Sharp Transitions in Making Squares (2009)
Ernie Croot, Andrew Granville, Robin Pemantle, Prasad Tetali
Supported by an NSF grant.
Andrew Granville, Thomas J. Tucker
In this age in which mathematicians are supposed to bring their research into the classroom, even at the most elementary level, it is rare that we can turn the tables and use our elementary teaching...
Square Root of Values of a Given Polynomial (2009)
Pamela Cutter, Andrew Granville, Thomas J. Tucker
Abstract. The abc-conjecture is applied to various questions involving the number of distinct fields Q ( √ f (n) ) , as we vary over integers n. 1
Andrew Granville, Thomas J. Tucker
In this age in which mathematicians are supposed to bring their research into the classroom, even at the most elementary level, it is rare that we can turn the tables and use our elementary teaching...
Sharp Transitions in Making Squares (2009)
Ernie Croot, Andrew Granville, Robin Pemantle, Prasad Tetali
2 Partiellement soutenu par une bourse de la Conseil de recherches en sciences naturelles et en génie du Canada. 3
Prime Factors of Dynamical Sequences (2009)
Faber, Xander, Granville, Andrew
Let f(t) be a rational function of degree at least 2 with rational coefficients. For a given rational number x_0, define x_{n+1}=f(x_n) for each nonnegative integer n. If this sequence is not...
ON THE EQUATIONS z m = F{x,y) AND Ax? + By * = Cz r (2009)
Henri Darmon, Andrew Granville
We investigate integer solutions of the superelliptic equation z m = F(x,y), (1) where F is a homogeneous polynomial with integer coefficients, and of the generalized Fermat equation Ax " +...
An introduction to additive combinatorics (2008)
This is a slightly expanded write-up of my three lectures at the Additive Combinatorics school. In the first lecture we introduce some of the basic material in Additive Combinatorics, and in the next...
Andrew Granville, Carl Pomerance, Andrew Granville, Carl Pomerance
Abstract. Erd}os [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x...
Sharp Transitions in Making Squares (2008)
Croot, Ernie, Granville, Andrew, Pemantle, Robin, Tetali, Prasad
In many integer factoring algorithms, one produces a sequence of integers (created in a pseudo-random way), and wishes to rapidly determine a subsequence whose product is a square (which we call a...
The distribution of the zeroes of random trigonometric polynomials (2008)
Granville, Andrew, Wigman, Igor
We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\to\infty$. It is known that as $N$ grows to infinity, the expected number...
Prime Possibilities and Quantum Chaos (2008)
David Eisenbud, Andrew Granville
and Non-Abelian Hodge Theory are drawing to a close. Once again I hear from departing members of all the work they did here, all the mathematical contacts they made, the excitement of the program,...
Smooth numbers: computational number theory and beyond (2008)
Abstract. A central topic in the analysis of number theoretic algorithms is the role usually played by integers which have only small prime factors. Such integers are known as “smooth numbers”....
Cycle lengths in a permutation are typically Poisson (2008)
The set of cycle lengths of almost all permutations in Sn are “Poisson distributed”: we show that this remains true even when we restrict the number of cycles in the permutation. The formulas we...
On the Least Prime in Certain Arithmetic Progressions (2008)
Andrew Granville, Carl Pomerance
We find infinitely many pairs of coprime integers, a and q, such that the least prime j a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other...
Subdesigns in Steiner Quadruple Systems (2008)
Andrew Granville, Alan Hartman
A Steiner quadruple system of order v, denoted SQS(v), is a pair (X; B) where X is a set of cardinality v, and B is a set of 4-subsets of X (called blocks), with the property that any 3-subset of X...
Running time predictions for factoring algorithms (2008)
Ernie Croot, Andrew Granville, Robin Pemantle, Prasad Tetali
Partiellement soutenu par une bourse de la Conseil de recherches en sciences naturelles et en génie du Canada. 3 Supported in part by NSF Grant DMS-01-03635. In 1994, Carl Pomerance proposed the...
UNIT FRACTIONS AND THE CLASS NUMBER OF A CYCLOTOMIC FIELD (2007)
Abstract. We further examine Kummer's incorrect conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field, h 1 (p). Whereas Kummer had...
