Abstract not found

The gcd-sum 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, gcd-sum. MSC2000 11A05, 11A25, 11M06, 11N37, 11N56. 1.

We consider the equation

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.

Consider the set Ln of convex polygons Γ with vertices on the integer lattice Z 2, non-negative 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 zeta-function. Key words: convex polygons, limit shape, limit theorems, Möbius inversion formula, Riemann zeta-function.

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 non-zero 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 L-functions 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 L-function attached to π is given by products of local factors for Re s> 1 (Godement and Jacquet [3]):

Applying famous Nesterenko’s result on algebraic independence of the numbers π, e π √ d, we show that the infinite sums like n=0 (±1) n n 2 + an + b, n=0 (−1) n (2n + a) n 2 + an + b, n=0

The gcd-sum 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.

the divisibility of odd perfect numbers by a high power of a prime ∗†‡

On the logarithmic factor in error term estimates in certain additive congruence problems