منابع مشابه
Benford’s Law for Coefficients of Newforms
Let f(z) = ∑∞ n=1 λf (n)e 2πinz ∈ S k (Γ0(N)) be a newform of even weight k ≥ 2 on Γ0(N) without complex multiplication. Let P denote the set of all primes. We prove that the sequence {λf (p)}p∈P does not satisfy Benford’s Law in any base b ≥ 2. However, given a base b ≥ 2 and a string of digits S in base b, the set Aλf (b, S) := {p prime : the first digits of λf (p) in base b are given by S} h...
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Let f = ∑ n≥1 λf (n)n (k1−1)/2qn and g = ∑ n≥1 λg(n)n (k2−1)/2qn be two newforms with real Fourier coeffcients. If f and g do not have complex multiplication and are not related by a character twist, we prove that #{n ≤ x | λf (n) > λg(n)} x.
متن کاملFactoring Polynomials with Rational Coefficients
In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial f e Q[X] in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q[X]. It is well known that this is equivalent to factoring primitive polynomials feZ[X] into irreducible factors in Z[X]. Here we call f ~ Z[X] primitive if the greatest comm...
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This paper presents a simple and efficient method for determining the solution of Riccati differential equation with coefficients rational. In case the differential Galois group of the differential equation (E l) : y = ry, r ∈ C(x) is reducible, we look for the rational solutions of Riccati differential equation θ + θ 2 = r, by reducing the number of check to be made and by accelerating the sea...
متن کاملSums of Squares of Polynomials with Rational Coefficients
We construct families of explicit polynomials f over Q that are sums of squares of polynomials over R, but not over Q. Whether or not such examples exist was an open question originally raised by Sturmfels. We also study representations of f as sums of squares of rational functions over Q. In the case of ternary quartics, we prove that our counterexamples to Sturmfels’ question are the only ones.
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ژورنال
عنوان ژورنال: The Ramanujan Journal
سال: 2009
ISSN: 1382-4090,1572-9303
DOI: 10.1007/s11139-008-9147-8