On binary correlations of multiplicative functions

We study logarithmically averaged binary correlations of bounded multiplicative functions $g_1$ and $g_2$. A breakthrough on these correlations was made by Tao, who showed that the correlation average is negligibly small whenever $g_1$ or $g_2$ does not pretend to be any twisted Dirichlet character, in the sense of the pretentious distance for multiplicative functions. We consider a wider class of real-valued multiplicative functions $g_j$, namely those that are uniformly distributed in arithmetic progressions to fixed moduli. Under this assumption, we obtain a discorrelation estimate, showing that the correlation of $g_1$ and $g_2$ is asymptotic to the product of their mean values. We derive several applications, first showing that the number of large prime factors of $n$ and $n+1$ are independent of each other with respect to the logarithmic density. Secondly, we prove a logarithmic version of the conjecture of Erd\H{o}s and Pomerance on two consecutive smooth numbers. Thirdly, we show that if $Q$ is cube-free and belongs to the Burgess regime $Q\leq x^{4-\varepsilon}$, the logarithmic average around $x$ of the real character $\chi \pmod{Q}$ over the values of a reducible quadratic polynomial is small.Comment: 33 pages; Referee comments incorporated; To appear in Forum Math. Sigm

The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures

We study the asymptotic behaviour of higher order correlations $$\mathbb{E}_{n \leq X/d} g_1(n+ah_1) \cdots g_k(n+ah_k)$$ as a function of the parameters $a$ and $d$, where $g_1,\dots,g_k$ are bounded multiplicative functions, $h_1,\dots,h_k$ are integer shifts, and $X$ is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all $X$ if $g_1 \cdots g_k$ does not (weakly) pretend to be a twisted Dirichlet character $n \mapsto \chi(n)n^{it}$, and behave asymptotically like a multiple of $d^{-it} \chi(a)$ otherwise. This extends our earlier work on the structure of logarithmically averaged correlations, in which the $d$ parameter is averaged out and one can set $t=0$. Among other things, the result enables us to establish special cases of the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we establish the $k$-point Chowla conjecture $\mathbb{E}_{n \leq X} \lambda(n+h_1) \cdots \lambda(n+h_k)=o(1)$ for $k$ odd or equal to $2$ for all scales $X$ outside of a set of zero logarithmic density.Comment: 48 pages, no figures. Referee comments incorporate

The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures

Let $g_0,\dots,g_k: {\bf N} \to {\bf D}$ be $1$-bounded multiplicative functions, and let $h_0,\dots,h_k \in {\bf Z}$ be shifts. We consider correlation sequences $f: {\bf N} \to {\bf Z}$ of the form $$f(a):= \widetilde{\lim}_{m \to \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_0(n+ah_0) \dots g_k(n+ah_k)}{n}$$ where $1 \leq \omega_m \leq x_m$ are numbers going to infinity as $m \to \infty$, and $\widetilde{\lim}$ is a generalised limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely that these sequences $f$ are the uniform limit of periodic sequences $f_i$. Furthermore, if the multiplicative function $g_0 \dots g_k$ "weakly pretends" to be a Dirichlet character $\chi$, the periodic functions $f_i$ can be chosen to be $\chi$-isotypic in the sense that $f_i(ab) = f_i(a) \chi(b)$ whenever $b$ is coprime to the periods of $f_i$ and $\chi$, while if $g_0 \dots g_k$ does not weakly pretend to be any Dirichlet character, then $f$ must vanish identically. As a consequence, we obtain several new cases of the logarithmically averaged Elliott conjecture, including the logarithmically averaged Chowla conjecture for odd order correlations. We give a number of applications of these special cases, including the conjectured logarithmic density of all sign patterns of the Liouville function of length up to three, and of the M\"obius function of length up to four.Comment: 41 pages, no figures. Submitted, Duke Math. J.. Referee changes incorporate

Odd order cases of the logarithmically averaged Chowla conjecture

A famous conjecture of Chowla states that the Liouville function $\lambda(n)$ has negligible correlations with its shifts. Recently, the authors established a weak form of the logarithmically averaged Elliott conjecture on correlations of multiplicative functions, which in turn implied all the odd order cases of the logarithmically averaged Chowla conjecture. In this note, we give a new and shorter proof of the odd order cases of the logarithmically averaged Chowla conjecture. In particular, this proof avoids all mention of ergodic theory, which had an important role in the previous proof.Comment: 15 pages, no figures, submitted, J. Numb. Thy. Bordeau

