4
4.0

Jun 30, 2018
06/18

by
Benjamin Lansdell; Kevin Ford; J. Nathan Kutz

texts

#
eye 4

#
favorite 0

#
comment 0

Prior to receiving visual stimuli, spontaneous, correlated activity called retinal waves drives activity-dependent developmental programs. Early-stage waves mediated by acetylcholine (ACh) manifest as slow, spreading bursts of action potentials. They are believed to be initiated by the spontaneous firing of Starburst Amacrine Cells (SACs), whose dense, recurrent connectivity then propagates this activity laterally. Their extended inter-wave intervals and shifting wave boundaries are the result...

Topics: Quantitative Biology, Neurons and Cognition

Source: http://arxiv.org/abs/1404.7549

40
40

Sep 19, 2013
09/13

by
Kevin Ford; Paul Pollack

texts

#
eye 40

#
favorite 0

#
comment 0

We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x(log x)^{-1+o(1)} numbers not exceeding x common to the ranges of Euler's function phi(n) and the sum-of-divisors function sigma(m).

Source: http://arxiv.org/abs/1010.5427v1

6
6.0

Jun 28, 2018
06/18

by
Sean Eberhard; Kevin Ford; Ben Green

texts

#
eye 6

#
favorite 0

#
comment 0

We say that permutations $\pi_1,\dots, \pi_r \in \mathcal{S}_n$ invariably generate $\mathcal{S}_n$ if, no matter how one chooses conjugates $\pi'_1,\dots,\pi'_r$ of these permutations, $\pi'_1,\dots,\pi'_r$ generate $\mathcal{S}_n$. We show that if $\pi_1,\pi_2,\pi_3$ are chosen randomly from $\mathcal{S}_n$ then, with probability tending to 1 as $n \rightarrow \infty$, they do not invariably generate $\mathcal{S}_n$. By contrast it was shown recently by Pemantle, Peres and Rivin that four...

Topics: Group Theory, Number Theory, Combinatorics, Mathematics

Source: http://arxiv.org/abs/1508.01870

55
55

Sep 20, 2013
09/13

by
Kevin Ford

texts

#
eye 55

#
favorite 0

#
comment 0

We give a relatively short proof of one of the central cases of the main theorem from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223. Namely, we determine the order of magnitude of the number of integers

Source: http://arxiv.org/abs/math/0607473v5

167
167

Sep 28, 2011
09/11

by
Mike Falcone, Thomas Green, Kevin Ford

audio

#
eye 167

#
favorite 0

#
comment 0

nostalgia podcast about pro wrestling action figures

Topics: wwf, wcw, galoob, hasbro, ljn, action figures, dolls, wrestling toys

86
86

Sep 21, 2013
09/13

by
Kevin Ford

texts

#
eye 86

#
favorite 0

#
comment 0

New version of my 1998 article. The method of proof of the main results follows the original, but there are many simplifications/streamlining of arguments, especially Lemma 3.6 (new Lemma 3.7). Fixed small error in proof of lower bound for V_k(x) (see the paragraph after (5.20)), fixed the statement and proof of Theorem 3. New, precise way to relate sums to volumes (Lemmas 3.1, 3.9) and provided full details in Section 6 of the proofs of Theorems 10-13. Slightly different versions of Theorems...

Source: http://arxiv.org/abs/1104.3264v2

81
81

Sep 23, 2013
09/13

by
Kevin Ford

audio

#
eye 81

#
favorite 0

#
comment 0

Marion Fontaine sits down with Kevin to discuss action figures and the EXTRAVAGANZA!

Topics: marion fontaine, wrestling podcast

38
38

Sep 21, 2013
09/13

by
Kevin Ford

texts

#
eye 38

#
favorite 0

#
comment 0

We give a simple method for estimating the average of singular series' associated with prime k-tuples.

