The sum of aΩ(n) over integers n ≤ x with all prime factors between a and y

For a > 0 let $\text{ψ}{}_{\mathrm{a}}\text{(x, y) =}\text{Σ}\text{a}{}^{\mathrm{\Omega (n)}}$, the sum taken over all n, 1 ≤ n ≤ x such that if p is prime and p|n then a < p ≤ y. It is shown for u < about ($\text{log log}\text{x}\text{log log log}\text{x}$) that $\text{ψ}{}_{\mathrm{a}}\text{(x, x}{}^{\mathrm{<mtext>1</mtext><mtext>u</mtext>}}\text{) ? x(}\text{log}\text{x)}{}^{\mathrm{a?1}}\text{p}{}_{\mathrm{a}}\text{(u)}$, where p_a(u) solves a delay differential equation much like that for the Dickman function p(u), and the asymptotic behavior of p_a(u) is worked out.     

Maximum zero strings of Bell numbers modulo primes

For any prime p, the sequence of Bell exponential numbers B_n is shown to have p ? 1 consecutive values congruent to zero (mod p), beginning with B_m, where $\text{m ≡ 1 ?}\text{(p}{}^{\mathrm{p}}\text{? 1)}\text{(p ? 1)}{}^2$ ($\text{mod}\text{(p}{}^{\mathrm{p}}\text{? 1)}\text{(p ? 1)}$). This is an improvement over previous results on the maximal strings of zero residues of the Bell numbers. Similar results are obtained for the sequence of generalized Bell numbers A_n generated by .     

On a function due to S. Chowla

Let θ(k, pⁿ) be the least s such that the congruence $\text{x}{}_1{}^{\mathrm{k}}\text{+ ? + x}{}_{\mathrm{s}}{}^{\mathrm{k}}\text{≡ 0}$ (mod pⁿ) has a nontrivial solution. It is shown that if k is sufficiently large and divisible by p but not by p ? 1, then $\text{θ(k, p}{}^{\mathrm{n}}\text{) ≤ k}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. We also obtain the average order of θ(k), the least s such that the above congruence has a nontrivial solution for every prime p and every positive integer n.     

Some divided difference inequalities for n-convex functions

Using results from the theory of B-splines, various inequalities involving the nth order divided differences of a function f with convex nth derivative are proved; notably, $\text{f}{}^{\mathrm{(n)}}\text{(z)}\text{n! ? [x}{}_0\text{,…, x}{}_{\mathrm{n}}\text{]f}\text{?}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n}}\text{(f}{}^{\mathrm{(n)}}\text{(x}{}_{\mathrm{i}}\text{)}\text{(n + 1)!)}$, where z is the center of mass $\text{(}\text{1}\text{(n + 1))}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n}}\text{x}{}_{\mathrm{i}}$.     

Elementary proof of the transformation formula for Lambert series involving generalized Dedekind sums

An elementary proof is given of the author's transformation formula for the Lambert series $\text{G}{}_{\mathrm{p}}\text{(x) =}\text{Σ}{}_{\mathrm{n?1}}{}^{\mathrm{<mtext>\infty </mtext>}}\text{n}{}^{\mathrm{?p}}\text{x}{}^{\mathrm{n}}\text{(1?x}{}^{\mathrm{n}}\text{)}$ relating G_p(e^2πiτ) to G_p(e^2πiAτ), where p > 1 is an odd integer and $\text{Aτ =}\text{(aτ + b)}\text{(cτ + d)}$ is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function $\text{log}\text{η(τ) =}\text{πiτ}\text{12}\text{? G}{}_1\text{(e}{}^{\mathrm{2\pi i\tau }}\text{)}$, and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions.     

The probability that k positive integers are relatively r-prime

If r, k are positive integers, then $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}$ denotes the number of k-tuples of positive integers (x₁, x₂, …, x_k) with 1 ≤ x_i ≤ n and (x₁, x₂, …, x_k)_r = 1. An explicit formula for $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}$ is derived and it is shown that $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}\text{n}{}^{\mathrm{k}}\text{=}\text{1}\text{ζ(rk)}$.If S = {p₁, p₂, …, p_a} is a finite set of primes, then 〈S〉 = {p₁^a₁p₂^a₂…p_s^a_s; p_i ∈ S and a_i ≥ 0 for all i} and $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}$ denotes the number of k-tuples (x₁, x₃, …, x_k) with 1 ≤ x_i ≤ n and (x₁, x₂, …, x_k)_r ∈ 〈S〉. Asymptotic formulas for $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}$ are derived and it is shown that $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}\text{n}{}^{\mathrm{k}}\text{=}\text{(p}{}_1\text{… p}{}_{\mathrm{a}}\text{)}{}^{\mathrm{rk}}\text{ζ(rk)(p}{}_1{}^{\mathrm{rk}}\text{? 1) … (p}{}_{\mathrm{s}}{}^{\mathrm{rk}}\text{? 1)}$.     

