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An asymptotic formula for the number of smooth values of a polynomial
 J. Number Theory
, 1999
"... Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from mult ..."
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Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from multiplicative questions to Waring’s problem to complexity
Lower bounds for the number of smooth values of a polynomial
, 1998
"... We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct ord ..."
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Cited by 3 (1 self)
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We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct order of magnitude has hitherto been established only in a few special cases. The purpose of this paper is to provide such a lower bound for an arbitrary polynomial. Various generalizations to subsets of the set of values taken by a polynomial are also obtained.
ON UPPER BOUNDS OF CHALK AND HUA FOR EXPONENTIAL SUMS
, 2001
"... Let f be a polynomial of degree d with integer coefficients, p any prime, m any positive integer and S(f, pm) the exponential sum S(f, pm)= � m p x=1 epm(f(x)). We establish that if f is nonconstant when read (mod p), then S(f,pm)  ≤ 4.41p m(1 − 1 d). Let t = ordp(f ′), let α be a zero of the co ..."
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Let f be a polynomial of degree d with integer coefficients, p any prime, m any positive integer and S(f, pm) the exponential sum S(f, pm)= � m p x=1 epm(f(x)). We establish that if f is nonconstant when read (mod p), then S(f,pm)  ≤ 4.41p m(1 − 1 d). Let t = ordp(f ′), let α be a zero of the congruence p−tf ′ (x) ≡ 0(modp) of multiplicity ν and let Sα(f, pm)bethe sum S(f, pm) with x restricted to values congruent to α (mod pm). We obtain Sα(f, pm) ≤ min {ν, 3.06}p t m(1 − 1 ν+1 p ν+1) for p odd, m ≥ t+2 and dp(f) ≥ 1. If, in addition, p ≥ (d − 1) (2d)/(d−2) , then we obtain the sharp upper bound Sα(f, pm m(1 − 1)≤p ν+1).
PAIRS OF DIAGONAL QUADRATIC FORMS AND LINEAR CORRELATIONS AMONG SUMS OF TWO SQUARES
"... Abstract. For suitable pairs of diagonal quadratic forms in 8 variables we use the circle method to investigate the density of simultaneous integer solutions and relate this to the problem of estimating linear correlations among sums of two squares. 1. ..."
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Abstract. For suitable pairs of diagonal quadratic forms in 8 variables we use the circle method to investigate the density of simultaneous integer solutions and relate this to the problem of estimating linear correlations among sums of two squares. 1.
FACTORING NEWPARTS OF JACOBIANS OF CERTAIN MODULAR CURVES
, 2009
"... We prove a conjecture of Yamauchi which states that the level N for which the new part of J0(N) is Qisogenous to a product of elliptic curves is bounded. We also state and partially prove a higherdimensional analogue of Yamauchi’s conjecture. In order to prove the above results, we derive a form ..."
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We prove a conjecture of Yamauchi which states that the level N for which the new part of J0(N) is Qisogenous to a product of elliptic curves is bounded. We also state and partially prove a higherdimensional analogue of Yamauchi’s conjecture. In order to prove the above results, we derive a formula for the trace of Hecke operators acting on spaces S new (N, k) of newforms of weight k and level N. We use this trace formula to study the equidistribution of eigenvalues of Hecke operators on these spaces. For any d ≥ 1, we estimate the number of normalized newforms of fixed weight and level, whose Fourier coefficients generate a number field of degree less than or equal to d.
THE POLYNOMIAL SIEVE AND EQUAL SUMS OF LIKE POLYNOMIALS
"... Dedicated to Étienne Fouvry on his sixtieth birthday Abstract. A new “polynomial sieve ” is presented and used to show that almost all integers have at most one representation as a sum of two values of a given polynomial of degree at least 3. 1. ..."
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Dedicated to Étienne Fouvry on his sixtieth birthday Abstract. A new “polynomial sieve ” is presented and used to show that almost all integers have at most one representation as a sum of two values of a given polynomial of degree at least 3. 1.