Source: http://arxiv.org/abs/1108.3861v1

8
8.0

Dec 23, 2014
12/14

by
B.J. Maxson with J. Martin Giesen, Kevin Ford

texts

#
eye 8

#
favorite 0

#
comment 0

Topics: Labor and Employment, Rehabilitation

59
59

Sep 22, 2013
09/13

by
Kevin Ford; Florian Luca; Carl Pomerance

texts

#
eye 59

#
favorite 0

#
comment 0

We show that the equation phi(a)=\sigma(b) has infinitely many solutions, where phi is Euler's totient function and sigma is the sum-of-divisors function. This proves a 50-year old conjecture of Erdos. Moreover, we show that there are infinitely many integers n such that phi(a)=n and sigma(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...

Source: http://arxiv.org/abs/0906.3380v2

44
44

Sep 19, 2013
09/13

by
Kevin Ford; Florian Luca

texts

#
eye 44

#
favorite 0

#
comment 0

We study the question of whether for each n there is another integer m with lambda(m)=lambda(n), where lambda is Carmichael's function. We give a "near" proof of the fact that this is the case unconditionally, and a complete conditional proof under the Extended Riemann Hypothesis. The main tool is a count of prime chains from paper arXiv:0906.3380.

Source: http://arxiv.org/abs/1010.5442v2

4
4.0

Jun 30, 2018
06/18

by
Kevin Ford; Florian Luca; Carl Pomerance

texts

#
eye 4

#
favorite 0

#
comment 0

We show that the counting function of the set of values of the Carmichael $\lambda$-function is $x/(\log x)^{\eta+o(1)}$, where $\eta=1-(1+\log\log 2)/(\log 2)=0.08607...$.

Topics: Mathematics, Number Theory

Source: http://arxiv.org/abs/1408.6506

47
47

Dec 30, 2013
12/13

by
Kevin Ford

audio

#
eye 47

#
favorite 0

#
comment 0

Fan to Fan #19

Topics: kevin ford, podcast

36
36

Sep 19, 2013
09/13

by
Kevin Ford

texts

#
eye 36

#
favorite 0

#
comment 0

We give sharp, uniform estimates for the probability that a random walk of n steps on the reals avoids a half-line [y,infinity) given that it ends at the point x. The estimates hold for general continuous or lattice distributions provided the 4th moment is finite.

Source: http://arxiv.org/abs/math/0610450v4

14
14

Jun 30, 2018
06/18

by
Kevin Ford; Ben Green; Sergei Konyagin; Terence Tao

texts

#
eye 14

#
favorite 0

#
comment 0

Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$ where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions...

Topics: Mathematics, Number Theory

Source: http://arxiv.org/abs/1408.4505

64
64

Nov 4, 2013
11/13

by
Kevin Ford

audio

#
eye 64

#
favorite 0

#
comment 0

@Wrestlefolks @NotThatTomGreen @kford13

Topics: Kevin Ford, podcast, wrestlefolks

2
2.0

Jun 29, 2018
06/18

by
Kevin Ford

texts

#
eye 2

#
favorite 0

#
comment 0

Let $N(x,y)$ denote the number of integers $n\le x$ which are divisible by a shifted prime $p-1$ with $p>y$, $p$ prime. Improving upon recent bounds of McNew, Pollack and Pomerance, we establish the exact order of growth of $N(x,y)$ for all $x\ge 2y\ge 4$.

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1604.00281

7
7.0

Jun 28, 2018
06/18

by
Kevin Ford; James Maynard; Terence Tao

texts

#
eye 7

#
favorite 0

#
comment 0

Let $p_n$ denote the $n$-th prime, and for any $k \geq 1$ and sufficiently large $X$, define the quantity $$ G_k(X) := \max_{p_{n+k} \leq X} \min( p_{n+1}-p_n, \dots, p_{n+k}-p_{n+k-1} ),$$ which measures the occurrence of chains of $k$ consecutive large gaps of primes. Recently, with Green and Konyagin, the authors showed that \[ G_1(X) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}\] for sufficiently large $X$. In this note, we combine the arguments in that paper with the...

