Proof of a conjecture of S. Chowla

Let θ(k, p) be the least s such that the congruence $\text{x}{}_1{}^{\mathrm{k}}\text{+ … + x}{}_{\mathrm{s}}{}^{\mathrm{k}}\text{≡ 0(}\text{mod}\text{p)}$ has a nontrivial solution. Let θ(k) = {max θ(k, p)| p > 1 + 2k}. The purpose of this note is to prove the following conjecture of S. Chowla: $\text{θ(k) = O(k}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>+?</mtext>}}\text{)}$.     

Votes and a half-binomial

In two party elections with popular vote ratio $\text{p}\text{q}\text{,}\text{1}\text{2}\text{≤p=1 ?q}$, a theoretical model suggests replacing the so-called MacMahon cube law approximation $\text{(}\text{p}\text{q}\text{)}{}^3$, for the ratio $\text{P}\text{Q}$ of candidates elected, by the ratio $\text{?}{}_{\mathrm{k}}\text{(p)}\text{?}{}_{\mathrm{k}}\text{(q)}$ of the two half sums in the binomial expansion of (p+q)^2k+1 for some k. This ratio is nearly $\text{(}\text{p}\text{q}\text{)}{}^3$ when k = 6. The success probability $\text{g}{}_{\mathrm{k}}\text{(p)=(}\text{p}{}^{\mathrm{a}}\text{(p}{}^{\mathrm{a}}\text{+q}{}^{\mathrm{a}}\text{)}$ for the power law $\text{(}\text{p}\text{q}\text{)}{}^{\mathrm{a}}\text{?}\text{P}\text{Q}$ is shown to so closely approximate $\text{?}{}_{\mathrm{k}}\text{(p)=Σ}{}_0{}^{\mathrm{k}}\text{(}{}_{\mathrm{r}}{}^{\mathrm{2k+1}}\text{)p}{}^{\mathrm{2k+1?r}}\text{q}{}^{\mathrm{r}}$, if we choose $\text{a = a}{}_{\mathrm{k}}\text{=}\text{(2k+1)!}\text{4}{}^{\mathrm{k}}\text{k!k!}$, that $\text{1≤}\text{?}{}_{\mathrm{k}}\text{(p)}\text{g}{}_{\mathrm{k}}\text{(p)}\text{≤1.01884086}$ for $\text{k≥1}\text{if}\text{1}\text{2}\text{≤p≤1}$. Computationally, we avoid large binomial coefficients in computing $\text{?}{}_{\mathrm{k}}\text{(p)}$ for k>22 by expressing $\text{2?}{}_{\mathrm{k}}\text{(p)?1}$ as the sum $\text{(p?q) Σ}{}_0{}^{\mathrm{k}}\text{(4pq)}{}^{\mathrm{s}}\text{a}{}_{\mathrm{s}}\text{(2s+1)}$, whose terms decrease by the factors $\text{(4pq)(1?}\text{1}\text{2s}\text{)}$. Setting K = 4k+3, we compute a_k for the large k using a continued fraction $\text{πa}{}_{\mathrm{k}}{}^2\text{=}\text{K+1}{}^2\text{(}\text{2K+3}{}^2\text{(}\text{2K+5}{}^2\text{(2K+…)}\text{)}\text{)}$ derived from the ratio of π to the finite Wallis product approximation.     

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.     

On the van der Waerden conjecture and zeros of polynomials

Let

$\text{F(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{A}{}_{\mathrm{n}}\text{≠ 0,}$

and

$\text{G(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{B}{}_{\mathrm{n}}\text{≠ 0,}$

be polynomials with real zeros satisfying A_n?1 = B_n?1 = 0, and let

$\text{H(x) =}\text{∑}\text{k=o}\text{n-2}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}\text{.}$

Using the recently proved validity of the van der Waerden conjecture on permanents, some results on the real zeros of H(x) are obtained. These results are related to classical results on composite polynomials.     

