Results 1  10
of
26
The GcdSum Function
, 2001
"... The gcdsum is an arithmetic function defined as the sum of the gcd's of the first n integers with n : g(n) = # n i=1 (i, n). The function arises in deriving asymptotic estimates for a lattice point counting problem. The function is multiplicative, and has polynomial growth. Its Dirichlet ser ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
The gcdsum is an arithmetic function defined as the sum of the gcd's of the first n integers with n : g(n) = # n i=1 (i, n). The function arises in deriving asymptotic estimates for a lattice point counting problem. The function is multiplicative, and has polynomial growth. Its Dirichlet series has a compact representation in terms of the Riemann zeta function. Asymptotic forms for values of partial sums of the Dirichlet series at real values are derived, including estimates for error terms. Keywords: greatest common divisor, Dirichlet series, lattice points, multiplicative, Riemann zeta function, gcdsum. MSC2000 11A05, 11A25, 11M06, 11N37, 11N56. 1.
A complete Vinogradov 3primes theorem under the Riemann hypothesis
 ERA Am. Math. Soc
, 1997
"... Abstract. We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation. 1. ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Abstract. We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation. 1.
Calculation of values of lfunctions associated to elliptic curves
 Mathematics of Computation 68
, 1999
"... Abstract. We calculated numerically the values of Lfunctions of four typical elliptic curves in the critical strip in the range Im(s) ≤ 400. We found that all the nontrivial zeros in this range lie on the critical line Re(s) =1andare simple except the one at s = 1. The method we employed in this ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. We calculated numerically the values of Lfunctions of four typical elliptic curves in the critical strip in the range Im(s) ≤ 400. We found that all the nontrivial zeros in this range lie on the critical line Re(s) =1andare simple except the one at s = 1. The method we employed in this paper is the approximate functional equation with incomplete gamma functions in the coefficients. For incomplete gamma functions, we continued them holomorphically to the right half plane Re(s)> 0, which enables us to calculate for large Im(s). Furthermore we remark that a relation exists between SatoTate conjecture and the generalized Riemann Hypothesis.
Limit theorems for random convex polygons
, 2003
"... Consider the set Ln of convex polygons Γ with vertices on the integer lattice Z 2, nonnegative inclination of the edges and fixed endpoints 0 = (0, 0) and n = (n1, n2). We study the asymptotic properties of the ensemble Ln, as n1, n2 → ∞, with respect to a certain parametric class of probability di ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Consider the set Ln of convex polygons Γ with vertices on the integer lattice Z 2, nonnegative inclination of the edges and fixed endpoints 0 = (0, 0) and n = (n1, n2). We study the asymptotic properties of the ensemble Ln, as n1, n2 → ∞, with respect to a certain parametric class of probability distributions Pn = P (r) n (0 < r < ∞) on the space Ln (in particular, including the uniform distribution). We identify the limit shape of the polygons (under the scaling (i, j) ↦ → (i/n1, j/n2)) and prove the corresponding law of large numbers and the central limit theorem. The laws of large numbers are also obtained for additive functionals of the polygons (e.g., the number of the polygon’s edges). The measure Pn is constructed as a conditional distribution induced by a suitable multiplicative statistic defined on the space L = ∪nLn of polygons with a free right endpoint. The proofs involve subtle analytic tools including the Möbius inversion formula and the properties of zeroes of the Riemann zetafunction. Key words: convex polygons, limit shape, limit theorems, Möbius inversion formula, Riemann zetafunction.
SELBERG’S ORTHOGONALITY CONJECTURE FOR AUTOMORPHIC LFUNCTIONS
"... Abstract. Let π and π ′ be automorphic irreducible unitary cuspidal representations of GLm(QA) and GLm ′(QA), respectively. Assume that π and π ′ are self contragredient. Under the Ramanujan conjecture on π and π ′, we deduce a prime number theorem for L(s, π × ˜π ′), which can be used to asymptotic ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. Let π and π ′ be automorphic irreducible unitary cuspidal representations of GLm(QA) and GLm ′(QA), respectively. Assume that π and π ′ are self contragredient. Under the Ramanujan conjecture on π and π ′, we deduce a prime number theorem for L(s, π × ˜π ′), which can be used to asymptotically describe whether π ′ ∼ ′ = π, or π ∼ iτ0 = π ⊗  det(·)  for some nonzero τ0 ∈ R, or π ′ ̸ ∼ = π ⊗  det(·)  it for any t ∈ R. As a consequence, we prove the Selberg orthogonality conjecture, in a more precise form, for automorphic Lfunctions L(s, π) and L(s, π ′), under the Ramanujan conjecture. When m = m ′ = 2 and π and π ′ are representations corresponding to holomorphic cusp forms, our results are unconditional. 1. Introduction. Let π be an irreducible unitary cuspidal representation of GLm(QA). Then the global Lfunction attached to π is given by products of local factors for Re s> 1 (Godement and Jacquet [3]):
On some properties of σ(N) ∗†
, 707
"... We show asymptotic upper and lower bounds for the greatest common divisor of N and σ(N). We also show that there are infinitely many integers N with fairly large g.c.d. of N and σ(N). 1 ..."
Abstract
 Add to MetaCart
We show asymptotic upper and lower bounds for the greatest common divisor of N and σ(N). We also show that there are infinitely many integers N with fairly large g.c.d. of N and σ(N). 1
Previous Up Next Article Citations From References: 0 From Reviews: 0
"... Irrationality of certain numbers that contain values of the di and trilogarithm. (English summary) Math. Z. 254 (2006), no. 2, 299–313. The authors prove that, for z ∈ {1/2, 2/3, 3/4, 4/5}, at least one of the two numbers Li2(z) + log(1 − z) log(z), Li3(z) + 1 2 log(1 − z) log(z)2, is irrational, w ..."
Abstract
 Add to MetaCart
Irrationality of certain numbers that contain values of the di and trilogarithm. (English summary) Math. Z. 254 (2006), no. 2, 299–313. The authors prove that, for z ∈ {1/2, 2/3, 3/4, 4/5}, at least one of the two numbers Li2(z) + log(1 − z) log(z), Li3(z) + 1 2 log(1 − z) log(z)2, is irrational, where Lis(z) = ∑∞ k=1 zk /ks. In the case z = 1/2, this improves the previous result of S. Fischler and the reviewer [J. Math. Pures Appl. (9) 82 (2003), no. 10, 1369–1394; MR2020926 (2004m:11114)] that at least one of the three numbers