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1511.04468

40
40

Sep 23, 2013
09/13

by
Kevin Ford

texts

#
eye 40

#
favorite 0

#
comment 0

Suppose P is a set of primes, such that for every p in P, every prime factor of p-1 is also in P. If P does not contain all primes, we apply a new sieve method to show that 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.

Source: http://arxiv.org/abs/1212.3498v2

85
85

Dec 2, 2013
12/13

by
Kevin Ford

audio

#
eye 85

#
favorite 0

#
comment 0

@keithlipinski - @kford13

Topics: kevin ford, podcast

51
51

Sep 19, 2013
09/13

by
Kevin Ford; Yong Hu

texts

#
eye 51

#
favorite 0

#
comment 0

We study the distribution of divisors of Euler's totient function and Carmichael's function. In particular, we estimate how often the values of these functions have "dense" divisors.

Source: http://arxiv.org/abs/math/0703535v3

42
42

Sep 21, 2013
09/13

by
Kevin Ford; Florian Luca; Pieter Moree

texts

#
eye 42

#
favorite 0

#
comment 0

For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n\le x such that q does not divide phi(n). Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the prime k-tuples conjecture and a conjecture of Ihara about the distribution of these Euler-Kronecker constants cannot be both true.

Source: http://arxiv.org/abs/1108.3805v3

36
36

Sep 18, 2013
09/13

by
Kevin Ford

texts

#
eye 36

#
favorite 0

#
comment 0

We apply recent bounds of the author (math.PR/0609224) for generalized Smirnov statistics to the distribution of integers whose prime factors satisfy certain systems of inequalities.

Source: http://arxiv.org/abs/0704.1789v2

44
44

Sep 20, 2013
09/13

by
Kevin Ford; Gerald Tenenbaum

texts

#
eye 44

#
favorite 0

#
comment 0

Let H(x,y,z) be the number of integers $\le x$ with a divisor in (y,z] and let H_1(x,y,z) be the number of integers $\le x$ with exactly one such divisor. When y and z are close, it is expected that H_1(x,y,z) H(x,y,z), that is, an integer with a divisor in (y,z] usually has just one. We determine necessary and sufficient conditions on y and z so that H_1(x,y,z) H(x,y,z). In doing so, we answer an open question from the paper "The distribution of integers with a divisor in a given...

Source: http://arxiv.org/abs/math/0607460v3

100
100

Sep 19, 2013
09/13

by
Harold G. Diamond; Kevin Ford

texts

#
eye 100

#
favorite 0

#
comment 0

We define a family {$\gamma(P)$} of generalized Euler constants indexed by finite sets of primes $P$ and study their distribution. These arise from partial sums of reciprocals of integers not divisible by any prime in $P$. An apparent monotonicity is investigated. We also prove that a certain property of these numbers is equivalent to the Riemann Hypothesis.

Source: http://arxiv.org/abs/math/0703508v2

42
42

Sep 23, 2013
09/13

by
Louigi Addario-Berry; Kevin Ford

texts

#
eye 42

#
favorite 0

#
comment 0

We determine, to within O(1), the expected minimal position at level n in certain branching random walks. The walks under consideration have displacement vector (v_1,v_2,...), where each v_j is the sum of j independent Exponential(1) random variables and the different v_i need not be independent. In particular, our analysis applies to the Poisson-Dirichlet branching random walk and to the Poisson-weighted infinite tree. As a corollary, we also determine the expected height of a random recursive...

Source: http://arxiv.org/abs/1012.2544v3

49
49

Sep 23, 2013
09/13

by
Kevin Ford; Paul Pollack

texts

#
eye 49

#
favorite 0

#
comment 0

Let phi(n) be Euler's totient function and let sigma(n) be the sum of the positive divisors of n. We show that most phi-values (integers in the range of phi) are not sigma-values and vice versa.

Source: http://arxiv.org/abs/1012.0080v1

5
5.0

Jun 30, 2018
06/18

by
Kevin Ford; D. R. Heath-Brown; Sergei Konyagin

texts

#
eye 5

#
favorite 0

#
comment 0

For any positive integer $k$, we show that infinitely often, perfect $k$-th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size $$ c_k \frac{\log p \log_2 p \log_4 p}{(\log_3 p)^2}, $$ where $p$ is the smaller of the two primes.

