29 research outputs found
A Creative Review on Coprime (Prime) Graphs
Coprime labelings and Coprime graphs have been of interest since 1980s and got popularized by the Entringer-Tout Tree Conjecture. Around the same time Newman's coprime mapping conjecture was settled by Pomerance and Selfridge. This result was further extended to integers in arithmetic progression. Since then coprime graphs were studied for various combinatorial properties. Here, coprimality of graphs for classes of graphs under the themes: Bipartite with special attention to Acyclicity, Eulerian and Regularity. Extremal graphs under non-coprimality and Eulerian properties are studied. Embeddings of coprime graphs in the general graphs, the maximum coprime graph and the Eulerian coprime graphs are studied as subgraphs and induced subgraphs. The purpose of this review is to assimilate the available works on coprime graphs. The results in the context of these themes are reviewed including embeddings and extremal problems
Orders and partitions of integers induced by arithmetic functions
We pursue the question how integers can be ordered or partitioned according
to their divisibility properties. Based on pseudometrics on , we
investigate induced preorders, associated equivalence relations, and quotient
sets. The focus is on metrics or pseudometrics on , the set of
divisors of a given modulus , that can be extended to
pseudometrics on .
Arithmetic functions can be used to generate such pseudometrics. We discuss
several subsets of additive and multiplicative arithmetic functions and various
combinations of their function values leading to binary metric functions that
represent different divisibility properties of integers.
We conclude this paper with numerous examples and review the most important
results. As an additional result, we derive a necessary condition for the truth
of the odd k-perfect number conjecture.Comment: 50 pages, 3 diagram
Small gaps in the spectrum of the rectangular billiard
We study the size of the minimal gap between the first N eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio α, in comparison to the corresponding quantity for a Poissonian sequence. If α is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size 1/N, which is essentially consistent with Poisson statistics. We also give related results for a set of α's of full measure. However, on a fine scale we show that Poisson statistics is violated for all α. The proofs use a variety of ideas of an arithmetical nature, involving Diophantine approximation, the theory of continued fractions, and results in analytic number theory
Mean values and correlations of multiplicative functions : the ``pretentious" approach
Le sujet principal de cette thĂšse est lâĂ©tude des valeurs moyennes et corrĂ©lations de fonctions
multiplicatives. Les résultats portant sur ces derniers sont subséquemment appliqués à la
résolution de plusieurs problÚmes.
Dans le premier chapitre, on rappelle certains résultats classiques concernant les valeurs
moyennes des fonctions multiplicatives. On y énonce également les théorÚmes principaux de
la thĂšse.
Le deuxiĂšme chapitre consiste de lâarticle âMean values of multiplicative functions over
the function fields". En se basant sur des résultats classiques de Wirsing, de Hall et de Tenenbaum
concernant les fonctions multiplicatives arithmétiques, on énonce et on démontre des
théorÚmes qui y correspondent pour les fonctions multiplicatives sur les corps des fonctions
Fq[x]. Ainsi, on résoud un problÚme posé dans un travail récent de Granville, Harper et
Soundararajan. On décrit dans notre thése certaines caractéristiques du comportement des
fonctions multiplicatives sur les corps de fonctions qui ne sont pas présentes dans le contexte
des corps de nombres. Entre autres, on introduit pour la premiĂšre fois une notion de
âsimulationâ pour les fonctions multiplicatives sur les corps de fonctions Fq[x].
