Kevin Ford

SHARP PROBABILITY ESTIMATES FOR GENERALIZED SMIRNOV STATISTICS (2008)

Abstract. We prove an explicit sieve upper bound based on the large sieve of Montgomery and Vaughan [MV], and apply it to show that σ(φ(m)) � m/39.4 for all positive integers m. 1.

On curves over finite fields with Jacobians with small exponent, Int (2008)

Dedicated to the memory of Walter Philipp Abstract. We give sharp, uniform estimates for the probability that the empirical distribution function for n uniform-[0, 1] random variables stays to one...

Kevin Ford, Igor Shparlinski

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

Abstract Localized large sums of random variables (2008)

Kevin Ford, Gérald Tenenbaum

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 WITH A DIVISOR IN (y, 2y] (2008)

The distribution of integers with at least two divisors in a short interval, Quart (2008)

Abstract. We determine, up to multiplicative constants, how many integers n ≤ x have a divisor in (y, 2y]. 1.

Kevin Ford, Gérald Tenenbaum

Abstract. We estimate the density of integers which have more than one divisor in an interval (y, z] with z ≈ y + y/(log y) log 4−1. As a consequence, we determine the precise range of z such...

Sharp probability estimates for random walks with barriers. 2006. Preprint available on the ArXiv at http://front.math.ucdavis.edu/math.PR/0610450 (2008)

On two conjectures of Sierpiński concerning the (2008)

Let X1, X2,... be independent, identically distributed random variables with mean EX1 = 0 and variance EX 2 1 = 1. Let S0 = T0 = 0 and for n ≥ 1 define

Kevin Ford, Sergei Konyagin

arithmetic functions σ and φ

On the distribution of imaginary parts of zeros of the Riemann zeta function, II (2008)

Ford, Kevin, Soundararajan, K., Zaharescu, Alexandru

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...

Diophantine approximation with arithmetic functions, I (2008)

Alkan, Emre, Ford, Kevin, Zaharescu, Alexandru

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.

THE DISTRIBUTION OF INTEGERS WITH AT LEAST TWO DIVISORS IN A SHORT INTERVAL (2007)

Generalized Smirnov statistics and the distribution of prime factors (2007)

We estimate the density of integers which have more than one divisor in an interval (y, z] with z ≈ y + y/(log y)log 4 − 1. As a consequence, we determine the precise range of z such that most...

On the largest prime factor of the Mersenne numbers (2007)

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.

Ford, Kevin, Luca, Florian, Shparlinski, Igor E.

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

Divisors of the Euler and Carmichael functions (2007)

Ford, Kevin, Hu, Yong

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.

Generalized Euler constants (2007)

Diamond, Harold G., Ford, Kevin

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...

Divisors of the Euler and Carmichael functions (2007)

Kevin Ford, Yong Hu

Two of the most studied functions in the theory of numbers are Euler’s totient function φ(n) and

Printed in the United Kingdom c ○ 2008 Cambridge Philosophical Society Generalized Euler constants (2007)

G. Diamond, Kevin Ford

We study the distribution of a family {γ(P)} of generalized Euler constants arising from integers sieved by finite sets of primes P. For P = Pr, the set of the first r primes, γ(Pr) → exp(−γ)...

Sharp probability estimates for random walks with barriers (2006)

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...

Sharp probability estimates for generalized Smirnov statistics (2006)

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...

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.

The distribution of integers with at least two divisors in a short interval (2006)

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...

We determine the true order of magnitude of the number of integers not exceeding x which have a prescribed number of divisors in a short interval, precisely defined in the text.

The distribution of integers with at least two divisors in a short interval (2006)

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...

We determine the true order of magnitude of the number of integers not exceeding x which have a prescribed number of divisors in a short interval, precisely defined in the text.

Ford, Kevin, Tenenbaum, Gerald

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...

The distribution of integers with at least two divisors in a short interval (2006)

Ford, Kevin, Tenenbaum, Gerald

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...

Integers with a divisor in (y,2y] (2006)

On curves over finite fields with Jacobians of small exponent (2006)

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...

Ford, Kevin, Shparlinski, Igor

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...

The distribution of integers with at least two divisors in a short interval (2006)

We determine the true order of magnitude of the number of integers not exceeding x which have a prescribed number of divisors in a short interval, precisely defined in the text.

The distribution of integers with at least two divisors in a short interval (2006)

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...

We determine the true order of magnitude of the number of integers not exceeding x which have a prescribed number of divisors in a short interval, precisely defined in the text.

SIEVING BY LARGE INTEGERS AND COVERING SYSTEMS OF CONGRUENCES (2006)

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...

Michael Filaseta, Sergei Konyagin, Gang Yu, Kevin Ford, Carl Pomerance

An old question of Erdős 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 ∈ S whose union is Z. We prove that if � n∈S...

Sieving by large integers and covering systems of congruences (2005)

Filaseta, Michael, Ford, Kevin, Konyagin, Sergei, Pomerance, Carl, Yu, Gang

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...

On the distribution of imaginary parts of zeros of the Riemann zeta function (2004)

Ford, Kevin, Zaharescu, Alexandru

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...

The distribution of integers with a divisor in a given interval (2004)

On Bombieri's asymptotic sieve (2004)

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...

Monatsh. Math. 146, 1–19 (2005) DOI 10.1007/s00605-005-0302-7 Values of the Euler Function in Various Sequences By (2004)

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}...

William D. Banks, Kevin Ford, Florian Luca

Abstract. Let ’ðnÞ and ðnÞ denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ’ðnÞ r ðnÞ s,wherer5s51are fixed positive integers. We also...

The prime number race and zeros of L-functions off the critical line (2002)

Ford, Kevin, Konyagin, Sergei

We examine the effects of certain hypothetical configurations of zeros of Dirichlet L-functions lying off the critical line on the distribution of primes in arithmetic progressions.

Zeros of Dirichlet L-Functions near the Real Axis and Chebyshev's Bias (2001)

Carter Bays, Kevin Ford, Richard H. Hudson, Michael Rubinstein

Regional Dendritic and Spine Variation in Human Cerebral Cortex: a Quantitative Golgi Study (2001)

Jacobs, Bob, Schall, Matthew, Prather, Melissa, Kapler, Elisa, Driscoll, Lori, Baca, Serapio, ...

The present study explored differences in dendritic/spine extent across several human cortical regions. Specifically, the basilar dendrites/spines of supragranular pyramidal cells were examined in...

The number of solutions of {$\phi(x)=m$} (1999)

The number of solutions of phi(x)=m (1999)

An old conjecture of Sierpiński 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....

The number of solutions of f (x) = m (1999)

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,...

Residue classes free of values of Euler’s function (1999)

Kevin Ford, Sergei Konyagin, Carl Pomerance

Dedicated to Andrzej Schinzel on his sixtieth birthday By a totient we mean a value taken by Euler’s function φ(n). Dence and Pomerance [DP] have established Theorem A. If a residue class contains...

DNA sequence preferences for an intercalating porphyin compound revealed by footprinting (1987)

Ford, Kevin, Fox, Keith R., Neidle, Stephen, Waring, Michael J.

The DNA sequence preferences of the compound peso -tetra- (4-N-methyl (pyrldyl) porphyrin and Its nickel complex have been Investigated by means of footprinting experiments on several DNA fragments,...

Localized large sums of random variables