Pairs of additive equations II. Large odd degree

Let k be an odd positive integer. Davenport and Lewis have shown that the equations

$\text{a}{}_1\text{x}{}_1{}^{\mathrm{k}}\text{+…+a}{}_{\mathrm{n}}\text{x}{}_{\mathrm{n}}{}^{\mathrm{k}}\text{=0}$

with integer coefficients, have a nontrivial solution in integers x₁,…, x_N provided that

$\text{N?[36klog6k]}$

Here it is shown that for any ? > 0 and k > k₀(?) the equations have a nontrivial solution provided that

$\text{N?}\text{8}\text{log 2}\text{+?}\text{k log k.}$

     

On the order of the error function of the (k,r)-integers

Let k and r be fixed integers such that 1 < r < k. Any positive integer n of the form n = a^kb, where b is r-free, is called a (k, r)-integer. In this paper we prove that if Q_k,r(x) denotes the number of (k, r)-integers ≤ x, then $\text{Q}{}_{\mathrm{k,r}}\text{(x) =}\text{xζ(k)}\text{ζ(r)}\text{+ Δ}{}_{\mathrm{k,r}}\text{(x)}$, where $\text{Δ}{}_{\mathrm{k,r}}\text{(x) = O(x}{}^{\mathrm{<mtext>1</mtext><mtext>r</mtext>}}\text{exp}\text{[?B}\text{log}{}^{\mathrm{<mtext>3</mtext><mtext>5</mtext>}}\text{x (}\text{log log}\text{x)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>5</mtext>}}\text{])}$, B being a positive constant depending on r and the O-estimate is uniform in k. On the assumption of the Riemann hypothesis, we improve the above order estimate of Δ_k,r(x) and prove that

$\text{1}\text{x}\text{∫}\text{1}\text{α}\text{δ}{}_{\mathrm{k,r}}\text{(t)dt=0(x}\text{1}\text{k}\text{ω(x))}\text{or}\text{0(x}{}^{\mathrm{3/(4r+1)}}\text{ω(x))}$

, according as $\text{k ≤}\text{(4r + 1)}\text{3}$ or $\text{k >}\text{(4r + 1)}\text{3}$, where ω(x) = exp [B log x(log log x)^?1].     

Product formulas for semigroups with elliptic generators

Real constant coefficient nth order elliptic operators, Q, which generate strongly continuous semigroups on L²($\text{R}$^k) are analyzed in terms of the elementary generator,

$\text{A = (?n)}{}^{\mathrm{<mtext>(</mtext><mtext>n</mtext><mtext>2</mtext><mtext>\; ?\; 1</mtext>}}\text{)(n!)}{}^{\mathrm{?1}}\text{∑}\text{k}\text{j = 1}\text{?}{}^{\mathrm{n}}\text{?x}{}_{\mathrm{j}}{}^{\mathrm{n}}$

, for n even. Integral operators are defined using the fundamental solutions p_n(x, t) to u_t = Au and using real polynomials q_l,…, q_k on $\text{R}$^m by the formula, for q = (q_l,…, q_k),

$\text{(F(t)?)(x) = ∫}$

_$\text{R}$m

$\text{?(x + q(z)) P}{}_{\mathrm{n}}\text{(z, t)dz}$

. It is determined when, strongly on L²($\text{R}$^k),

$\text{e}{}^{\mathrm{tQ}}\text{=}\text{lim}\text{j → ∞}\text{F}\text{t}\text{j}{}^{\mathrm{j}}$

. If n = 2 or k = 1, this can always be done. Otherwise the symbol of Q must have a special form.     

Author index

A matrix T=(t_ik) is introduced, the coefficients of which are defined by $\text{k}{}_{\mathrm{ik}}\text{:= (}\text{i}{}^{\mathrm{k}}\text{(ik)!)}\text{Σ}{}_{\mathrm{x?S}}{}_{\mathrm{n}}\text{a}{}_{\mathrm{i}}\text{(x)}{}^{\mathrm{k}}\text{, i, k?N={1, 2, 3,…,}}$, where a_i(x) denotes the s the number of i cycles in the element x of the symmetric group S_n. It is shown that these numbers are natural numbers, that they are easy to evaluate, and that they serve very well in order to formulate an infinite number of characterizations of multiply transitive subgroups of symmetric groups in terms of the cycle structure of their elements.     

