where Aj are complex k × k matrices, and ∑ i∈Zd βi‖Ai ‖ < ∞, for a fixed sequence of weights {βi} i∈Zd subject to 1 ≤ βi+j ≤ βiβj. A further generalization, also studied in the paper under review, consists of considering the biinfinite matrices A with the additional requirement that Ai = 0 for i ̸ ∈ J, where J is a fixed subgroup of Zd. The factorizations are considered with respect to a total order in Zd. One result: If A is positive definite, satisfies the additional requirement, and the symbol of A is invertible on the maximal ideal space, then A has a Cholesky factorization, where the factor L and its inverse also satisfy the additional requirement and belong to the Wiener class with the trivial weights βi ≡ 1; in the scalar case k = 1 the factor L in fact belongs to the weighted Wiener class. Convergence of projection methods for computing the Cholesky factorization is proved and illustrated by an example. A comparison is made with Krein’s method, which is based on the additive decomposition of the logarithm of A (when applicable).
doi:10.1112/S0025579313000235 THE ERROR TERM IN THE COUNT OF ABUNDANT NUMBERS
"... Abstract. A natural number n is called abundant if the sum of the proper divisors of n exceeds n. For example, 12 is abundant, since 1+ 2+ 3+ 4+ 6 = 16. In 1929, BesselHagen asked whether or not the set of abundant numbers possesses an asymptotic density. In other words, if A(x) denotes the count o ..."
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Abstract. A natural number n is called abundant if the sum of the proper divisors of n exceeds n. For example, 12 is abundant, since 1+ 2+ 3+ 4+ 6 = 16. In 1929, BesselHagen asked whether or not the set of abundant numbers possesses an asymptotic density. In other words, if A(x) denotes the count of abundant numbers belonging to the interval [1, x], does A(x)/x tend to a limit? Four years later, Davenport answered BesselHagen’s question in the affirmative. Calling this density 1, it is now known that 0.24761<1< 0.24766, so that just under one in four numbers are abundant. We show that A(x)−1x < x/exp((log x)1/3) for all large x. We also study the behavior of the corresponding error term for the count of socalled αabundant numbers. §1. Introduction. Let σ(n):=∑dn d be the usual sumofdivisors function. It is traditional to call the natural number n abundant if the sum of its proper divisors exceeds n, that is, if σ(n)> 2n. The abundant numbers have been of interest for over two thousand years. However, it was only comparatively recently, in 1929, that BesselHagen [3, p. 1571] posed the question of whether
Journal of Integer Sequences, Vol. 16 (2013), Article 13.5.5 On the Middle Prime Factor of an Integer
"... Given an integer n ≥ 2, let pm(n) denote the middle prime factor of n. We obtain an estimate for the sum of the reciprocals of pm(n) for n ≤ x. 1 ..."
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Given an integer n ≥ 2, let pm(n) denote the middle prime factor of n. We obtain an estimate for the sum of the reciprocals of pm(n) for n ≤ x. 1