Algorithms for determining integer complexity

We present three algorithms to compute the complexity $\Vert n\Vert$ of all natural numbers $n\le N$. The first of them is a brute force algorithm, computing all these complexities in time $O(N^2)$ and space $O(N\log^2 N)$. The main problem of this algorithm is the time needed for the computation. In 2008 there appeared three independent solutions to this problem: V. V. Srinivas and B. R. Shankar [11], M. N. Fuller [7], and J. Arias de Reyna and J. van de Lune [3]. All three are very similar. Only [11] gives an estimation of the performance of its algorithm, proving that the algorithm computes the complexities in time $O(N^{1+\beta})$, where $1+\beta =\log3/\log2\approx1.584963$. The other two algorithms, presented in [7] and [3], were very similar but both superior to the one in [11]. In Section 2 we present a version of these algorithms and in Section 4 it is shown that they run in time $O(N^\alpha)$ and space $O(N\log\log N)$. (Here $\alpha = 1.230175$). In Section 2 we present the algorithm of [7] and [3]. The main advantage of this algorithm with respect to that in [11] is the definition of kMax in Section 2.7. This explains the difference in performance from $O(N^{1+\beta})$ to $O(N^\alpha)$. In Section 3 we present a detailed description a space-improved algorithm of Fuller and in Section 5 we prove that it runs in time $O(N^\alpha)$ and space $O(N^{(1+\beta)/2}\log\log N)$, where $\alpha=1.230175$ and $(1+\beta)/2\approx0.792481$.Comment: 21 pages. v2: We improved the computations to get a better bound for $\alpha

Some bounds and limits in the theory of Riemann's zeta function

For any real a>0 we determine the supremum of the real \sigma\ such that \zeta(\sigma+it) = a for some real t. For 0 1 the results turn out to be quite different.} We also determine the supremum E of the real parts of the `turning points', that is points \sigma+it where a curve Im \zeta(\sigma+it) = 0 has a vertical tangent. This supremum E (also considered by Titchmarsh) coincides with the supremum of the real \sigma\ such that \zeta'(\sigma+it) = 0 for some real t. We find a surprising connection between the three indicated problems: \zeta(s) = 1, \zeta'(s) = 0 and turning points of \zeta(s). The almost extremal values for these three problems appear to be located at approximately the same height.Comment: 28 pages 1 figur

On the exact location of the non-trivial zeros of Riemann's zeta function

In this paper we introduce the real valued real analytic function kappa(t) implicitly defined by exp(2 pi i kappa(t)) = -exp(-2 i theta(t)) * (zeta'(1/2-it)/zeta'(1/2+it)) and kappa(0)=-1/2. (where theta(t) is the function appearing in the known formula zeta(1/2+it)= Z(t) * e^{-i theta(t)}). By studying the equation kappa(t) = n (without making any unproved hypotheses), we will show that (and how) this function is closely related to the (exact) position of the zeros of Riemann's zeta(s) and zeta'(s). Assuming the Riemann hypothesis and the simplicity of the zeros of zeta(s), it will follow that the ordinate of the zero 1/2 + i gamma_n of zeta(s) will be the unique solution to the equation kappa(t) = n.Comment: 28 pages, 9 figures. Added a referenc

Constraints, recent change, objective and subjective well-being: urban, rural-nonfarm, and rural-farm households in Poland

This study is in response to the findings of Winter et al. (1999) that in 1994 Poland, compared to urban households, rural households experienced worse domain living conditions yet rated their household situation as better. Using the same data, which were collected in the province of Lublin in Poland from primarily female respondents, 592 households are analyzed;The relationship between socioeconomic and demographic characteristics of the household (constraints), recent change in the household\u27s situation, conditions (objective well-being), and satisfaction (subjective well-being) is assessed globally and within five specific domains: health, housing, household equipment, food, and Transportation; Parallel ordinary least squares regression analyses are performed and total effects are decomposed for urban and rural residents, and further, for rural-nonfarm and rural-farm households. The means of the exogenous and endogenous variables are compared for urban, rural-nonfarm and rural-farm households;The results of the comparison-of-means procedures indicate that respondents in urban areas tend to have the highest levels of education, and urban households are apt to have the best conditions. Respondents from rural-nonfarm households tend to be the oldest, and rural-farm households are likely to have the most household members, children, and workers;The findings of the regression analyses and the decomposition of total effects indicate that, of the household characteristics entered into the analyses, age of the respondent, education of the respondent, and total household income are the most consistent predictors of conditions, and of satisfaction, indirectly through their influence on conditions;As expected, constraints and recent change affect objective well-being, and constraints, recent change, and objective well-being affect subjective well-being. A finding that was not expected, however, is that recent change, rather than objective well-being, is a more consistent and a stronger predictor of subjective well-being. It is possible that the measures of recent change and conditions are entered into the model in reverse order; rather than reported recent change leading to conditions, it is the perception of recent change that is influenced by current conditions, which, in turn, affects satisfaction. It also is likely that recent change and conditions influence one another, and it is this relationship that affects satisfaction