A divided difference inequality for n-convex functions

For a₍₁₎ ? a₍₂₎ ? ··· ? a_(n) ? 0, b₍₁₎ ? b₍₂₎ ? ··· ? b_(n) ? 0, the ordered values of a_i, b_i, i = 1, 2,…, n, m fixed, m ? n, and p ? 1 it is shown that

$\text{∑}\text{1}\text{n}\text{a}{}_{\mathrm{i}}\text{b}{}_{\mathrm{i}}\text{?}\text{∑}\text{1}\text{m}\text{a}{}^{\mathrm{p}}{}_{\mathrm{(i)}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{∑}\text{1}\text{m?k?1}\text{b}{}^{\mathrm{q}}{}_{\mathrm{(i)}}\text{+}\text{b}{}^{\mathrm{q}}{}_{\mathrm{[m?k]}}\text{(k+1)}{}^{\mathrm{<mtext>q</mtext><mtext>p</mtext>}}{}^{\mathrm{<mtext>1</mtext><mtext>q</mtext>}}$

where $\text{1}\text{p}\text{+}\text{1}\text{q}\text{= 1, b}{}_{\mathrm{[j]}}\text{= b}{}_{\mathrm{(j)}}\text{+ b}{}_{\mathrm{(j\; +\; 1)}}\text{+ ··· + b}{}_{\mathrm{(n)}}\text{,}\text{and}\text{k}$ is the integer such that $\text{b}{}_{\mathrm{(m\; ?\; k\; ?\; 1)}}\text{?}\text{b[m ? k]}\text{(k + 1)}$ and $\text{b}{}_{\mathrm{(m\; ?\; k)}}\text{<}\text{b}{}_{\mathrm{[m\; ?\; k\; +\; 1]}}\text{k}$. The inequality is shown to be sharp. When p < 1 and a_(i)'s are in increasing order then the inequality is reversed.     

On a conjecture by D. Ž. D̄okovic

It is shown that if $\text{A?Ω}{}_{\mathrm{n}}\text{?{J}{}_{\mathrm{n}}\text{}}$ satisfies

$\text{nkσ}{}_{\mathrm{k}}\text{(A)?(n?k+1)}{}^2\text{σ}{}_{\mathrm{k?1}}\text{(A)}$

$\text{(k=1,2,…,n)}$

, where σ_k(A) denotes the sum of all kth order subpermanent of A, then Per[λJ_n+(1?λ)A] is strictly decreasing in the interval 0<λ<1.     

A note on Ramsey numbers

Upper bounds are found for the Ramsey function. We prove $\text{R(3, x) <}\text{cx}{}^2\text{ln}\text{x}$ and, for each k ? 3, $\text{R(k, x) <}\text{c}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k\; ?\; 1}}\text{(}\text{ln}\text{x)}{}^{\mathrm{k\; ?\; 2}}$ asymptotically in x.     

Derivations of Witt vectors with application to K2 of truncated polynomial rings and Laurent series

The absolute Kähler module $\text{Ω}{}_{\mathrm{w}}{}_{\mathrm{n}}{}^{\mathrm{(k)}}$ of the truncated generalized Witt vectors of a field k of positive characteristic is zero if and only if k is perfect. This recovers known information on $\text{K}{}_2\text{(}\text{k[t]}\text{(t}{}^{\mathrm{n}}\text{)}\text{)}$ with which the structure of K₂(k((t))) can be studied.     

A probability model for computer system overhead

Two models for computer system overhead are developed by considering the number of jobs in the system as an immigration-death process. The models developed relate the death rates to the state of the process. The first model uses the total numver of jobs in the system as the state of the processe. The second model classifies the jobs in the system according to the priority classes unsed in the computer system.In the model based on the total number of jobs in the system, the death rate when the system is in the state i, μ_i, is μ_i = μ min(i, x₀) where μ and x₀ are parameters to be estimated by maximum likelihood. In the second model, the death rate for jobs in the k^th priority class when the state of the system is i = (i₁…,i_p), μ_i^(k) is given by $\text{μ}{}_{\mathrm{i}}{}^{\mathrm{(k)}}\text{=μ}{}^{\mathrm{(k)}}\text{i}{}_{\mathrm{k}}\text{min}\text{(1,}\text{x}{}_0\text{i·1}\text{)}$. Computational difficulties in finding the maximum likelihood estimates of the parameters for this model are noted.     