On the Liouville function at polynomial arguments

Let $\lambda$ denote the Liouville function. A problem posed by Chowla and by Cassaigne-Ferenczi-Mauduit-Rivat-S\'ark\"ozy asks to show that if $P(x)\in \mathbb{Z}[x]$, then the sequence $\lambda(P(n))$ changes sign infinitely often, assuming only that $P(x)$ is not the square of another polynomial. We show that the sequence $\lambda(P(n))$ indeed changes sign infinitely often, provided that either (i) $P$ factorizes into linear factors over the rationals; or (ii) $P$ is a reducible cubic polynomial; or (iii) $P$ factorizes into a product of any number of quadratics of a certain type; or (iv) $P$ is any polynomial not belonging to an exceptional set of density zero. Concerning (i), we prove more generally that the partial sums of $g(P(n))$ for $g$ a bounded multiplicative function exhibit nontrivial cancellation under necessary and sufficient conditions on $g$. This establishes a "99% version" of Elliott's conjecture for multiplicative functions taking values in the roots of unity of some order. Part (iv) also generalizes to the setting of $g(P(n))$ and provides a multiplicative function analogue of a recent result of Skorobogatov and Sofos on almost all polynomials attaining a prime value.Comment: Simplified some proof

The Experience and Beauty in the Cultural Heritage Discourse: Reflections from Two Case Studies

The cultural heritage in the built environment is developing discursively, and the   concept is today exposed more variously than a decade ago when I explained in the doctoral dissertation through a case study how the place, the process and the experience were arising in the Foucaldian discourses. My on-going research on the change of the cultural heritage discourse (kulttuuriympäristö) is showing how the designations are changing and the concepts re-defined. The national strategy on the cultural heritage (2014) is emphasizing everybody’s right on the good cultural heritage environment and also the responsibilities to take care of that. When we now share the idea that the cultural heritage in the built environment is belonging to all of us, the places and experiences of all are also important. Nevertheless, in the end is the experience or the aesthetic experience exactly, really important when opinions are contradictory and the crucial decision has to be made: to preserve or to dismantle the building? The absence of aesthetics in decision-making has been extremely explicit since the recession of the 1990s and public discussions and political decision-making seem to involve mostly economic arguments. Architects are using the experience, meaning the aesthetic, bodily experience or referring to art, in their professional speech but to speak about “the beauty experience” or able to emphasize the meaning of beauty in architecture, and also in the environment is usually left outside the discussions. The experience, together with reflection, following Dewey, is very important in the speech of teaching architects. In the cultural heritage discourse, narratives, experiences and local stories from bottom-up are arising but do we talk about the aesthetics or the experiences of beauty in the built environment? In this paper, the aim is to discuss about the meaning of experiences and the role of beauty in cultural heritage discourse. The method used here is the case study research, and two local cases from different decades will be introduced to demonstrate how miniscule or completely absent aesthetic argumentation in decision making processes can remain, and how different the solutions ended up, though both cases concerned the question of built cultural heritage. The central question in my on-going research project on the changing cultural heritage discourse is: How “the aesthetic experience” is appearing today in the cultural heritage discourses? This paper aims to cast light on that and tries to answer especially this: How did the cultural heritage discourse evolve from different experiences; and how did ugliness become important rather than beauty in the case studies

