arXiv subjects
Publications and source records attributed to N. A. Carella.
The possible values of the nth Fourier coefficients a(n) of some cusp forms f(z) of weight k => 12 are studied in this article. In particular, the values of the tau function are investigated in some details, and proved that tau(p) =! 0 for all primes p => p_0.
This note simplifies the proof of a recent result on the oscillation of the prime product in Martens Theorem, and provides a quantitative expression for the error term. In addition, the corresponding oscillation results for the finite sums of the reciprocal of the primes up to a fixed number and other finite sums and products are also given.
A method for estimating the merit factors of sequences will be provided. The result is also effective in determining the nonexistence of certain infinite collections of cyclic difference sets and cyclic matrices and associated binary sequences.
This is an introduction to the theory of normal bases of finite fields. The first few chapters cover a wide range of topics on the theory of normal bases of finite fields. Most standard definitions and results, including proofs are given. The last few chapters cover the theory of guassian and period normal bases of finite fields of low degrees. The last chapter presents the asymptotic proofs of the existence of primitive polynomials of degree n with approximately n/2 arbitrary coefficients, and primitive normal polynomials of arbitrary traces.
This work proposes a proof of the simplest cubic primes counting problem. It shows that the subset of primes {p = n^3 + 2 is prime : n => 1} is an infinite subset of primes. Further, the expected order of magnitude of the cubic primes counting function is established.
This short note contributes a new zero-free region of the zeta function. This zero-free region has the form {s : Re(s) > a}, where a = 21/40.
A unified proof of the irrationality of the special values L(n, X), n > 1 an integer, of the beta L-function is put forward in this note. The first case of n = 2 seems to confirm that the Catalan constant L(2, X) is an irrational number.
This work proposes elementary proofs of several related primes counting problems, based on an elementary weighted sieve. The subsets of primes considered here are the followings: the subset of twin primes PT = {p and p + 2 are primes}, the subset of Germain primes PG = {p and 2p + 1 are primes}, and the subset of quadratic primes Pf = {p = n^2 + 1 primes}. These subsets of primes are widely believed to be infinite subsets of prime numbers.
A few elementary estimates of a basic character sum over the prime numbers are derived here. These estimates are nontrivial for character sums modulo large q. In addition, an omega result for character sums over the primes is also included.
This note shows that the Dedekind psi function achieves its extreme values on the subset of primorial integers N_k = 2*3*5*...*p_k, where p_k is the kth prime. In particular, the inequality psi(N_k) > cloglog N_k, where c = 1.08... is a universal constant, holds for all large primorial integers N_k = 2*3*5*...*p_k unconditionally.
The collection of cyclic Hadamard matrices {H = (a_{i - j}) : 0 <= i, j < n, and a_i = -1, 1} of order n is characterized by the orthogonality relation HH^T = nI. Only two of such matrices are currently known. It will be shown that this collection consists of precisely two matrices. An application of this result implies that there are exactly seven Barker sequences over the binary set {-1, 1}.
It will be demonstrated that there is a thin basis of order four of minimal cardinality #A(x) = O(x^.25). The current literature shows the existence of a thin basis of order four of cardinality #A(x) = O(x^(.25+ε), ε > 0, but speculates on the existence of a thin basis of order four of minimal cardinality.
The orders of magnitudes of the summatory Liouville function L(x), and the summatory Mobius function M(x), are unconditionally proven to be of the forms L(x) = O(x^.5)), and M(x) = O(x^.5) respectively. Furthermore, applications of these estimates to zeta functions and L-functions are also considered.
This article is a short introduction to the theory of the groups of points of elliptic curves over finite fields. It is concerned with the elementary theory and practice of elliptic curves cryptography, the new generation of public key systems. The material and coverage are focused on the groups of points of elliptic curves and algebraic curves, but not exclusively.
This note provides new closed forms evaluations of a few classes of exponential sums associated with elliptic curves and hyperelliptic curves.
A study of the Dedekind psi function concludes that its extreme values are supported on the subset of primorial integers N_k = 2*3***p_k, where p_k is the kth prime. In particular, the inequality psi(N_k) > cloglogN_k, c > 0 constant, holds for all large primorial integers.
A perfect number is a number whose divisors add up to twice the number itself. The existence of odd perfect numbers is a millennia-old unsolved problem. This note proposes a proof of the nonexistence of odd perfect numbers. More generally, the same analysis seems to generalize to a proof of the nonexistence of odd multiperfect numbers.
The theoretical aspects of four integer factorization algorithms are discussed in details in this note. The focus is on the performances of these algorithms on the subset of hard to factor balanced integers N = pq, p < q < 2p. The running time complexity of these algorithms ranges from deterministic exponential time complexity O(N^(1/2)) to heuristic and unconditional logarithmic time complexity O((log N)^c), c > 0 constant.