Andrew Granville, Carl Pomerance, Paul Erdos
Abstract. Erdos [8] conjectured that there are x
It is currently very much in vogue to study sums of the form i(p 1; p 2; : : : ; p g):= X
Product of integers in an interval, modulo squares (2007)
Andrew Granville, J. L. Selfridge
Abstract: We prove a conjecture of Irving Kaplansky which asserts that between any pair of consecutive positive squares there is a set of distinct integers whose product is twice a square. This...
A BINARY ADDITIVE PROBLEM OF ERD OS AND THE ORDER OF 2 MOD p 2 (2007)
Andrew Granville, K. Soundararajan
We'd like to thank Paul Erdos for the questions Abstract. We show that the problem of representing every odd positive integer as the sum of a squarefree number and a power of 2, is strongly...
An Upper Bound on the Least Inert Prime in a Real Quadratic Field (2007)
Andrew Granville, R. A. Mollin, H. C. Williams
It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant D? 3705, there is always at least one prime p! p D=2 such that the Kronecker symbol
Bradley W. Brock, Andrew Granville
On average, there are q r + o(q r=2) F q r-rational points on curves of genus g defined over F q r. This is also true if we restrict our average to genus g curves defined over Fq, provided r is odd...
Andrew Granville, Daniel Shiu, Peter Shiu
Given a prime p and distinct non-zero integers a 1; a 2; : : : ; a k (mod p), we investigate
THE LEAST COMMON MULTIPLE AND LATTICE POINTS ON HYPERBOLAS (2007)
Abstract. We bound, from below, the least common multiple of k integers from a short interval. This is used to bound the length of an arc \Gamma of the hyperbola, xy = N, containing k integer lattice...
In my paper [1], we studied Pascal's Triangle modulo 2, 4, 8 and 16; and, in particular, its self-similar structure. It is well-known that the number of entries j 1 mod 2 in the nth row of...
Dedicated to the memory of Paul Erdos Abstract. Assuming the abc-conjecture we show that there are only finitely many powerful binomial coefficients \Gamma
Oesterl'e and Masser's abc-conjecture asserts that for any given " ? 0, if a; b and
Andrew Granville, K. Soundararajan
Dedicated to the memory of S. D. Chowla Abstract. In this article, we describe and motivate some of the results and notions from our ongoing project [2]. The results stated here are substantially new...
The Distribution of values of L(1 (2007)
Andrew Granville, K. Soundararajan
The values of L(1;), where is a non-principal character (mod q), occupy a central role in number theory. In particular, significant improvements to Siegel's (as
Andrew Granville, Richard A. Mollin
In the late eighteenth century both Euler and Legendre noticed that n
Abstract. The Cunningham project seeks to factor numbers of the form b n (2007)
Andrew Granville, Peter Pleasants
\Sigma 1 with b = 2; 3; : : : small. One of the most useful techniques is Aurifeuillian Factorization whereby such a number is partially factored by replacing b by a polynomial in such a way that...
Zeros of Fekete polynomials (2007)
Brian Conrey, Andrew Granville, Bjorn Poonen, K. Soundararajan
Dirichlet noted that, from the formula \Gamma(s) = n
DECAY OF MEAN-VALUES OF MULTIPLICATIVE FUNCTIONS (2007)
Andrew Granville, K. Soundararajan
Given a multiplicative function f with jf(n)j 1 for all n, we are concerned with obtaining
Andrew Granville, K. Soundararajan
2 The natural and logarithmic densities of m-th power residues
Decay of Mean-Values of Multiplicative Functions (2007)
Andrew Granville, K. Soundararajan
Introduction Given a multiplicative function f with jf(n)j 1 for all n, we are concerned with obtaining explicit upper bounds on the mean-value 1 x j P nx f(n)j. Ideally, one would like to give a...
The Lattice Points of an N-Dimensional Tetrahedron (2007)
this paper we consider the set of integer lattice points inside or on the boundary of the n--dimensional tetrahedron bounded by the hyperplanes (1:1) X 1 = 0; X 2 = 0; : : : ; Xn = 0 and (1:2) w 1 X...