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 → ∞.     

On the series Σk = 1∞(k2k)−1k−n and related sums

Series of the form $\text{Σ}{}_{\mathrm{k\; =\; 1}}{}^{\mathrm{\infty }}\text{(}\text{2k}\text{2k}\text{)}{}^{\mathrm{?1}}\text{k}{}^{\mathrm{?n}}$ may be expressed as log sin integrals and are shown to be summable exactly in terms of Dirichlets L-series for values of n up to and including 5. Other related series are also discussed and several exact results are given.     

Sur la densité divisorielle d'une suite d'entiers

The condition $\text{Σ}{}_{\mathrm{k<x}}\text{|Σ}{}_{\mathrm{n<x}}\text{(χ(n) ? z)}\text{4}{}^{\mathrm{\Omega (n)}}\text{n}\text{| = o(√}\text{log}\text{x)}$, where Ω(n) stands for the number of prime factors, counted according to multiplicity, of the positive integer n, is shown to be necessary and sufficient for the integer sequence with characteristic function χ to have divisor density z, i.e., Σ_d|nχ(d) = (z + o(1)) Σ_d|n 1 when n → ∞ if one neglects a sequence of asymptotic density zero. Among the applications, the following result, first conjectured by R. R. Hall, is proved: given any positive α, we have, for almost all n's, and uniformly with respect to z in |0, 1|,

$\text{card}\text{{d:d|n, (}\text{log}\text{d)}{}^{\mathrm{\alpha }}\text{< z (}\text{mod}\text{1)}=(z+o(1))}\text{∑}\text{d|n}\text{1.}$

     

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.     

Minimax estimators for a multinormal precision matrix

Let S_p×p ～ Wishart (Σ, k), Σ unknown, k > p + 1. Minimax estimators of Σ^?1 are given for L₁, an Empirical Bayes loss function; and L₂, a standard loss function (R_i ≡ E(L_i ∣ Σ), i = 1, 2). The estimators are $\text{Σ}\text{?}{}^{\mathrm{?1}}\text{=}\text{a}\text{S}{}^{\mathrm{?1}}\text{+}\text{br}\text{(S)I}{}_{\mathrm{p\times p}}$, a, b ≥ 0, r(·) a functional on $\text{R}{}^{\mathrm{<mtext>p(p+2)</mtext><mtext>2</mtext>}}$. Stein, Efron, and Morris studied the special cases $\text{Σ}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{=}\text{a}\text{S}{}^{\mathrm{?1}}\text{(E}\text{Σ}\text{?}{}_{\mathrm{<mtext>k?p?1</mtext>}}{}^{\mathrm{?1}}\text{= Σ}{}^{\mathrm{?1}}\text{)}$ and $\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{=}\text{a}\text{S}{}^{\mathrm{?1}}\text{+ (b/tr S)I}$, for certain, a, b. From their work $\text{R}{}_1\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) ≤ R}{}_1\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{; S) (?Σ)}$, a = k ? p ? 1, b = p² + p ? 2; whereas, we prove $\text{R}{}_2\text{(Σ}{}^{\mathrm{?1}}\text{Σ}\text{?}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{; S) ≤}\text{R}{}_2\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) (?Σ)}$. The reversal is surprising because $\text{L}{}_1\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) →}\text{L}{}_2\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S)}$ a.e. (for a particular L₂). Assume $\text{R}$ (compact) ? $\text{S}$, $\text{S}$ the set of p × p p.s.d. matrices. A “divergence theorem” on functions F_p×p : $\text{R}$ → $\text{S}$ implies identities for R_i, i = 1, 2. Then, conditions are given for $\text{R}{}_{\mathrm{i}}\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}^{\mathrm{?1}}\text{; S) ≤}\text{R}{}_{\mathrm{i}}\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) ≤}\text{R}{}_{\mathrm{i}}\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{; S) (?Σ)}$, i = 1, 2. Most of our results concern estimators with r(S) = t(U)/tr(S), U = p ∣S∣^1/p/tr(S).     

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.