Randomness implies order

Let τ: [0, 1] → [0, 1] possess a unique invariant density $\text{f}{}^1$. Then given any ? > 0, we can find a density function p such that $\text{∥ p ? f}{}^1\text{∥ < ?,}\text{and}\text{p}$ is the invariant density of the stochastic difference equation x_{n + 1} = τ(x_n) + W, where W is a random variable. It follows that for all starting points $\text{x}{}_0\text{? [0, 1],}\text{lim}{}_{\mathrm{n\to \infty }}\text{(1}\text{n)}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n\; ?\; 1}}\text{χ}{}_{\mathrm{B}}\text{(x}{}_{\mathrm{i}}\text{) = ∝}{}_{\mathrm{B}}\text{p(ξ) dξ}$.     

A note on character sums

Let $\text{S(k) = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{p?1}}\text{(}\text{n}\text{p}\text{)n}{}^{\mathrm{k}}$ where p is a prime ≡ 3 mod 4 and k is an integer ≥ 3. Then S(k) frequently takes large values of each sign.     

On the probability that k positive integers are relatively prime II

Given a set S of positive integers let $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t)}$ denote the number of k-tuples 〈m₁, …, m_k〉 for which $\text{m}{}_{\mathrm{i}}\text{∈ S ? [1, t]}$ and (m₁, …, m_k) = 1. Also let $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n)}$ denote the probability that k integers, chosen at random from $\text{S ? [1, n]}$, are relatively prime. It is shown that if P = {p₁, …, p_r} is a finite set of primes and S = {m : (m, p₁ … p_r) = 1}, then $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t) = (td(S))}{}^{\mathrm{k}}\text{Π}{}_{\mathrm{\nu ?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) + O(t}{}^{\mathrm{k?1}}\text{)}$ if k ≥ 3 and $\text{Z}{}_2{}^{\mathrm{S}}\text{(t) = (td(S))}{}^2\text{Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^2\text{) + O(t}\text{log}\text{t)}$ where d(S) denotes the natural density of S. From this result it follows immediately that $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) = (ζ(k))}{}^{\mathrm{?1}}\text{Π}{}_{\mathrm{p\in P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$ as n → ∞. This result generalizes an earlier result of the author's where $\text{P = ?}$ and S is then the whole set of positive integers. It is also shown that if S = {p₁^x₁ … p_r^x_r : x_i = 0, 1, 2,…}, then $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → 0}$ as n → ∞.     

A note on polynomial matrix functions over a finite field

Let F_n denote the ring of n×n matrices over the finite field F=GF(q) and let A(x)=A_Nx^N+ ?+ A₁x+A₀?F_n[x]. A function $\text{?:F}{}_{\mathrm{n}}\text{→F}{}_{\mathrm{n}}$ is called a right polynomial function iff there exists an A(x)?F_n[x] such that $\text{?(B)=A}{}_{\mathrm{N}}\text{B}{}^{\mathrm{N}}\text{+?+A}{}_1\text{B+ A}{}_0$ for every B?F_n. This paper obtains unique representations for and determines the number of right polynomial functions.     

Asymptotic results for partitions (II) and a conjecture of Bateman and Erdös

Asymptotic results are obtained for p_A^(k)(n), the kth difference of the function p_A(n) which is the number of partitions of n into integers from A. Under certain restrictions on A it is shown that

$\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k+1)}}\text{(n)}\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k)}}\text{(n)}\text{= O(n}{}^{\mathrm{?1/2}}\text{) (n→ ∫)}$

thereby verifying for these A a conjecture of Bateman and Erdös.     

The asymptotic eigenvalue-distribution for a certain class of random matrices

For an n × n Hermitean matrix A with eigenvalues λ₁, …, λ_n the eigenvalue-distribution is defined by $\text{G(x, A) :=}\text{1}\text{n}$ · number {λ_i: λ_i ? x} for all real x. Let A_n for n = 1, 2, … be an n × n matrix, whose entries a_ik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, $\text{R}$, p) with the same distribution F_a. Suppose that all moments $\text{E}$ | a | ^k, k = 1, 2, … are finite, $\text{E}$a=0 and $\text{E}$ | a | ². Let

$\text{M(A)=}\text{∑}\text{σ=1}\text{s}\text{θ}{}_{\mathrm{\sigma }}\text{P}{}_{\mathrm{\sigma }}\text{(A,A}{}^1\text{)}$

with complex numbers θ_σ and finite products P_σ of factors A and $\text{A}{}^1$ (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G₀(x) = G₁x) + G₂(x), where G₁ is a step function and G₂ is absolutely continuous, such that with probability $\text{1 G(x, M(}\text{A}{}_{\mathrm{n}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{))}$ converges to G₀(x) as n → ∞ for all continuity points x of G₀. The density g of G₂ vanishes outside a finite interval. There are only finitely many jumps of G₁. Both, G₁ and G₂, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples $\text{A}{}_{\mathrm{r}}\text{A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{+ A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{A}{}^{\mathrm{1r}}\text{± A}{}^{\mathrm{1r}}\text{A}{}^{\mathrm{r}}$, r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form

$\text{lim sup}\text{n→∞}\text{∑}\text{i=1}{}^{\mathrm{n}}\text{|γ}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{|}{}^2\text{?6A}{}_{\mathrm{n}}\text{6}{}^2\text{? 0.8228…}$

of Schur's inequality for the eigenvalues λ_i⁽ⁿ⁾ of A_n holds. Consequently random matrices do not tend to be normal matrices for large n.     

