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We establish a uniform asymptotic formula with a power saving error term for the average of the divisor function τ(n) := ∑ k|n 1 over values of the quadratic polynomial x 2 + |D| where D < 0 is a fundamental discriminant.

Let M be a surface with conical singularities, and consider a degenerating family of surfaces obtained from M by removing disks of smaller and smaller radius around a subset of the conical singularities. Such families arise naturally in the study of the moduli space of flat metrics on higher-genus surfaces with boundary. In particular, they have been used by Khuri to prove that the determinant of the Laplacian is not a proper map on this moduli space when the genus of M is positive. Khuri's work is closely related to the isospectral compactness results of Osgood, Phillips, and Sarnak. Our main theorem is an asymptotic formula for the determinant of the Laplacian on the degenerating family of surfaces, up to terms which vanish in the singular limit. We then apply this theorem to extend and sharpen the results of Khuri. The proof uses the determinant gluing formula of Burghelea, Friedlander, and Kappeler along with an observation of Wentworth on the asymptotics of Dirichlet-to-Neumann operators.

We re-prove the Hardy-Ramanujan asymptotic formula for the partition function without using the circle method. We derive our result from recent work of Bruinier and Ono on harmonic weak Maass forms.

Let $K$ be a fixed number field, and assume that $K$ is Galois over $\qq$. Previously, the author showed that when estimating the number of prime ideals with norm congruent to $a$ modulo $q$ via the Chebotar\"ev Density Theorem, the mean square error in the approximation is small when averaging over all $q\le Q$ and all appropriate $a$. In this article, we replace the upper bound by an asymptotic formula. The result is related to the classical Barban-Davenport-Halberstam Theorem in the case $K=\qq$., Comment: Preprint of an old paper

We obtain an asymptotic formula for the 2k-th power mean of odd primitive character sums over the interval [1, q/4).

We consider the nonlinear problem arising in population dynamics: - u"(t) + u(t) p = λu(t), u(t) > 0, t ∈ I:= (0,1), u(0) = u(1) = 0, where p > 1 is a constant and λ > 0 is a positive parameter. We establish the crucial asymptotic formula for the branch of positive solutions λ q (α) in L q -framework as α → oo, where a:= ∥u λ ∥ q (1 < q < ∞). Especially, for the original logistic equation, namely the case where p = 2 and q = 1, we obtain not only the asymptotic expansion formula for λ 1 (α) but also the remainder estimate. Such a formula for the bifurcation branch seems to be new.

We prove an asymptotic formula for the number of sextic number fields with Galois group S 3 and absolute discriminant < X. In addition, we give an interpretation of the constant in the formula in terms of the asymptotic densities of given local completions among these sextic fields. Our proof gives analogous results when we count S 3 -sextic extensions of any number field, and also when finitely many local completions have been specified for the sextic extensions.

Let μ k (Ω) be the kth Neumann eigenvalue of a bounded domain Ω with piecewisely smooth boundary in R". In 1992, P. Kroger proved that k - n+2 n Σ k j=1 μ j ≤ 4nπ 2 Ν+2(ω n V) -2/n , where the upper bound is sharp in view of Weyl's asymptotic formula. The aim of this paper is twofold. First, we will improve this estimate by multiplying a factor in terms of k to its right-hand side which approaches strictly from below to 1 as k tends to infinity. Second, we will generalize Kroger's estimate to the case when Ω is a compact Euclidean submanifold.

Recently Wu proved that for all primes q q , \[ ∑ f L ( 1 , sym 2 f ) = π 4 432 q + O ( q 27 / 28 log B ⁡ q ) \sum _{f} L(1, \mbox {sym}^2f) =\frac {\pi ^4}{432}q +O(q^{27/28}\log ^B q) \] where f f runs over all normalized newforms of weight 2 and level q q . Here we show that 27 / 28 27/28 can be replaced by 9 / 10 9/10 .

By modifying Beukers’ proof of Apéry’s theorem that ζ ( 3 ) \zeta (3) is irrational, we derive criteria for irrationality of Euler’s constant, γ \gamma . For n > 0 n>0 , we define a double integral I n I_n and a positive integer S n S_n , and prove that with d n = LCM ⁡ ( 1 , … , n ) d_n=\operatorname {LCM}(1,\dotsc ,n) the following are equivalent: 1. The fractional part of log ⁡ S n \log S_n is given by { log ⁡ S n } = d 2 n I n \{\log S_n\}=d_{2n}I_n for some n n . 2. The formula holds for all sufficiently large n n . 3. Euler’s constant is a rational number. A corollary is that if { log ⁡ S n } ≥ 2 − n \{\log S_n\}\ge 2^{-n} infinitely often, then γ \gamma is irrational. Indeed, if the inequality holds for a given n n (we present numerical evidence for 1 ≤ n ≤ 2500 ) 1\le n\le 2500) and γ \gamma is rational, then its denominator does not divide d 2 n ( 2 n n ) d_{2n}\binom {2n}{n} . We prove a new combinatorial identity in order to show that a certain linear form in logarithms is in fact log ⁡ S n \log S_n . A by-product is a rapidly converging asymptotic formula for γ \gamma , used by P. Sebah to compute γ \gamma correct to 18063 decimals.