Smoothing "Smooth" Numbers (2007)
Andrew Granville, John B. Friedlander, John B. Friedl, Andrew Granville
: An integer is called y-smooth if all of its prime factors are y. An important problem is to show that the y-smooth integers up to x are equi-distributed amongst short intervals. In particular, for...
Integers, without large prime factors, in arithmetic progressions, II (2007)
: We show that, for any fixed " ? 0, there are asymptotically the same number of integers up to x, that are composed only of primes y, in each arithmetic progression (mod q), provided that y q...
Least Primes in Arithmetic Progressions (2007)
For a fixed non-zero integer a and increasing function f , we investigate the lower density of the set of integers q for which the least prime in the arithmetic progression a(mod q) is less than...
On The Number Of Solutions To The Generalized Fermat Equation (2007)
. We discuss the maximum number of distinct non-trivial solutions that a generalized Fermat equation Ax n + By n = Cz n might possibly have. The abc- conjecture implies that it can never have more...
The Kummer-Wieferich-Skula Approach to the First Case of Fermat's Last Theorem (2007)
this paper that (1.1) does have solutions and then deduce a variety of implausible consequences. For example, in 1857 Kummer showed that if (1.1) has a solution then, for all n in the range 2 n p...
On the Number of Co-Prime-Free Sets. (2007)
Neil J. Calkin, Andrew Granville
: For a variety of arithmetic properties P (such as the one in the title) we investigate the number of subsets of the positive integers x, that have that property. In so doing we answer some...
: For a given polynomial f we use `local' methods to find exponents k for which there are no non--trivial integer solutions x 1 ; x 2 ; : : : ; xn to the Diophantine equation f(x k 1 ; x k 2 ; :...
On Sparse Languages Such That LL= \Sigma* (2007)
Per Enflo, Andrew Granville, Jeffrey Shallit, Sheng Yu
. A language L 2 \Sigma is said to be sparse if L contains a vanishingly small fraction of all possible strings of length n in \Sigma . C. Ponder asked if there exists a sparse language L such that...
Integers, without large prime factors, in arithmetic progressions, I (2007)
: We give a variety of estimates for the number of integers, free of large prime factors, in arithmetic progressions. In particular we show that we get approximately the same number of integers up to...
On Positive Integers ≤x with Prime Factors ≤t log x (2007)
. It is not difficult to estimate the function /(x; y), which counts integers x, free of prime factors ? y, by "smooth" functions whenever y log 1=2 x or y is a fixed power of x. This can...
A Note on Sums of Primes (2007)
: Under the assumption of the prime k--tuplets conjecture we show that it is possible to construct an infinite sequence of integers, such that the average of any two is prime. Recently Pomerance,...
Andrew Granville, J. L. Selfridge
Abstract: We prove a conjecture of Irving Kaplansky which asserts that between any pair of consecutive positive squares there is a set of distinct integers whose product is twice a square. Along...
ERY-LIKE FORMULAE FOR i(4n + 3) (2007)
Gert Almkvist, Andrew Granville
The Riemann zeta-function is defined by i(s):=
Andrew Granville, J. L. Selfridge
Abstract: We prove a conjecture of Irving Kaplansky which asserts that between any pair of consecutive positive squares there is a set of distinct integers whose product is twice a square. Along...
F (x Y, Henri Darmon, Andrew Granville
Abstract: We investigate integer solutions of the superelliptic equation (1) z m = F (x; y); where F is a homogenous polynomial with integer coecients, and of the generalized Fermat equation (2) Ax
Doron Zeilberger, David Bressoud, Gaurav Bhatnagar, Anders Bjorner, Jonathan Borwein, Francesco Brenti, ...
Two stones build two houses. Three build six houses. Four build four and twenty houses. Five build hundred and twenty houses. Six build Seven hundreds and twenty houses. Seven build five thousands...
On Finite Sets Which Tile the Integers (2007)
Andrew Granville Izabella, Andrew Granville, Yang Wang
this paper we will assume that A is finite. It is well known (see [7]) that any tiling of Z by a finite set A must be periodic: C = B +MZ for some finite set B Z such that = M . W then write A#B =...
by Andrew Granville, Daniel Shiu and Peter Shiu (2007)
Leicestershire Le Tu, Andrew Granville, Andrew Granville, Andrew Granville, Daniel Shiu, ...
this paper is to investigate the spectrum of possible values, and the accumulation points of the spectrum, for k = 3 and 4. Our methods do not yield much for larger k
Multiplicative functions in arithmetic progressions (2007)
Balog, Antal, Granville, Andrew, Soundararajan, K.
