We prove an exact formula for the expected density of distribution of complex level crossings of random polynomials spanned by orthogonal polynomials on the unit circle and unit disk. We then inquire into the consequences of the asymptotical evaluations.
A new method for solving quartic equations due to Luo and Lin is investigated both computationally and theoretically. As a result, a completely straightforward elementary method is given for solving Bumby's equation 3X^4-2Y^2=1, along with a conjecture, which if resolved, would enable a similar proof for a possibly infinite family of similar equations.
We prove that the sequence of the last nonzero digits of factorials in every integer base b>2 is not eventually periodic. We also extend the Adamczewski–Bugeaud criterion, originally formulated for integer base expansions, to Cantor base expansions associated with a periodic Cantor base. As an application, we show that a certain real number expressed through a Cantor base expansion is transcendental when the Cantor base and the digit sequence satisfy suitable conditions.
Let R & lowast;` denote the number of overpartitions of n where nonoverlined parts are -regular pound and overlined parts are unrestricted. In this paper, we establish several Ramanujan like congruences and infinite families of congruences for R & lowast;` for pound = 2k, 5, 6 and 8k. For example, for all n 0 R & lowast;184 (16n + 15) equivalent to 0 (mod 1472).
For every real number x > 5, we define the partial sum S-Lambda(x) of the von Mangoldt function Lambda involving the greatest common divisor by Sigma(mn <= x) Lambda(gcd(m, n)). In this paper, we consider precise behavior of asymptotic formulas for S-Lambda(x), and derive the mean square estimate for the error term of S-Lambda(x) under the Riemann Hypothesis.
Let Q be a nonempty set and let A be a a-ring of subsets of Q containing all singletons. Denote by L(A) the Riesz space of real-valued A-measurable functions on Q. This space is locally convex-solid when equipped with the topology tau p of pointwise convergence on Q. We present several conditions equivalent to the nonexistence of other Hausdorff locally solid [respectively, locally convex-solid] topologies on L(A). Some of them assert the nonexistence of a-order continuous submeasures or two-valued measures on A vanishing on singletons.
We denote by l(n) the minimal length of an addition chain leading to n and we define the counting function F(m, r) := # {n is an element of [2(m), 2(m+1)) : l(n) <= m + r} , where m is a positive integer and r >= 0 is a real number. We show that for 0 < c < log 2 and for any epsilon > 0, we have as m -> infinity, F (m, cm/log m) < exp (cm + epsilon m log log m/log m ) and F(m, cm/log m) > exp (cm - (1 + epsilon)cm log log m/log m ) This extends a result of Erd & odblac;s which says that for almost all n, as n -> infinity, l(n) = log n/log 2 + (1 + o(1)) log n/log log n .
We explicitly describe the splitting of odd integral primes in the radical extension ℚ(√(a)), where x^n-a is an irreducible polynomial in ℤ[x]. Our motivation is to classify common index divisors, the primes whose splitting provides a local obstruction to the existence of a power integral basis for the ring of integers of ℚ(√(a)). Among other results, we show that if p is such a prime, even or otherwise, then p divides n.
Shintani's celebrated invariants are conjectured to generate abelian extensions of real quadratic number fields, offering a potential solution to Hilbert's 12th problem in that setting. In this note, we derive new expressions for Shintani's invariants by generalizing an observation of Yamamoto, who showed that these invariants - originally formulated using the double sine function - can be expressed in terms of the q-Pochhammer symbol.
We obtain asymptotic results on the average numbers of Goldbach representations of an interger as the sum of two primes in different arithmetic progressions. We also prove an omega-result showing that the asymptotic result is essentially the best possible.
Let p be prime, e, k be positive integers, and let R = Zp[e root p]. We calculate the Waring numbers gR(k) for many values of p, e, and k, and investigate how the Waring numbers for p = 2 change as e and k vary.
Let lambda be the Barban-Vehov weights, defined in (1). Let X >= z(1) >= 100 and z(2) = z(1)(2). We prove that Sigma(n <= X) 1/ n (Sigma (d|n) lambda(d))(2) <= 30 log X/ log(z(2)/z(1)), saving more than a factor of 5 on what was the best known constant in such an inequality. Two related estimates are also provided for X >= z(1) >= 100 and z(2) = z(1)(tau) for some tau > 1.
Let (X-k)(k >= 1) and (Y-k)(k >= 1) be sequences of X and Y-coordinates of the positive integer solutions (x, y) of the equation x(2 )- dy(2) = t. In this paper, we completely describe these recurrence sequences such that the sums of two terms of recurrence sequences in the solution sets of generalized Pell equations are infinitely many. Further, we give an upper bound for the number of such terms when there are only finitely many of them. This work is motivated by the recent paper of Hajdu and Sebesty & eacute;n (Int. J. Number Theory 18 (2022), 1605-1612).