Let λ k (Ω) be the kth Dirichlet eigenvalue of a bounded domain Ω in R n . According to Weyl's asymptotic formula we have λ k (Ω) ∼ C n (k/V(Ω)) 2/n . The optimal in view of this asymptotic relation lower estimate for the sums Σ λ j (Ω) has been proven by P.Li and S.T.Yau (Comm. Math. Phys. 88 (1983), 309-318). Here we will improve this estimate by adding to its righthand side a term of the order of k that depends on the ratio of the volume to the moment of inertia of Ω.

In this work we consider a general class of hypoelliptic operators, for which we give an estimate of the remainder of the so-called Weyl asymptotic formula for the eigenvalues.

LetAAbe a nonempty finite set of relatively prime positive integers, and letpA(n)p_A(n)denote the number of partitions ofnnwith parts inAA. An elementary arithmetic argument is used to prove the asymptotic formula\[pA(n)=(1∏a∈Aa)nk−1(k−1)!+O(nk−2).p_A(n) = \left (\frac {1}{\prod _{a\in A}a}\right ) \frac {n^{k-1}}{(k-1)!} + O\left ( n^{k-2}\right ).\]

We prove an effective mean-value theorem for the values of a non-degenerate, algebraic exponential polynomial in several variables. These objects generalise simultaneously the fundamental examples of linear recurrence sequences and sums of S S -units. The proof is based on an effective, uniform estimate for the deviation of the exponential polynomial from its expected value. This estimate is also used to obtain a non-effective asymptotic formula counting the norms of these values below a fixed bound.

Some estimations for the "juxtaposition function" h F {h_F} and an asymptotic formula for the function h F / h G {h_F}/{h_G} , where F , G F,\;G are central symmetric convex bodies, are given. Hadwiger and Grünbaum gave for h F ( 1 ) {h_F}(1) the bounds n 2 + n ⩽ h F ( 1 ) ⩽ 3 n − 1 {n^2} + n \leqslant {h_F}(1) \leqslant {3^n} - 1 . Grünbaum conjectured (and proved for n = 2 n = 2 in Pacific J. Math. 11 (1961), 215-219) that for every even r r between these bounds there exists in E n {E^n} an oval F F such that h F ( 1 ) = r {h_F}(1) = r . Lower bounds for h F {h_F} could be derived in the same way as in Theorems 1 and 2 from a good estimate of packing numbers on a Minkowski sphere, that is, from solutions to a Tammes-type problem in a Banch space.

A second-order linear differential equation with two irregular singular points, of which one has unit rank, is investigated. A simple limit formula is obtained for the characteristic exponent of the multiplicative solutions as well as a more detailed asymptotic formula which is suitable for getting accurate numerical values. Only the recursively known coefficients of the formal power series solutions at the irregular singular point of unit rank enter our formulas.

If x = 0 x = 0 is a contractive fixed point for the function F F , then under certain conditions, the iterates F k ( a ) {F_k}(a) are asymptotically equal to the numbers ξ k {\xi _k} defined by k = ∫ ξ k a d u u − F ( u ) k = \int _{{\xi _k}}^a {\frac {{du}}{{u - F(u)}}} . Using somewhat different hypotheses, we give a more precise bound on F k ( a ) / ξ k {F_k}(a)/{\xi _k} .

Let P ( x ) = ∏ j = 2 k ( x + δ j ) P(x) = \prod \nolimits _{j = 2}^k {(x + {\delta _j})} be a polynomial with real coefficients and Re ⁡ δ j > − 1 ( j = 1 , … , k ) \operatorname {Re} {\delta _j} > - 1(j = 1, \ldots ,k) . Define the zeta function Z p ( s ) {Z_p}(s) associated with the polynomial P ( x ) P(x) as \[ Z P ( s ) = ∑ n = 1 ∞ 1 P ( n ) s , Re ⁡ s > 1 / k . {Z_P}(s) = \sum \limits _{n = 1}^\infty {\frac {1}{{P{{(n)}^s}}}} ,\operatorname {Re} s > 1/k. \] Z P ( s ) Z_P(s) is holomorphic for Re ⁡ s > 1 / k \operatorname {Re} s > 1/k and it has an analytic continuation in the whole complex s s -plane with only possible simple poles at s = j / k ( j = 1 , 0 , − 1 , − 2 , − 3 , … ) s = j/k(j = 1,0, - 1, - 2, - 3, \ldots ) other than nonpositive integers. In this paper, we shall obtain the explicit value of Z P ( − m ) {Z_P}( - m) for any non-negative integer m m , the asymptotic formula of Z P ( s ) {Z_P}(s) at s = 1 / k s = 1/k , the value Z P ′ ( 0 ) {Z’_P}(0) and its application to the determinants of elliptic operators.

