Additive functions at consecutive integers

Additive functions f ₀,…,f _k satisfying the relation $$\liminf_{x\to\infty}\frac{1}{x}\sum_{n\leqq x} \|f_0(n)+ \dots+f_k(n+k) +\Gamma\| =0 $$ are characterized for those f _j for which ∥f _j (p)∥≦η holds for every prime p, and η is a suitable small positive number, $\|z\| =\min_{k\in\mathbb{Z}} {|z-k|}$ .     

On the multiplicative representation of integers

Leta ₁<a ₂<··· be an infinite sequence of integers. Denote byg(n) the number of solutions ofn=a _i···a _j. Ifg(n)>0 for a sequencen of positive upper density then lim supg(n)=∞. Dedicated to my friend A. D. Wallace on the occasion of his 60th birthday.     

On the longest length of consecutive integers

Choose m numbers from the set {1, 2, …, n} at random without replacement. In this paper we first establish the limiting distribution of the longest length of consecutive integers and then apply the result to test randomness of selecting numbers without replacement.     

On a division property of consecutive integers

Pillai and Brauer proved that form≧17 we can find blocksB _m ofm consecutive integers such that no element in the block is pairwise prime with each of the other elements. The following basic generalization is proved: For eachd>1 there is a numberG(d) such that for everym≧G(d) there exist infinitely many blocksB _m ofm consecutive integers, such that for eachr∈B _m there existss∈B _m , (r,s)≧d.     

On the ratio of two blocks of consecutive integers

Under certain assumptions, it is shown that eq. (2) has only finitely many solutions in integersx≥0,y≥0,k≥2,l≥0. In particular, it is proved that (2) witha=b=1, l=k implies thatx=7,y=0,k=3.     

On the multiplicative group generated by a dense sequence of integers

The subgroup of the positive rationals, generated by a sequence of upper density , has index essentially not exceeding ^–1.In honor of the seventieth birthday of Professor E. HlawkaPartially supported by N.S.F. Contract No. DMS-8500949.     

On pseudorandom properties of multiplicative functions

In an earlier paper [3] Cassaigne et al studied the pseudorandom properties of the Liouville function. In this paper some of their results are generalized and sharpened considerably. This revised version was published online in June 2006 with corrections to the Cover Date.     

Prime factors of consecutive integers

This note contains a new algorithm for computing a function introduced by Erdős to measure the minimal gap size in the sequence of integers at least one of whose prime factors exceeds . This algorithm enables us to show that is not monotone, verifying a conjecture of Ecklund and Eggleton.

     

On multiplicative functions which are small on average

Let f be a completely multiplicative function that assumes values inside the unit disc. We show that if ${\sum_{n \leq x}f(n)\ll x/(\rm log x)^A}$ ∑ n ≤ x f ( n ) ? x / ( l o g x ) A , ${x \geq 2}$ x ≥ 2 , for some A > 2, then either f(p) is small on average or f pretends to be ${\mu(n)n^{it}}$ μ ( n ) n i t for some t.     

On a multiplicative inequaluty for derived functions

В работе доказываетс я следующее неравенс тво. Пусть α₀, α₁, α₂, - произво льные неотрицательн ые числа α₀≠α₂. Тогда, еслиx(t) люб ая функция, для которой п роизводнаяx непреры вна и функция \(x^{a_0 } (\dot x)^{a_1 } (\ddot x)^{a_2 } \) принадлежи т пространствуL _∞[0, 1], то (*) $$\left\| x \right\|_{H^r [0,1]} \leqq c\left\| {x^{a_0 } (\dot x)^{a_1 } (\ddot x)^{a_2 } } \right\|L_{\infty [0,1]} ,$$ где ∥ · ∥H ^r [0,1] - норма в кл ас се функций на отрезке [0, 1], обладающих в простра нствеL _∞[0, 1] дробной производной гельдеровского типа порядкаr=(α₁+2α₂)/(α₀+α₁+α₂);с - конста нта, зависящая только от α₀, α₁, α₂. Это неравенство является точным в том смысле, что показател ьr есть максимальный, п ри котором неравенст во (*) имеет место с конечной конс тантойс. При α₀=α_? появляются логарифмические доб авки. Хорошо известно, что д ля непрерывной на [0, 1] фу нкции частные суммы Фурье п о тригонометрической системе равномерно с уммируются к ней методом (С, 1). И. Пра йс доказал, что для любой неограниченной последовательности целых положительных чисел {P _k} _k ^∞ =1 и Для любогоa∈[0, 1] существует непрерыв ная на [0, 1] функция, ряд Фурье которой по ортонормированно й мультипликативной системе (OHMC) не суммируется методом (С, 1) в точкеx=a. СССР, МОСКВА 103 055 УЛ. ОБРАЗЦОВА 15 МОСКОВСКИЙ ИНСТИТУТ ИНЖЕНЕРОВ ЖЕЛЕЗНОДО РОЖНОГО ТРАНСТПОРТА     

Large strings of consecutive smooth integers

In this note we improve an algorithm from a recent paper by Bauer and Bennett for computing a function of Erdös that measures the minimal gap size f(k) in the sequence of integers at least one of whose prime factors exceeds k. This allows us to compute values of f(k) for larger k and obtain new values of f(k).     

On mean values of multiplicative functions on the symmetric group

Mean values of nonnegative multiplicative functions defined on the symmetric group are explored in the paper. The result gives a sharp quantitative upper bound for their Cesàro mean. An approach that originated in number theory is adopted. It can be further applied for mappings defined on general decomposable structures, in particular, for estimating mean values with respect to multiplicative measures defined on additive partitions of a natural number.     

On multiplicative functions which are additive on sums of primes

Let ${f : \mathbb{N} \to \mathbb{C}}$ be a multiplicative function satisfying f(p ₀) ≠ 0 for at least one prime number p ₀, and let k ≥ 2 be an integer. We show that if the function f satisfies f(p ₁ + p ₂ + . . . + p _k ) = f(p ₁) + f(p ₂) + . . . + f(p _k ) for any prime numbers p ₁, p ₂, . . . ,p _k then f must be the identity f(n) = n for each ${n \in \mathbb{N}}$ . This result for k = 2 was established earlier by Spiro, whereas the case k = 3 was recently proved by Fang. In the proof of this result for k ≥ 6 we use a recent result of Tao asserting that every odd number greater than 1 is the sum of at most five primes.     

Remarks on the structure of the multiplicative monoid of integers modulom

This note presents some results concerningH-classes, Schützenberger groups, and regular elements in the multiplicative monoid ℤ _m of integers modulom. It also shows that in ℤ _m , the product of twoH-classes is anH-class.     

On the difference between the number of prime divisors at consecutive integers

Suppose thatg(n) is equal to the number of divisors ofn, counting multiplicity, or the number of divisors ofn, a≠0 is an integer, andN(x,b)=|{n∶n≤x, g(n+a)−g(n)=b orb+1}|. In the paper we prove that sup _b N(x,b)≤C(a)x)(log log 10 _x )^−1/2 and that there exists a constantC(a,μ)>0 such that, given an integerb |b|≤μ(log logx)^1/2,x≥x _o, the inequalityN(x,b)≥C(a,μ)x(log logx(^−1/2) is valid. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 579–595, October, 1999.