We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.
An uncertainty principle for arithmetic sequences (2007)
Granville, Andrew, Soundararajan, K.
Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are ¿well-distributed¿ in some appropriate sense. In various discrepancy problems, combinatorics...
Rational and Integral Points on Quadratic Twists of a Given Hyperelliptic Curve (2007)
We show that the abc-conjecture implies that few quadratic twists of a given hyperelliptic curve have any non-trivial rational or integral points; and indicate how these considerations dovetail with...
On the Least Prime in Certain Arithmetic Progressions (2006)
Granville, Andrew, Pomerance, Carl
We find infinitely many pairs of coprime integers, a and q, such that the least prime congruent to a (modulo q) is unusually large. In so doing we also consider the question of approximating...
Estimates for representation numbers of quadratic forms (2006)
Blomer, Valentin, Granville, Andrew
Let $f$ be a primitive positive integral binary quadratic form of discriminant $-D$, and let $r_f(n)$ be the number of representations of $n$ by $f$ up to automorphisms of $f$. In this article, we...
Pretentious multiplicative functions and an inequality for the zeta-function (2006)
Granville, Andrew, Soundararajan, K.
We note how several central results in multiplicative number theory may be rephrased naturally in terms of multiplicative functions $f$ that pretend to be another multiplicative function $g$. We...
Lattice points on circles, squares in arithmetic progressions and sumsets of squares (2006)
Cilleruelo, Javier, Granville, Andrew
Rudin conjectured that there are never more than c N^(1/2) squares in an arithmetic progression of length N. Motivated by this surprisingly difficult problem we formulate more than twenty conjectures...
Sieving and the Erd{\H o}s-Kac theorem (2006)
Granville, Andrew, Soundararajan, K.
We give a relatively easy proof of the Erd\H os-Kac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in...
Aurifeuillian Factorization (2006)
Granville, Andrew, Pleasants, Peter
The Cunningham project seeks to factor numbers of the form bn±1 with b = 2, 3, . . . small. One of the most useful techniques is Aurifeuillian Factorization whereby such a number is partially...
Aurifeuillian Factorization (2006)
Granville, Andrew, Pleasants, Peter
The Cunningham project seeks to factor numbers of the form bn±1 with b = 2, 3, . . . small. One of the most useful techniques is Aurifeuillian Factorization whereby such a number is partially...
Sharp Transitions in Making Squares (2006)
Ernie Croot, Andrew Granville, Prasad Tetali
In many integer factoring algorithms, one produces a sequence of integers (created in a pseudo-random way), and wishes to determine a subsequence whose product is a square. A good model for how this...
Sharp Transitions in Making Squares (2006)
Ernie Croot, Andrew Granville, Prasad Tetali
In many integer factoring algorithms, one produces a sequence of integers (created in a pseudo-random way), and wishes to determine a subsequence whose product is a square. A good model for how this...
Negative values of truncations to $L(1,\chi)$ (2005)
Granville, Andrew, Soundararajan, K.
For a given $x$ we consider the minimum of $\sum_{n\le x} \chi(n)/n$ as $\chi$ ranges over all quadratic Dirichlet characters. For all large $x$, this minimum is negative and we give upper and lower...
Large character sums: Pretentious characters and the Polya-Vinogradov Theorem (2005)
Granville, Andrew, Soundararajan, K.
In 1918 Polya and Vinogradov gave an upper bound for the maximal size of character sums which still remains the best known general estimate. One of the main results of this paper provides a...
Extreme values of $|\zeta(1+it)|$ (2005)
Granville, Andrew, Soundararajan, K.
Improving a result of N. Levinson, we exhibit large and small values of $|\zeta(1+it)|$.
It is easy to determine whether a given integer is prime (2005)
Dedicated to the memory of W. ‘Red ’ Alford, friend and colleague Abstract. “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime...
Granville, Andrew, Martin, Greg
This is a survey article on prime number races. Chebyshev noticed in the first half of the nineteenth century that for any given value of x, there always seem to be more primes of the form 4n+3 less...
