In this paper we address some variants for the products of Hadamard and Patterson. We prove that all zeros of the Riemann $\Xi$--function are real. We also prove that the Riemann hypothesis is true. The equivalence theorems associated with the Riemann zeta--function are obtained in detail.

The basic idea is to expand the completed zeta function $\xi(s)$ in MacLaurin series (infinite polynomial), which can be further expressed as infinite product (Hadamard product) by conjugate complex roots. Finally, the functional equation $\xi(s)=\xi(1-s)$ leads to $(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}$ with solution $\alpha_i= \frac{1}{2}, i \in \mathbb{N}$, where $\alpha_i$ are the real parts of the zeros of $\xi(s)$, i.e., $s_i =\alpha_i\pm j\beta_i, i\in \mathbb{N}$. Therefore, a proof of the Riemann Hypothesis is achieved.

Background: Suicide is a major health problem. The aim of this study was a comparative study of the causes and methods of completed suicide referred to legal Medicine Organization in Bushehr Province during the years 2006- 2015. Methods: This consideration has been done as a retrospective cross-sectional study; the data were collected by the check list. After coordination to the medico legal 343 cases of completed suicide lead to death over a ten year period from 2006 to 2015 were extracted and the data were analyses by SPSS software. Results: The rate of completed suicide was 2.43 and 1.12 suicides per 100,000 people in male and female, respectively. The mean age were 29.25 and 27.61 years in male and female, respectively p>0.05. A higher completed suicide rate was found among single people. The most common ways of a suicide among people who live in Bushehr were hanging out 184(53.6%), self-Immolation 83(24.2%) and firearms 37(10.8%). Motive of suicide was psychological disorders, Physical disease, Conflict with family and substance abuse. 43(12.5%) people smoking and 22(6.4%) people were both smoking and opioid. During the ten year period of studied, throughout the all years that frequency of hanging out has been higher than others. Conclusion: Due to the high incidence of completed suicide in young people and unmarried individuals also hanging out in male and self-burning in females are prevalent. So determination of preventive measures appears to be necessary. Social factors such as psychological disorders and family conflicts have a noticeable role from .Bushehr people.

Background The link between age at marriage and first birth in explaining completed family size is not always direct, due to heterogeneity in circumstances, that compel individual women to marry or initiate childbearing at a particular age. We analyzed data for 1020 women aged 45-49 in 2014 of the 1965-1969 birth cohort from the 2013-14 Zambia Demographic and Health Survey (ZDHS). Methods We fitted a bivariate and multivariate multinomial logistic regression to establish the relationship between a mother’s age at first marriage and at first birth on completed family size (CFS). Chi-square test of proportions measuring differences in proportions and relative risk ratios (RRR) with confidence intervals at 95% are reported. Results Our results show that the average CFS was 6.7 (95% CI: 6.5 – 6.9) among women completing their reproductive span in 2014 with mean age at first marriage and birth being 18.3 years (95% CI: 18.0 – 18.5) and 18.9 years (95% CI: 18.7 – 19.1) respectively. Women marrying at younger ages and having their first birth at younger ages were more likely (RRR: 1.187; 95% CI: 1.138-1.239 and RRR: 0.195; 95% CI: 0.074-0.511 respectively) to have higher CFS than their compatriots that initiated both marriage and childbearing at or after age 22 controlling for covariates as presented in the controlled model. The independent effects model shows that the risk of having 1-3 children compared to 6 or more children was lower (RRR: 0.073; 95% CI: 0.009-0.611; RRR: 0.136; 95% CI: 0,046-0.402 and RRR: 0.421; 05% CI: 0.135-1.312) for women whose ages at first marriage were <15, 15-18 and 19-21 respectively relative to 22+. Women with 1-3 children were 2.5 times more likely to use contraception than women with 6 or more children. Conclusion Having no education, being a rural resident and having a medium household wealth all increase the risk of having higher CFS. Women that marry before age 19 have a higher likelihood of having 6 or more children by the end of their reproductive period. The study concludes that apart from a woman’s age at first marriage and first birth, a complex network of factors interact to determined CFS.