Related papers: The asymptotic formula in Waring's problem: higher…

Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by placing divisibility conditions on the square-free number.

The paper extends the classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide the uniform asymptotic expansion in terms of the Hermite polynomials. We explicitly obtain expressions for a few…

Let $p_{1}<p_2<... <p_{\nu}<...$ be the sequence of prime numbers and let $m$ be a positive integer. We give a strong asymptotic formula for the distribution of the set of integers having prime factorizations of the form…

We derive the complete asymptotic expansion in terms of powers of $N$ for the Riesz $s$-energy of $N$ equally spaced points on the unit circle as $N\to \infty$. For $s\ge -2$, such points form optimal energy $N$-point configurations with…

Let $k \ge 2$ and $s$ be positive integers, and let $n$ be a large positive integer subject to certain local conditions. We prove that if $s \ge k^2+k+1$ and $\theta > 31/40$, then $n$ can be expressed as a sum $p_1^k + \dots + p_s^k$,…

We give asymptotic expressions for the number of commuting matrices over finite fields. For this, we use product expansions for the corresponding generating functions.

We prove an asymptotic formula for the number of representations of squares by nonsingular cubic forms in six or more variables. The main ingredients of the proof are Heath-Brown's form of the Circle Method and various exponential sum…

Let N_{a,b}(x) count the number of primes p<=x with p dividing a^k+b^k for some k>=1. It is known that asymptotically N_{a,b}(x) grows like c(a,b)x/log x for some rational number c(a,b) that depends in a rather intricate way on a and b. A…

I compute several terms of the asymptotic expansion of the number of connected labelled graphs with n nodes and m edges, for small k=m-n.

Sums over inverse s-th powers of semiprimes and k-almost primes are transformed into sums over products of powers of ordinary prime zeta functions. Multinomial coefficients known from the cycle decomposition of permutation groups play the…

We use the spectral theory of Hilbert-Maass forms for real quadratic fields to obtain the asymptotics of some sums involving the number of representations as a sum of two squares in the ring of integers.

We reconsider the classical problem of representing a finite number of forms of degree d in n+1 variables as sums of powers of linear forms. We define a geometric construct called a `grove', which, in a number of cases allows us to…

We consider the asymptotic expansion of the Wright function \[W_{\lambda,\mu}(z)=\sum_{n=0}^\infty\frac{z^n}{n! \Gamma(\lambda n+\mu)}\qquad (\lambda>-1)\] for large (positive and negative) variable and large parameter $\mu$. The analysis…

In this paper, we first establish asymptotic expansions of the Humbert function $\Psi_1$ for one large variable. The resulting expansions are then used to derive an asymptotic expansion of Saran's hypergeometric function $F_K$ when two of…

Let $k\geq2$ and $s$ be positive integers. Let $\theta\in(0,1)$ be a real number. In this paper, we establish that if $s>k(k+1)$ and $\theta>0.55$, then every sufficiently large natural number $n$, subjects to certain congruence conditions,…

In this paper, we computed the first three coefficients of the asymptotic expansion of Zelditch. We also proved that in general, the $k$-th coefficient is a polynomial of the curvature and its derivative of weight $k$.

We obtain an explicit simple formula for the coefficients of the asymptotic expansion for the factorial of a natural number,in terms of derivatives of powers of an elementary function. The unique explicit expression for the coefficients…

We derive an asymptotic formula which counts the number of abelian extensions of prime degrees over rational function fields. Specifically, let $\ell$ be a rational prime and $K$ a rational function field $\Bbb F_q(t)$ with $\ell \nmid q$.…

We prove an asymptotic formula as $x\to +\infty$ for the number of algebraic integers $\alpha$ belonging to a fixed CM number field and satisfying $\alpha\overline{\alpha}\leq x$. This problem is related to the height zeta function…

In the semisimple case, we derive (asymptotic) formulas for the growth rate of the number of summands in tensor powers of the generating object in diagram/interpolation categories.