An uncertainty principle for arithmetic sequences (2004)
Granville, Andrew, Soundararajan, K.
Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are ``well-distributed'' in some appropriate sense. In various discrepancy problems, combinatorics...
Errata to: “The distribution of values of L(1, χ d )”, in GAFA 13:5 (2003) (2004)
Granville, Andrew, Soundararajan, K.
((no abstract.))
and Lasting Consequences (2004)
David Eisenbud, John W. Morgan, Margaret H. Wright, Thomas C. Hales, Andrew Granville, Margaret H. Wright
Interior methods are a pervasive feature of the optimization landscape today, but it was not always so. Although interior-point techniques, primarily in the form of barrier methods, were widely used...
and Lasting Consequences (2004)
David Eisenbud, John W. Morgan, Margaret H. Wright, Thomas C. Hales, Andrew Granville, Margaret H. Wright
Interior methods are a pervasive feature of the optimization landscape today, but it was not always so. Although interior-point techniques, primarily in the form of barrier methods, were widely used...
The distribution of values of L(1, χ d ) (2003)
Granville, Andrew, Soundararajan, K.
((no abstract))
The number of unsieved integers up to x (2003)
Granville, Andrew, Soundararajan, Kannan
Typically, one expects that there are around x\prod_{p\not\in P, p
The distribution of values of L(1,chi_d) (2002)
Granville, Andrew, Soundararajan, Kannan
In this paper we investigate the distribution of values of L(1,chi) as chi ranges over primitive real characters. In particular we focus on the extent to which this distribution may be approximated...
The number of fields generated by the square root of values of a given polynomial (2002)
Pamela Cutter, Andrew Granville, Thomas J. Tucker
Abstract. The abc-conjecture is applied to various questions involving the number of distinct fields Q( p f(n)), as we vary over integers n.
The number of fields generated by the square root of values of a given polynomial (2002)
Pamela Cutter, Andrew Granville, Thomas J. Tucker
Abstract. The abc-conjecture is applied to various questions involving the number of distinct fields Q( p f(n)), as we vary over integers n.
A characterization of finite sets that tile the integers (2001)
Granville, Andrew, Laba, Izabella, Wang, Yang
We consider the problem of characterizing finite sets which tile the integers by translations. Coven and Meyerowitz (J. Algebra 1999) found necessary and sufficient conditions for a finite set A to...
Upper bounds for |L(1,chi)| (2001)
Granville, Andrew, Soundararajan, Kannan
Given a non-principal Dirichlet character chi mod q, an important problem in number theory is to obtain good estimates for the size of L(1,chi). In this paper we focus on sharpening the upper bounds...
Zeros of Fekete polynomials (2000)
Conrey, Brian, Granville, Andrew, Poonen, Bjorn, Soundararajan, K.
The least common multiple and lattice points on hyperbolas (2000)
Granville, Andrew, Jiménez-urroz, Jorge
We bound, from below, the least common multiple of k integers from a short interval. This is used to bound the length of an arc Γ of the hyperbola, xy = N, containing k integer lattice points.
Decay of mean-values of multiplicative functions (1999)
Granville, Andrew, Soundararajan, K.
Given a multiplicative function f satisfying |f(n)|
The spectrum of multiplicative functions (1999)
Granville, Andrew, Soundararajan, K.
Let S be a subset of the unit disk, and let F(s) denote the class of completely multiplicative functions f such that f(p) is in S for all primes p. The authors' main concern is which numbers arise as...
Zeros of Fekete polynomials (1999)
Conrey, J. Brian, Granville, Andrew, Poonen, Bjorn, Soundararajan, K.
The authors study the distribution of zeros of the Fekete polynomial f_p(t) (defined for p prime) as p -> infinity. They show that asymptotically a constant fraction of the zeros lie on the unit...
Granville, Andrew, Soundararajan, K.
Assuming the Generalized Riemann Hypothesis, the authors study when a character sum over all n infinity and q -> infinity (q is the size of the finite field).
Borwein and Bradley's Apéry-like formulae for {$\zeta(4n+3)$} (1999)
Almkvist, Gert, Granville, Andrew
We prove a formula for $\zeta(4n+3)$ discovered by Borwein and Bradley (Experimental Mathematics 6:3 (1997), 181-194).