Topics: Mathematics, Number Theory

Source: http://arxiv.org/abs/1411.6543

4
4.0

Jun 30, 2018
06/18

by
Kevin Ford; Ben Green; Sergei Konyagin; James Maynard; Terence Tao

texts

#
eye 4

#
favorite 0

#
comment 0

Let $p_n$ denotes the $n$-th prime. We prove that $$\max_{p_{n+1} \leq X} (p_{n+1}-p_n) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}$$ for sufficiently large $X$, improving upon recent bounds of the first three and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the R\"odl nibble method.

Topics: Mathematics, Number Theory, Combinatorics

Source: http://arxiv.org/abs/1412.5029

4
4.0

Jun 28, 2018
06/18

by
Kevin Ford; Moubariz Garaev; Sergei Konyagin

texts

#
eye 4

#
favorite 0

#
comment 0

Let $p$ be a prime and $p_1,\ldots, p_r$ be distinct prime divisors of $p-1$. We prove that the smallest positive integer $n$ which is a simultaneous $p_1,\ldots,p_r$-power nonresidue modulo $p$ satisfies $$ n

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1511.08428

46
46

Sep 19, 2013
09/13

by
Michael Filaseta; Kevin Ford; Sergei Konyagin; Carl Pomerance; Gang Yu

texts

#
eye 46

#
favorite 0

#
comment 0

An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\sum_{n\in S} 1/n$ is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirming a conjecture of Erdos and Selfridge. We also prove a conjecture of Erdos and Graham, that, for each fixed number K>1, the complement in the integers of any union of...

Source: http://arxiv.org/abs/math/0507374v3

37
37

Sep 22, 2013
09/13

by
Kevin Ford; Jason Sneed

texts

#
eye 37

#
favorite 0

#
comment 0

Under two assumptions, we determine the distribution of the difference between two functions each counting the numbers < x that are in a given arithmetic progression modulo q and the product of two primes. The two assumptions are (i) the Extended Riemann Hypothesis for Dirichlet L-functions modulo q, and (ii) that the imaginary parts of the nontrivial zeros of these L-functions are linearly independent over the rationals. Our results are analogs of similar results proved for primes in...

Source: http://arxiv.org/abs/0908.0093v2

140
140

Sep 18, 2013
09/13

by
Kevin Ford

texts

#
eye 140

#
favorite 0

#
comment 0

We determine the order of magnitude of H(x,y,z), the number of integers n\le x having a divisor in (y,z], for all x,y and z. We also study H_r(x,y,z), the number of integers n\le x having exactly r divisors in (y,z]. When r=1 we establish the order of magnitude of H_1(x,y,z) for all x,y,z satisfying z\le x^{0.49}. For every r\ge 2, $C>1$ and $\epsilon>0$, we determine the the order of magnitude of H_r(x,y,z) when y is large and y+y/(\log y)^{\log 4 -1 - \epsilon} \le z \le...

Source: http://arxiv.org/abs/math/0401223v5

70
70

Sep 21, 2013
09/13

by
Kevin Ford

texts

#
eye 70

#
favorite 0

#
comment 0

An old conjecture of Sierpinski asserts that for every integer k \ge 2, there is a number m for which the equation \phi(x)=m has exactly k solutions. Here \phi 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 Sierpinski's conjecture. The proof uses many results from sieve theory, in particular the famous theorem of Chen.

Source: http://arxiv.org/abs/math/9907204v1

142
142

Sep 18, 2013
09/13

by
Kevin Ford

texts

#
eye 142

#
favorite 0

#
comment 0

If a sequence $(a_n)$ of non-negative real numbers has ``best possible'' distribution in arithmetic progressions, Bombieri showed that one can deduce an asymptotic formula for the sum $\sum_{n\le x} a_n \Lambda_k(n)$ for $k\ge 2$. By constructing appropriate sequences, we show that any weakening of the well-distribution property is not sufficient to deduce the same conclusion.

