Some binary representations of real numbers and odd solutions of the functional equation ... .
Some Borel measures associated with the generalized Collatz mapping
This paper is a continuation of a recent paper [2], in which the authors studied some Markov matrices arising from a mapping T:ℤ → ℤ, which generalizes the famous 3x+1 mapping of Collatz. We extended T to a mapping of the polyadic numbers and construct finitely many ergodic Borel measures on which heuristically explain the limiting frequencies in congruence classes, observed for integer trajectories.
Some characterizations of specially multiplicative functions.
Some characterizations of totients.
Some classes of Diophantine equations connected with McFarland's and Ma's conjectures
In this paper we consider some special classes of Diophantine equations connected with McFarland's and Ma's conjectures about difference sets in abelian groups and we obtain an extension of known results.
Some classes of irreducible polynomials
Some classes of numbers and derivatives.
Some computational formulas for -Nörlund numbers.
Some computations with Hecke rings and deformation rings. With an appendix by Amod Agashe and William Stein.
Some congruence properties of binomial coefficients and linear second order recurrences.
Some congruence property of modular forms.
Some congruences concerning the Bell numbers.
Some congruences for 3-component multipartitions
Let p3(n) denote the number of 3-component multipartitions of n. Recently, using a 3-dissection formula for the generating function of p3(n), Baruah and Ojah proved that for n ≥ 0, p3(9n + 5) ≡ 0 (mod 33) and p3 (9n + 8) ≡ 0 (mod 34). In this paper, we prove several congruences modulo powers of 3 for p3(n) by using some theta function identities. For example, we prove that for n ≥ 0, p3 (243n + 233) ≡ p3 (729n + 638) ≡ 0 (mod 310).
Some congruences for binomial coefficients
Some congruences for the partial Bell polynomials.
Some congruences involving binomial coefficients
Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that , where the central trinomial coefficient Tₙ is the constant term in the expansion of . We also prove three congruences modulo p³ conjectured by Sun, one of which is . In addition, we get some new combinatorial identities.
Some conjectures concerning Gauss sums.
Some conjectures on the zeros of approximates to the Riemann ≡-function and incomplete gamma functions
Riemann conjectured that all the zeros of the Riemann ≡-function are real, which is now known as the Riemann Hypothesis (RH). In this article we introduce the study of the zeros of the truncated sums ≡N(z) in Riemann’s uniformly convergent infinite series expansion of ≡(z) involving incomplete gamma functions. We conjecture that when the zeros of ≡N(z) in the first quadrant of the complex plane are listed by increasing real part, their imaginary parts are monotone nondecreasing. We show how this...
Some consequences of the Riemann hypothesis
Some counter-examples in the theory of the Galois module structure of wild extensions
Considering the ring of integers in a number field as a -module (where is a galois group of the field), one hoped to prove useful theorems about the extension of this module to a module or a lattice over a maximal order. In this paper it is show that it could be difficult to obtain, in this way, parameters which are independent of the choice of the maximal order. Several lemmas about twisted group rings are required in the proof.