On the sum powers of matrices

A theorem of Lagrange says that every natural number is the sum of 4 squares. M. Newman proved that every integral n by n matrix is the sum of 8 (-1)ⁿ squares when n is at least 2. He asked to generalize this to the rings of integers of algebraic number fields. We show that an n by n matrix over a a commutative R with 1 is the sum of squares if and only if its trace reduced modulo 2Ris a square in the ring R/2R. It this is the case (and n is at least 2), then the matrix is the sum of 6 squares (5 squares would do when n is even). Moreover, we obtain a similar result for an arbitrary ring R with 1. Answering another question of M. Newman, we show that every integral n by n matrix is the sum of ten k-th powers for all sufficiently large n. (depending on k).     

Waring-Goldbach problem for fourth powers in short intervals

We prove that almost all integers N satisfying some necessary congruence conditions are the sum of almost equal fourth prime powers.     

On the sum of a prime and a k-th power of prime in short intervals

Let H_k\mathcal{H}_{k} denote the set {n∣2|n, n\not o 1 (mod p)n\not\equiv 1\ (\mathrm{mod}\ p) ∀ p>2 with p−1|k}. We prove that when X^{\frac1120(1-\frac12k) +e}\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{2k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n ? \allowbreak H_k ?(X, X+H]n\in\allowbreak {\mathcal{H}_{k} \cap (X, X+H]} can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when X^{\frac1120(1-\frac1k) +e}\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n∈(X,X+H] can be represented as the sum of a prime and a k-th power of integer for k≧3.     

On the mean value of a sum analogous to character sums over short intervals

The main purpose of this paper is to study the mean value properties of a sum analogous to character sums over short intervals by using the mean value theorems for the Dirichlet L-functions, and to give some interesting asymptotic formulae. This work is supported by the N.S.F. (60472068) of P.R. China.     

On the divisor function in short intervals

Let d(n) denote the number of positive divisors of the natural number n. The aim of this paper is to investigate the validity of the asymptotic formula

$\begin{array}{lll}\sum \limits_{x < n \leq x+h(x)}d(n)\sim h(x)\log x\end{array}$

for \({x \to + \infty,}\) assuming a hypothetical estimate on the mean

$\begin{array}{lll} \int \limits_X^{X+Y}(\Delta(x+h(x))-\Delta (x))^2\,{d}x, \end{array}$

which is a weakened form of a conjecture of M. Jutila.

     

On solving equations of algebraic sum of equal powers

It is well known that a system of equations of sum of equal powers can be converted to an algebraic equation of higher degree via Newton's identities. This is the Viete-Newton theorem. This work reports the generalizations of the Viete-Newton theorem to a system of equations of algebraic sum of equal powers. By exploiting some facts from algebra and combinatorics, it is shown that a system of equations of algebraic sum of equal powers can be converted in a closed form to two algebraic equations, whose degree sum equals the number of unknowns of the system of equations of algebraic sum of equal powers.     

On square-full numbers in short intervals

It is shown that the number of square-full numbers in the interval is asymptotically equal to for every in the range 1/6>0.14254, which extends P.Shiu's range 1/6>0.1526.     

On the sum of powers of Laplacian eigenvalues of bipartite graphs

For a bipartite graph G and a non-zero real α, we give bounds for the sum of the αth powers of the Laplacian eigenvalues of G using the sum of the squares of degrees, from which lower and upper bounds for the incidence energy, and lower bounds for the Kirchhoff index and the Laplacian Estrada index are deduced.     

On the sum of two primes and k powers of two

Let X be a large integer. We prove that, for any fixed positiveinteger k, a suitable asymptotic formula for the number of representationsof an even integer N [1,X] as the sum of two primes and k powersof 2 holds with at most exceptions.     

On the Dirichlet divisor problem in short intervals

We present several new results involving Δ(x+U)?Δ(x), where U=o(x) and $$\varDelta(x):=\sum_{n\leq x}d(n)-x\log x-(2\gamma-1)x $$ is the error term in the classical Dirichlet divisor problem.     

The algebraic independence of the sum of divisors functions

Let σj(n)=_∑d|ndj be the sum of divisors function, and let I be the identity function. When considering only one input variable n, we show that the set of functions is algebraically independent. With two input variables, we give a non-trivial identity involving the sum of divisors function, prove its uniqueness, and use it to prove that any perfect number n must have the form n=rσ(r)/(2r−σ(r)), with some restrictions on r. This generalizes the known forms for both even and odd perfect numbers.     

On the normal density of primes in short intervals

Selberg has shown on the basis of the Riemann hypothesis that for every ε > 0 most intervals |x,x+x^?| of length x^? contain approximately $\text{x}{}^{\mathrm{?}}\text{log}\text{x}$ primes. Here by “most” we mean “for a set of values of x of asymptotic density one.” Prachar has extended Selberg's result to primes in arithmetic progressions. Both authors noted that if we assume the quasi Riemann hypothesis, that ζ(s) has no zeros in the domain {$\text{σ>}\text{1}\text{2}\text{+δ}$} for some $\text{δ<}\text{1}\text{2}$, then the same conclusions hold, provided that ε > 2 δ. Here we give a simple proof of these theorems in a general context, where an arbitrary signed measure takes the place of d[ψ(x)?x]. Then we show by a counterexample that this general theorem is the best of its kind: the condition ε > 2δ cannot be replaced by ε = 2δ. In our example, the associated Dirichlet integral is an entire function which remains bounded on the domain {$\text{σ≥}\text{1}\text{2}\text{+δ}$}. Thus its growth and regularity properties are better than those of $\text{ζ′(s)}\text{ζ(s)}$. Nevertheless the corresponding signed measure behaves badly.     

On the sum of powers of terms of a linear recurrence sequence

Let (F _n ) _n??0 be the Fibonacci sequence given by F _n+2 = F _n+1 + F _n , for n ?? 0, where F ₀ = 0 and F ₁ = 1. There are several interesting identities involving this sequence such as F _n ² + F _n+1 ² = F _2n+1, for all n ?? 0. In a very recent paper, Marques and Togbé proved that if F _n ^s + F _n+1 ^s is a Fibonacci number for all sufficiently large n, then s = 1 or 2. In this paper, we will prove, in particular, that if (G _m ) _m is a linear recurrence sequence (under weak assumptions) and G _n ^s + ... + G _n+k ^s ?? (G _m ) _m , for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on k and the parameters of G _m .