Source: http://arxiv.org/abs/math/0401215v1

41
41

Sep 20, 2013
09/13

by
Kevin Ford; Igor Shparlinski

texts

#
eye 41

#
favorite 0

#
comment 0

We show that finite fields over which there is a curve of a given genus g with its Jacobian having a small exponent, are very rare. This extends a recent result of W. Duke in the case g=1. We also show when g=1 or g=2 that our bounds are best possible.

Source: http://arxiv.org/abs/math/0607474v3

232
232

Dec 3, 2017
12/17

by
Big Punisher; Beenie Man; Boyz II Men; Chanté Moore; Diana King; Jazzie B; Kevin Ford; Rufus Blaq; K‐Ci & JoJo; Lady Saw; Nadine Sutherland; Mary J. Blige; Maxi Priest; Me’Shell NdegéOcello; Shaggy; Janet Jackson; Soul II Soul; Wyclef Jean; Stevie Wonder

audio

#
eye 232

#
favorite 2

#
comment 0

Tracklist: 1. Jazzie B (intro) - Jazzie B 2. Mastablasta '98 - Wyclef Jean; Stevie Wonder 3. Luv Me, Luv Me - Shaggy; Janet Jackson 4. Beautiful - Mary J. Blige 5. Never Say Never Again - K‐Ci & JoJo 6. Makes Me Sweat - Big Punisher; Beenie Man 7. Your Home Is in My Heart (Stella's Love Theme) - Boyz II Men; Chanté Moore 8. Free Again - Soul II Soul 9. Make My Body Hot - Diana King 10. The Art of Seduction - Maxi Priest 11. Let Me Have You - Me’Shell NdegéOcello 12. Dance for Me -...

Topic: Compilation

Source: CD

47
47

Sep 22, 2013
09/13

by
Kevin Ford; K. Soundararajan; Alexandru Zaharescu

texts

#
eye 47

#
favorite 0

#
comment 0

We continue our investigation of the distribution of the fractional parts of $a \gamma$, where $a$ is a fixed non-zero real number and $\gamma$ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to pair correlation functions and the distribution of primes in short intervals. We also discuss analogous results for a more general L-function. This is a sequel to the paper math.NT/0405459.

Source: http://arxiv.org/abs/0805.2745v3

179
179

Sep 21, 2013
09/13

by
Yvonne Buttkewitz; Christian Elsholtz; Kevin Ford; Jan-Christoph Schlage-Puchta

texts

#
eye 179

#
favorite 0

#
comment 0

We determine asymptotically the maximal order of log d(d(n)), where d(n) is the number of positive divisors of n. This solves a problem first put forth by Ramanujan in 1915.

Source: http://arxiv.org/abs/1108.1815v1

56
56

Sep 20, 2013
09/13

by
Kevin Ford; Alexandru Zaharescu

texts

#
eye 56

#
favorite 0

#
comment 0

We investigate the distribution of the fractional parts of ag, where a is a fixed non-zero real number and g runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. The revision includes several minor corrections.

Source: http://arxiv.org/abs/math/0405459v2

98
98

Jul 20, 2013
07/13

by
Kevin Ford; Trevor D. Wooley

texts

#
eye 98

#
favorite 0

#
comment 0

We enhance the efficient congruencing method for estimating Vinogradov's integral for moments of order $2s$, with $1\le s\le k^2-1$. In this way, we prove the main conjecture for such even moments when $1\le s\le \tfrac{1}{4}(k+1)^2$, showing that the moments exhibit strongly diagonal behaviour in this range. There are improvements also for larger values of $s$, these finding application to the asymptotic formula in Waring's problem.