Topics in Multiplicative Number Theory

This thesis is comprised of four articles in multiplicative number theory, a subfield of analytic number theory that studies questions related to prime numbers and multiplicative functions. A central principle in multiplicative number theory is that multiplicative structures, such as the primes or the values of a multiplicative function, should not correlate with additive structures of various types. The results in this thesis can be interpreted as instances of this principle. In the first article, we consider the problem of finding almost primes in almost all short intervals, which is a natural approximation to the problem of finding primes in short intervals. We show that almost all intervals of nearly optimal length contain a product of exactly three primes. For products of exactly two primes, we improve a result of Harman. The proofs are based on careful analysis of Dirichlet polynomials related to almost primes. The second article is about the Goldbach problem for a sparse subset of the primes. Vinogradov famously showed that any large odd number is the sum of three primes, so it is natural to study the same problem with the summands coming from a subset of the primes. Improving a result of Matomäki, we show that a special set of primes, consisting of primes representable as one plus the sum of two squares, satisfies the ternary Goldbach problem. We also establish a number of other additive results for this same set of primes. The proofs use sieve methods and transference principles for additive equations in primes. We also study the Möbius function and its autocorrelations. A famous conjecture of Chowla asserts that products of shifts of the Möbius function should have mean zero. In the third article, together with T. Tao we settle a logarithmic version of this conjecture in all the cases involving an odd number of shifts. This complements Tao's earlier result that the two-point Chowla conjecture holds with logarithmic weights. Lastly, in the fourth article, we study binary correlations of multiplicative functions with logarithmic weights. We prove an asymptotic formula for these correlations for a wide class of multiplicative functions, extending an earlier result of Tao. We then derive a number of applications regarding the largest prime factors of consecutive integers, including a logarithmic version of a conjecture of Erdős and Turán. Moreover, we prove a new estimate for character sums over reducible quadratic polynomials.Tämä väitöskirja koostuu neljästä artikkelista multiplikatiivisessa lukuteoriassa, joka on alkulukuja ja multiplikatiivisia funktioita tutkiva analyyttisen lukuteorian haara. Keskeinen periaate multiplikatiivisessa lukuteoriassa on, että multiplikatiivisten objektien (kuten alkulukujen tai multiplikatiivisten funktioiden arvojen) ei pitäisi korreloida additiivisten objektien kanssa. Tämän väitöskirjan tulokset voidaankin tulkita kyseisen periaatteen ilmentyminä. Ensimmäisessä artikkelissa tarkastelemme melkein alkulukujen löytämistä melkein kaikilta lyhyiltä väleiltä; tämä on luonnollinen approksimaatio alkulukujen löytämiselle lyhyiltä väleiltä. Osoitamme, että melkein kaikki välit, joiden pituus on lähes optimaalisen lyhyt, sisältävät tasan kolmen alkuluvun tulon. Tasan kahden alkuluvun tulojen tapauksessa parannamme Harmanin tulosta. Todistukset perustuvat melkein alkulukuihin liitettyjen Dirichlet'n polynomien tarkkaan analysointiin. Toinen artikkeli koskee Goldbach-ongelmaa eräälle harvalle osajoukolle alkulukuja. Vinogradov osoitti kuuluisassa työssään, että jokainen riittävän suuri pariton luku on kolmen alkuluvun summa, joten on luonnollista tarkastella vastaavaa ongelmaa alkulukujen osajoukoille. Parantaen Matomäen tulosta osoitamme, että vastaus ternääriseen Goldbach-oneglmaan on positiivinen niiden alkulukujen joukolle, jotka voidaan esittää ykkösen ja kahden neliöluvun summana. Osoitamme myös useita muita additiivisia tuloksia samalle alkulukujen osajoukolle. Todistukset käyttävät seulamenetelmiä sekä ns. transferenssiperiaatteita additiivisille yhtälöille alkulukujen joukossa. Tutkimme myös Möbiuksen funktiota ja sen autokorrelaatioita. Chowlan kuuluisa konjektuuri väittää, että Möbiuksen funktioiden translaatioiden tuloilla pitäisi olla keskiarvo nolla. Kolmannessa artikkelissa yhdessä T. Taon kanssa ratkaisemme logaritmisen version tästä konjektuurista kaikissa tapauksissa, joissa translaatioiden määrä on pariton. Tämä täydentää Taon aikaisempaa tulosta, jonka mukaan kahden pisteen Chowlan konjektuuri pätee logaritmisilla painoilla. Lopuksi neljännessä artikkelissa tutkimme multiplikatiivisten funktioiden binäärisiä korrelaatioita logaritmisilla painoilla. Todistamme asymptoottisen kaavan näille korrelaatioille, joka pätee laajalle luokalle multiplikatiivisia funktioita ja parantaa Taon aikaisempaa tulosta. Johdamme sitten useita sovelluksia koskien peräkkäisten lukujen suurimpia alkutekijöitä -- mukaan lukien logaritmisen version eräästä Erdősin ja Turánin konjektuurista. Lisäksi todistamme uuden arvion karakterisummille yli jaollisen toisen asteen polynomin arvojen

Organizational Culture: Case of the Finnish Construction Industry

Academic literature has long recognized the correlation between a company’s organizational culture and its quality performance. The Finnish construction industry is still a highly human powered industry, and thus, organizational culture is seen to have a significant effect on an organization’s efficiency as well. The aim of this study is to examine and determine organizational cultural profiles of organizations in the Finnish construction industry as they are currently perceived and preferred by professionals themselves. In all, 121 professionals working in organizations in the Finnish construction industry were surveyed using the Organizational Culture Assessment Instrument (OCAI). The reliability of characteristics was tested by calculating Cronbach alpha reliability coefficients, and the found differences between the response characteristics were analysed in-depth with paired and independent t-test analyses. The findings show that, on average, construction industry organizations in Finland currently operate with a mixture of clan and hierarchy cultures. Thus, the current organizational culture stresses the point of view of internal focus and integration. However, the organizations desired to emphasize more flexibility and discretion toward individuals. The novelty value of this paper is presenting existing and preferred culture profiles in the Finnish construction industry. These found profiles have the potential to improve management of organizations, which results in better efficiency of the industry through better performance of organizations in the construction industry

The 1950s and 1960s Modern Home: Magazines as research material

   This article is based on my keynote lecture at the architectural research symposium held at Aalto University on October 25, 2018. The lecture dealt with my doctoral dissertation: Modern Home. Single-family housing ideals as presented in Finnish architecture and interior design magazines in the 1950s and 1960s. (Sanaksenaho, 2017)&nbsp