Les chapitres 3 et 4 comprennent plusieurs rĂ©sultats de lâarticle âCorrelations of multiplicative
functions and applications". Dans cet article, on détermine une formule asymptotique
pour les corrélations
X
n6x
f1(P1(n)) · · · fm(Pm(n)),
oĂč f1, . . . ,fm sont des fonctions multiplicatives de module au plus ou Ă©gal Ă 1 âsimulatricesâ
qui satisfont certaines hypothĂšses naturelles, et P1, . . . ,Pm sont des polynomes ayant des coefficients
positifs. On dĂ©duit de cette formule plusieurs consĂ©quences intĂ©ressantes. Dâabord,
on donne une classification des fonctions multiplicatives f : N ! {â1,+1} ayant des sommes
partielles uniformĂ©ment bornĂ©es. Ainsi, on rĂ©soud un problĂšme dâErdos datant de 1957 (dans
la forme conjecturée par Tao). Ensuite, on démontre que si la valeur moyenne des écarts
|f(n + 1) â f(n)| est zĂ©ro, alors soit |f| a une valeur moyenne de zĂ©ro, soit f(n) = ns avec
iii
Re(s) < 1. Ce résultat affirme une ancienne conjecture de Kåtai. Enfin, notre théorÚme principal
est utilisĂ© pour compter le nombre de reprĂ©sentations dâun entier n en tant que somme
a+b, oĂč a et b proviennent de sous-ensembles multiplicatifs fixĂ©s de N. Notre dĂ©monstration
de ce rĂ©sultat, dĂ» Ă lâorigine Ă BrĂŒdern, Ă©vite lâusage de la âmĂ©thode du cercle".
Les chapitres 5 et 6 sont basĂ©s sur les rĂ©sultats obtenus dans lâarticle âEffective asymptotic
formulae for multilinear averages and sign patterns of multiplicative functions," un
travail conjoint avec Alexander Mangerel. DâaprĂšs une mĂ©thode analytique dans lâesprit du
théorÚme des valeurs moyennes de Halåsz, on détermine une formule asymptotique pour les
moyennes multidimensionelles
xâl
X
n2[x]l
Y
16j6k
fj(Lj(n)),
lorsque x ! 1, oĂč [x] := [1,x] et L1, . . . ,Lk sont des applications linĂ©aires affines qui satisfont
certaines hypothÚses naturelles. Notre méthode rend ainsi une démonstration neuve
dâun rĂ©sultat de Frantzikinakis et Host avec, Ă©galement, un terme principal explicite et un
terme dâerreur quantitatif. On applique nos formules Ă la dĂ©monstration dâun phĂ©nomĂšne
local-global pour les normes de Gowers des fonctions multiplicatives. De plus, on découvre
et explique certaines irrégularités dans la distribution des suites de signes de fonctions
multiplicatives f : N ! {â1,+1}. Visant de tels rĂ©sultats, on dĂ©termine les densitĂ©s asymptotiques
des ensembles dâentiers n tels que la fonction f rend une suite fixĂ©e de 3 ou 4 signes
dans presque toutes les progressions arithmétiques de 3 ou 4 termes, respectivement, ayant
n comme premier terme. Ceci mÚne à une généralisation et amélioration du travail de Buttkewitz
et Elsholtz, et donne un complĂ©ment Ă un travail rĂ©cent de MatomĂ€ki, RadziwiĆĆ et
Tao sur les suites de signes de la fonction de Liouville.The main theme of this thesis is to study mean values and correlations of multiplicative
functions and apply the corresponding results to tackle some open problems.
The first chapter contains discussion of several classical facts about mean values of multiplicative
functions and statement of the main results of the thesis.
The second chapter consists of the article âMean values of multiplicative functions over
the function fields". The main purpose of this chapter is to formulate and prove analog of
several classical results due to Wirsing, Hall and Tenenbaum over the function field Fq[x],
thus answering questions raised in the recent work of Granville, Harper and Soundararajan.
We explain some features of the behaviour of multiplicative functions that are not present
in the number field settings. This is accomplished by, among other things, introducing the
notion of âpretentiousness" over the function fields.
Chapter 3 and Chapter 4 include results of the article âCorrelations of multiplicative
functions and applications". Here, we give an asymptotic formula for correlations
X
n_x
f1(P1(n))f2(P2(n)) · · · · · fm(Pm(n))
where f . . . ,fm are bounded âpretentious" multiplicative functions, under certain natural
hypotheses. We then deduce several desirable consequences. First, we characterize all multiplicative
functions f : N ! {â1,+1} with bounded partial sums. This answers a question
of Erdos from 1957 in the form conjectured by Tao. Second, we show that if the average
of the first divided difference of multiplicative function is zero, then either f(n) = ns for
Re(s) < 1 or |f(n)| is small on average. This settles an old conjecture of KĂĄtai. Third, we
apply our theorem to count the number of representations of n = a + b where a,b belong to
some multiplicative subsets of N. This gives a new "circle method-free" proof of the result of
BrĂŒdern.