The set of differences of a given set (1999)
Andrew Granville, Friedrich Roesler
A central problem of combinatorial geometry and additive number theory is to understand the set of sums or differences of a given set of vectors. For example, given a set of m arbitrary vectors A,...
Andrew Granville, K. Soundararajan
this paper we investigate the distribution of the size of character sums, and in particular in what range the estimate (1) should hold. For example on this question we prove:
ABC allows us to count squarefrees (1998)
Dedicated to the memory of Paul Erdos Abstract. We show several consequences of the abc-conjecture for questions in analytic number theory which were of interest to Paul Erdos: For any given...
Defect zero p\Gammablocks for finite simple groups (1996)
Abstract. We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a p-block with defect 0, completing an investigation of many authors. The only finite simple groups...
Explicit Bounds on Exponential Sums and the Scarcity of Squarefree Binomial Coefficients (1996)
Andrew Granville, Olivier Ramaré
This paper fills what we believe to be a lacuna in the existing literature concerning upper bounds on exponential sums. Although it has always been evident that many of the known estimates can be...
Values of Bernoulli polynomials (1996)
this paper to determine the value of
Harald Cramér and the distribution of prime numbers (1995)
“It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began...
Proof Of The Alternating Sign Matrix Conjecture (1995)
Doron Zeilberger, Gert Almkvist, Noga Alon, George Andrews, Dror Bar-natan, Francois Bergeron, ...
: The number of n n matrices whose entries are either -1, 0, or 1, whose row- and column- sums are all 1, and such that in every row and every column the non-zero entries alternate in sign, is proved...
On the equations z^m = F(x,y) and Ax^p + By^q = Cz^r (1995)
Henri Darmon, Andrew Granville
We investigate integer solutions of the superelliptic equation (1) z m = F (x; y); where F is a homogeneous polynomial with integer coefficients, and of the generalized Fermat equation (2) Ax p + By...
On the equations z m = F (x, y) and Ax p + By q = Cz r (1995)
Henri Darmon, Andrew Granville
Abstract: We investigate integer solutions of the superelliptic equation (1) zm = F (x, y), where F is a homogenous polynomial with integer coefficients, and
An upper bound in Goldbach's problem (1993)
Jean-Marc Deshouillers, Andrew Granville, Wladyslaw Narkiewicz, Carl Pomerance
: It is clear that the number of distinct representations of a number n as the sum of two primes is at most the number of primes in the interval [n=2; n \Gamma 2]. We show that 210 is the largest...
: We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that...
Squares in Arithmetic Progressions (1992)
Enrico Bombieri, Andrew Granville, János Pintz
this paper we improve the above upper bound, though we are still far from proving Rudin's conjecture that Q(N) i p
Limitations to the Equi-distribution of Primes III (1992)
John Friedlander, Andrew Granville
: In an earlier paper [FG] we showed that the expected asymptotic formula ß(x; q; a) ¸ ß(x)=OE(q) does not hold uniformly in the range q ! x= log N x, for any fixed N ? 0. There are several...
The prime factors of Wendt's Binomial Circulant Determinant (1991)
: Wm , Wendt's binomial circulant determinant, is the determinant of an m by m circulant matrix of integers, with (i; j)th entry i m ji\Gammajj j whenever 2 divides m but 3 does not. We explain...
On the Size of the First Factor of the Class Number of a Cyclotomic Field (1990)
We show that Kummer's conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field is untrue under the assumption of two well-known and widely...
A polynomial Time Algorithm for (1982)
A.Shamir, W. R. Alford, Andrew Granville, Carl Pomerance
this paper we show that C(x) ? x
Brian Conrey, Andrew Granville, Bjorn Poonen, K. Soundararajan
this paper we shall study the complex zeros of f p (t). Using zero locating software one finds that, for primes p up to 1000, about half of the zeros lie on the unit circle; leading one to expect...
The Spectrum of Multiplicative Functions
Andrew Granville, K. Soundararajan
this paper is to understand the spectrum. Although we can determine the spectrum explicitly only in one interesting case (where S = [\Gamma1; 1]), we are able, in general, to qualitatively describe...