Diagonal forms over finite fields

We shall establish for all finite fields GF(pⁿ) the following result of Chowla: given a positive integer m greater than one and the finite field GF(p), p a prime, such that x^m = ?1 is solvable in GF(p), then there exists an absolute positive constant c, $\text{c ≤}\text{10}\text{ln}\text{2}$, such that for each set of s nonzero elements a_i of GF(p), $\text{a}{}_1\text{x}{}_1{}^{\mathrm{m}}\text{+ ? + a}{}_{\mathrm{s}}\text{x}{}_{\mathrm{s}}{}^{\mathrm{m}}$ has a non-trivial zero in GF(p) if s ≥ c ln m.     

The asymptotic equivalence of Bayes and maximum likelihood estimation

Let (X, $\text{A}$) be a measurable space, Θ ? $\text{R}$ an open interval and P_Ω ∥ $\text{A}$, Ω ? Θ, a family of probability measures fulfilling certain regularity conditions. Let $\text{Ω}{}_{\mathrm{n}}$ be the maximum likelihood estimate for the sample size n. Let λ be a prior distribution on Θ and let $\text{R}{}_{\mathrm{<mtext>n,</mtext><mtext>x</mtext>}}$ be the posterior distribution for the sample size n given $\text{x}\text{? X}{}^{\mathrm{n}}$. $\text{L:}\text{Θ}\text{× Θ →}\text{R}$ denotes a loss function fulfilling certain regularity conditions and T_n denotes the Bayes estimate relative to λ and L for the sample size n. It is proved that for every compact K ? Θ there exists c_K ≥ 0 such that

This theorem improves results of Bickel and Yahav [3], and Ibragimov and Has'minskii [4], as far as the speed of convergence is concerned.     

Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$

     

On the number of groups of a given order

Letting G(n) denote the number of nonisomorphic groups of order n, it is shown that for square-free n, G(n) ≤ ?(n) and G(n) ≤ (log n)^c on a set of positive density. Letting F_k(x) denote the number of n ≤ x for which G(n) = k, it is shown that $\text{F}{}_2\text{(x) =}\text{O(x(}\text{log}{}_4\text{x)}\text{(}\text{log}{}_3\text{x)}{}^2\text{)}$, where log_rx denotes the r-fold iterated logarithm.     

Kolmogorov-Landau inequalities for monotone functions

Presented in this report are two further applications of very elementary formulae of approximate differentiation. The first is a new derivation in a somewhat sharper form of the following theorem of V. M. Olovyani?nikov: LetN_n (n ? 2) be the class of functionsg(x) such thatg(x), g′(x),…, g⁽ⁿ⁾(x) are ? 0, bounded, and nondecreasing on the half-line ?∞ < x ? 0. A special element ofN_nis

$\text{g}{}_1\text{(x) = 0}\text{if}\text{?∞ < x < ?1, g}{}_1\text{(x) = (1 + x)}{}^{\mathrm{n}}\text{if}\text{?1 ? x ? 0}$

. Ifg(x) ∈ N_nis such that

$\text{g(0) ? g}{}_1\text{(0) = 1, g}{}^{\mathrm{(n)}}\text{(0) ? g}{}_1{}^{\mathrm{(n)}}\text{(0) = n!}$

, then

$\text{g}{}^{\mathrm{(v)}}\text{(0) ? g}{}_1{}^{\mathrm{(v)}}\text{(0)}$

for

1$\text{v = 1,…, n ? 1}$

. Moreover, if we have equality in (1) for some value of v, then we have there equality for all v, and this happens only if $\text{g(x) = g}{}_1\text{(x)}$ in (?∞, 0].The second application gives sufficient conditions for the differentiability of asymptotic expansions (Theorem 4).     

Least quadratic non-residues in algebraic number fields

Let νp denote a totally positive integer of an algebraic number field K such that νp is a least quadratic non-residue modulo a prime ideal p of K, least in the sense that N(νp) is minimal. Then the following result is shown: For x ≥ 2 and ε > 0,

$\text{|p;Np?x}\text{and}\text{N(vp)>Np}{}^{\mathrm{\epsilon }}\text{|=O}{}_{\mathrm{\epsilon }}\text{(}\text{log log}\text{3x).}$

     

Positive operators on the n-dimensional ice cream cone

Let $\text{K}{}_{\mathrm{n}}\text{= {x ? R}{}^{\mathrm{n}}\text{: (x}{}_1{}^2\text{+ · +x}{}^2{}_{\mathrm{n?1}}\text{)}\text{1}\text{2}\text{? x}{}_{\mathrm{n}}\text{}}$ be the n-dimensional ice cream cone, and let Γ(K_n) be the cone of all matrices in $\text{R}$ⁿⁿ mapping K_n into itself. We determine the structure of Γ(K_n), and in particular characterize the extreme matrices in Γ(K_n).