Chapters 5 and Chapter 6 are based on the results obtained in the article âEffective
asymptotic formulae for multilinear averages and sign patterns of multiplicative functions,"
joint with Alexander Mangerel. Using an analytic approach in the spirit of HalĂĄszâ mean
v
value theorem, we compute multidimensional averages
xâl
X
n2[x]l
Y
16j6k
fj(Lj(n))
as x ! 1, where [x] := [1,x] and L1, . . . ,Lk are affine linear forms that satisfy some natural
conditions. Our approach gives a new proof of a result of Frantzikinakis and Host that is
distinct from theirs, with explicit main and error terms.
As an application of our formulae, we establish a local-to-global principle for Gowers norms
of multiplicative functions. We reveal and explain irregularities in the distribution of the
sign patterns of multiplicative functions by computing the asymptotic densities of the sets
of integers n such that a given multiplicative function f : N ! {â1, 1} yields a fixed sign
pattern of length 3 or 4 on almost all 3- and 4-term arithmetic progressions, respectively,
with first term n. The latter generalizes and refines the work of Buttkewitz and Elsholtz and
complements the recent work of Matomaki, RadziwiĆĆ and Tao.
We conclude this thesis by discussing some work in progress
A Generalised abc Conjecture and Quantitative Diophantine Approximation
The abc Conjecture and its number field variant have huge implications across a wide
range of mathematics. While the conjecture is still unproven, there are a number of
partial results, both for the integer and the number field setting. Notably, Stewart
and Yu have exponential abc bounds for integers, using tools from linear forms in
logarithms, while GyĆry has exponential abc bounds in the number field
case, using methods from S-unit equations [20]. In this thesis, we aim to combine
these methods to give improved results in the number field case. These results are
then applied to the effective Skolem-Mahler-Lech problem, and to the smooth abc
conjecture.
The smooth abc conjecture concerns counting the number of solutions to a+b = c
with restrictions on the values of a, b and c. this leads us to more general methods
of counting solutions to Diophantine problems. Many of these results are asymptotic
in nature due to use of tools such as Lemmas 1.4 and 1.5 of Harman's "Metric Number Theory". We make these
lemmas effective rather than asymptotic other than on a set of size ÎŽ > 0, where ÎŽ is
arbitrary. From there, we apply these tools to give an effective Schmidtâs Theorem,
a quantitative Koukoulopoulos-Maynard Theorem (also referred to as the Duffin-
Schaeffer Theorem), and to give effective results on inhomogeneous Diophantine
Approximation on M0-sets, normal numbers and give an effective Strong Law of
Large Numbers. We conclude this thesis by giving general versions of Lemmas 1.4
and 1.5 of Harman's "Metric Number Theory"
Small gaps in the spectrum of the rectangular billiard
We study the size of the minimal gap between the first N eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio α, in comparison to the corresponding quantity for a Poissonian sequence. If α is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size 1/N, which is essentially consistent with Poisson statistics. We also give related results for a set of α's of full measure. However, on a fine scale we show that Poisson statistics is violated for all α. The proofs use a variety of ideas of an arithmetical nature, involving Diophantine approximation, the theory of continued fractions, and results in analytic number theory
A Pythagorean Introduction to Number Theory : Right Triangles, Sums of Squares, and Arithmetic
In the ?rst section of this opening chapter we review two different proofs of the PythagoreanTheorem,oneduetoEuclidandtheotheroneduetoaformerpresident oftheUnitedStates,JamesGar?eld.Inthesamesectionwealsoreviewsomehigher dimensional analogues of the Pythagorean Theorem. Later in the chapter we de?ne Pythagorean triples; explain what it means for a Pythagorean triple to be primitive; and clarify the relationship between Pythagorean triples and points with rational coordinates on the unit circle. At the end we list the problems that we will be interested in studying in the book. In the notes at the end of the chapter we talk about Pythagoreans and their, sometimes strange, beliefs. We will also brie?y review the history of Pythagorean triples