Source: http://arxiv.org/abs/1304.6917v2

13
13

Jun 28, 2018
06/18

by
Sean Eberhard; Kevin Ford; Ben Green

texts

#
eye 13

#
favorite 0

#
comment 0

Let $i(n,k)$ be the proportion of permutations $\pi\in\mathcal{S}_n$ having an invariant set of size $k$. In this note we adapt arguments of the second author to prove that $i(n,k) \asymp k^{-\delta} (1+\log k)^{-3/2}$ uniformly for $1\leq k\leq n/2$, where $\delta = 1 - \frac{1 + \log \log 2}{\log 2}$. As an application we show that the proportion of $\pi\in\mathcal{S}_n$ contained in a transitive subgroup not containing $\mathcal{A}_n$ is at least $n^{-\delta+o(1)}$ if $n$ is even.

Topics: Group Theory, Combinatorics, Mathematics

Source: http://arxiv.org/abs/1507.04465

In 2009 and in 2011, the Library of Congress made two of its largest authority files â€“ Subject Headings and Names â€“ available as linked data via LCâ€™s Linked Data Service, ID.LOC.GOV. Both are offered in MADS/RDF and SKOS. It is LCâ€™s objective, in 2012, to publish another of its largest authority files as linked data: LC Classification. Whereas the source records for Subject Headings and Names are encoded in the MARC Authority format, from which there is a relatively...

Topics: Library linked data, Library of Congress Classification, Ontology, SKOS, MARC21, Authority control

Source: https://www.jlis.it/article/view/5465

54
54

Jul 20, 2013
07/13

by
Kevin Ford

texts

#
eye 54

#
favorite 0

#
comment 0

We give sharp, uniform estimates for the probability that the empirical distribution function for n uniform-[0,1] random variables stays to one side of a given line.

Source: http://arxiv.org/abs/math/0609224v5

53
53

Sep 21, 2013
09/13

by
Kevin Ford; Sergei Konyagin; Youness Lamzouri

texts

#
eye 53

#
favorite 0

#
comment 0

We show, for any $q\ge 3$ and distinct reduced residues $a,b \pmod q$, the existence of certain hypothetical sets of zeros of Dirichlet $L$-functions lying off the critical line implies that $\pi(x;q,a)

Source: http://arxiv.org/abs/1204.6715v1

66
66

Sep 22, 2013
09/13

by
Emre Alkan; Kevin Ford; Alexandru Zaharescu

texts

#
eye 66

#
favorite 0

#
comment 0

We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.

Source: http://arxiv.org/abs/0805.2867v1

43
43

Sep 23, 2013
09/13

by
Kevin Ford; Sergei V. Konyagin; Florian Luca

texts

#
eye 43

#
favorite 0

#
comment 0

We study the distribution of prime chains, which are sequences p_1,...,p_k of primes for which p_{j+1}\equiv 1\pmod{p_j} for each j. We give estimates for the number of chains with p_k\le x (k variable), and the number of chains with p_1=p and p_k \le px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with p_k=p, which is also the height of the Pratt tree for p. We show H(p)\ge c\log\log p and H(p)\le (\log p)^{1-c'} for almost all p, with c,c'...

Source: http://arxiv.org/abs/0904.0473v4

83
83

Sep 18, 2013
09/13

by
Kevin Ford; Florian Luca; Igor E. Shparlinski

texts

#
eye 83

#
favorite 0

#
comment 0

Let P(k) be the largest prime factor of the positive integer k. In this paper, we prove that the series $\sum_{n\ge 1}\frac{(\log n)^a}{P(2^n-1)}$ is convergent for each constant a

Source: http://arxiv.org/abs/0704.1327v1

32
32

May 27, 2021
05/21

by
Kevin Ford

texts

#
eye 32

#
favorite 0

#
comment 0

Let S(n) be the smallest integer k so that nlk!. This is known as the Smarandache function and has been studied by many authors. If P( n) denotes the largest prime factor of n, it is clear that S(n)≥P(n).

49
49

Sep 20, 2013
09/13

by
Kevin Ford; Gerald Tenenbaum

texts

#
eye 49

#
favorite 0

#
comment 0

We study large partial sums, localized with respect to the sums of variances, of a sequence of centered random variables. An application is given to the distribution of prime factors of typical integers.

Source: http://arxiv.org/abs/math/0608411v2