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The number of solutions of Φ(x) = m
"... An old conjecture of Sierpiński asserts that for every integer k � 2, there is a number m for which the equation φ(x) = m has exactly k solutions. Here φ is Euler’s totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an uncondit ..."
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Cited by 9 (2 self)
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An old conjecture of Sierpiński asserts that for every integer k � 2, there is a number m for which the equation φ(x) = m has exactly k solutions. Here φ is Euler’s totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an unconditional proof of Sierpiński’s conjecture. The proof uses many results from sieve theory, in particular the famous theorem of Chen.
An uncertainty principle for arithmetic sequences
, 2004
"... Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are “welldistributed ” in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when working with ..."
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Cited by 6 (3 self)
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Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are “welldistributed ” in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when working with the “uncertainty principle”. In this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory, formulating a general principle, and giving several examples.
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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Cited by 4 (3 self)
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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.
SUMS OF PRIMES AND SQUARES OF PRIMES IN SHORT INTERVALS
, 801
"... Abstract. Let H2 denote the set of even integers n ̸ ≡ 1 (mod 3). We prove that when H ≥ X 0.33, almost all integers n ∈ H2 ∩ (X, X + H] can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes. 1. ..."
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Abstract. Let H2 denote the set of even integers n ̸ ≡ 1 (mod 3). We prove that when H ≥ X 0.33, almost all integers n ∈ H2 ∩ (X, X + H] can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes. 1.
ON PRIMES REPRESENTED BY QUADRATIC POLYNOMIALS
, 2008
"... This is a survey article on the HardyLittlewood conjecture about primes in quadratic progressions. We recount the history and quote some results approximating this hitherto unresolved conjecture. ..."
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This is a survey article on the HardyLittlewood conjecture about primes in quadratic progressions. We recount the history and quote some results approximating this hitherto unresolved conjecture.
ADDITIVE PROBLEMS WITH PRIME VARIABLES THE CIRCLE METHOD OF HARDY, RAMANUJAN AND LITTLEWOOD
, 2012
"... In these lectures we give an overview of the circle method introduced by Hardy and Ramanujan at the beginning of the twentieth century, and developed by Hardy, Littlewood and Vinogradov, among others. We also try and explain the main difficulties in proving Goldbach’s conjecture and we give a sket ..."
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In these lectures we give an overview of the circle method introduced by Hardy and Ramanujan at the beginning of the twentieth century, and developed by Hardy, Littlewood and Vinogradov, among others. We also try and explain the main difficulties in proving Goldbach’s conjecture and we give a sketch of the proof of Vinogradov’s threeprime Theorem.
BUBBLES OF CONGRUENT PRIMES
"... Abstract. In [15], Shiu proved that if a and q are arbitrary coprime integers, then there exist arbitrarily long strings of consecutive primes which are all congruent to a modulo q. We generalize Shiu’s theorem to imaginary quadratic fields, where we prove the existence of “bubbles ” containing arbi ..."
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Abstract. In [15], Shiu proved that if a and q are arbitrary coprime integers, then there exist arbitrarily long strings of consecutive primes which are all congruent to a modulo q. We generalize Shiu’s theorem to imaginary quadratic fields, where we prove the existence of “bubbles ” containing arbitrarily many primes which are all, up to units, congruent to a modulo q. 1.
MAIER MATRICES BEYOND Z
"... This paper is an expanded version of a talk given at the 2007 Integers Conference, giving an overview of the Maier matrix method and surveying the author’s work in extending it beyond the integers. 1. Maier Matrices Loosely speaking, a Maier matrix is a combinatorial device used to prove the existen ..."
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This paper is an expanded version of a talk given at the 2007 Integers Conference, giving an overview of the Maier matrix method and surveying the author’s work in extending it beyond the integers. 1. Maier Matrices Loosely speaking, a Maier matrix is a combinatorial device used to prove the existence of irregular or interesting patterns in the distribution of primes or related sequences. We will illustrate the technique with two particularly interesting examples. The first is Maier’s 1985 proof [6] that “unexpected ” irregularities exist in the distribution of primes in short intervals. In particular, Maier proved that for any A> 0 there exists a constant δA> 0 such that lim sup n→∞ π(n + log A n) − π(n) log A−1 n ≥ 1 + δA, lim inf n→∞ π(n + log A n) − π(n) log A−1 n ≤ 1 − δA. (0.1) These irregularities are unexpected in the sense that they contradict probabilistic heuristics for A> 2. The proof is as follows. For a variable y, let Q = ∏ p<y p, let x1 = QD for some fixed large D, and let C be a parameter to be determined later. Consider the following matrix of integers: ⎡