An asymptotic estimate for the lattice of a Fuchsian group with quotient of finite area is discussed. The estimate is used to obtain an asymptotic formula for the Taylor coefficients of holomorphic automorphic forms.

Under the assumption of Riemann’s Hypothesis, a general asymptotic formula for sums ∑ n ≤ x a ( n ) \sum \nolimits _{n \leq x} {a\left ( n \right )} is established which applies (e.g.) to arithmetic functions a ( n ) a\left ( n \right ) defined by a kind of convolution of the Möbius function with some divisor function.

The following question was posed by C. A. Nicol: Given an arbitrary set B B of positive integers, find the extremal denominators of regular continued fractions with partial denominators from B B , each element occurring a given number of times. Partial solutions have been given by T. S. Motzkin and E. G. Straus, and later by T. W. Cusick. We derive the general solutions from a purely combinatorial theorem about the set of permutations of a vector with components from an arbitrary linearly ordered set. We also consider certain halfregular continued fractions. Here the maximizing arrangements have to be described in terms of an algorithmic procedure, as their combinatorial structure is exceptionally complicated. Its investigation leads to a connection with the well-known Markov spectrum. Finally we obtain an asymptotic formula for the ratio of extremal continuants and some sharp (essentially analytic) inequalities concerning cyclic continuants.

1. In a previous paper [1] a coprime chain was defined to be an increasing sequence { a,, .., ak } of integers greater than 1 which contains exactly one multiple of each prime equal to or less than ak. We let s(n), n> 1, denote the number of coprime chains with largest member n. For convenience we define s(1) = 1. In this paper we will obtain a partial recursion formula for s(n) and an asymptotic formula for log s(n). A table of values of s(n), n < 113, is also provided.

()a n (n)[y(n) +1]* [y(n) + kn -1J (3 ) = E k=-1k+2k+-k. . ,kZ,k$O n1 kni Thus the partition k = lki+2k2+ +Ikt of k into ki ones, k2 twos, and so on, is weighted in a certain way by the product shown, and various partition functions are obtained by choosing the function 'y(n) appropriately. For example, if 'y(n) =1 for all n, then ay(k) becomes the unrestricted partition function; if 'y(n) = n for all n, then a,,(k) becomes the plane partition of Wright [2];' if 'y(n) =1 for rth powers and 0 otherwise, then a.,(k) is the number of partitions of k into rth powers; and so on. The purpose of this paper is to establish the asymptotic formula

In his paper, L. Hörmander has used the Weyl calculus to study the Fourier integral operator theory. In the present paper, the author considers the correspondences W τ {W_\tau } , τ ∈ R \tau \in R ( R R is the set of the real numbers), which mean the standard correspondence of symbol and operator if τ = 0 \tau = 0 , and the correspondence of Weyl type if τ = 1 / 2 \tau = 1/2 , and shows the explicit asymptotic formula which describes the deviation of W σ ( W τ ) − 1 {W_\sigma }{\left ( {{W_\tau }} \right )^{ - 1}} from the automorphisms as Lie algebra, and makes some remarks on the above formula.

and the boundary conditions u(O) = 0, u(1) = 0, the function g(x) being assumed continuous on 0 ? x < 1. He used the asymptotic formula for the kth eigenfunction (1) uk(x) = (2)1/2 [sin k7rx + (1/k)4k(x)], | k(X) | _ C. Comparing series of these functions with corresponding series of the functions (2) Uk(X) = (2)1/2 sin kirx, he proved that if a function in L2 is expanded in terms of both sets of functions, then the series of term-by-term differences converges uniformly and absolutely to zero on 0? x ?1. This result, as Walsh points out, is closely related to that of Haar [1], who considered the Sturm-Liouville problem with boundary conditions u'+Hu =0, and compared the series of the resulting functions with the Fourier cosine series. He obtained equiconvergence in

An asymptotic formula is derived for block Toeplitz determinants which generalizes Szeg6's well-known formula for ordinary Toeplitz determinants. The predecessor of this paper [5] was concerned, among other things, with the limiting behavior of the block Toeplitz determinants DN[.] = det TNh[01 where TN[s] = (sb.X), i, j= 0, *?, N. Here +(Z) =k E Xk l z |=1 is an r x r matrix-valued function satisfying

Let c(n) denote the number of square-full divisors of n. In this note an asymptotic formula for >n,X oc(n) is established.