The distribution of the variance of primes in arithmetic progressions

Hooley conjectured that the variance V(x;q) of the distribution of primes up to x in the arithmetic progressions modulo q is asymptotically x log q, in some unspecified range of q\leq x. On average over 1\leq q \leq Q, this conjecture is known unconditionally in the range x/(log x)^A \leq Q \leq x; this last range can be improved to x^{\frac 12+\epsilon} \leq Q \leq x under the Generalized Riemann Hypothesis (GRH). We argue that Hooley's conjecture should hold down to (loglog x)^{1+o(1)} \leq q \leq x for all values of q, and that this range is best possible. We show under GRH and a linear independence hypothesis on the zeros of Dirichlet L-functions that for moderate values of q, \phi(q)e^{-y}V(e^y;q) has the same distribution as that of a certain random variable of mean asymptotically \phi(q) log q and of variance asymptotically 2\phi(q)(log q)^2. Our estimates on the large deviations of this random variable allow us to predict the range of validity of Hooley's Conjecture.Comment: 26 pages; Modified Definition 2.1, the error term for the variance in Theorem 1.2 and its proo

Residue classes containing an unexpected number of primes

We fix a non-zero integer $a$ and consider arithmetic progressions $a \bmod q$, with $q$ varying over a given range. We show that for certain specific values of $a$, the arithmetic progressions $a \bmod q$ contain, on average, significantly fewer primes than expected.Comment: 18 pages. Added a few remarks, changed the numbering of sections, slightly improved results, and made a few correction

A conditional determination of the average rank of elliptic curves

Under a hypothesis which is slightly stronger than the Riemann Hypothesis for elliptic curve $L$-functions, we show that both the average analytic rank and the average algebraic rank of elliptic curves in families of quadratic twists are exactly $\frac 12$. As a corollary we obtain that under this last hypothesis, the Birch and Swinnerton-Dyer Conjecture holds for almost all curves in our family, and that asymptotically one half of these curves have algebraic rank $0$, and the remaining half $1$. We also prove an analogous result in the family of all elliptic curves. A way to interpret our results is to say that nonreal zeros of elliptic curve $L$-functions in a family have a direct influence on the average rank in this family. Results of Katz-Sarnak and of Young constitute a major ingredient in the proofs.Comment: 27 page

Teachers' Burnout Profile- Risk and protective factors

Background: Burnout syndrome represents a factual risk for school teachers during their career. Several factors have been analyzed as stress sources enabled to menace teachers’ general well-being; nevertheless, protective factors mostly related to their personal resources may differently characterize teachers’ profiles. Objectives: The current study aimed to define different teachers’ profiles based on their burnout levels and attitudes towards job (i.e., job satisfaction, self-efficacy, attitudes toward professional growth, collective efficacy, positive and negative emotions, and hedonic balance). attitudes towards job Methods: Participants were 266 school teachers (F=69.1%) ranging from 26 to 65 years old (M=48.95; SD=8.31), with teaching experience ranged from 1 to 41 years (M=21.72; SD=10.36). Data were collected by three self-report questionnaires: Copenhagen Burnout Inventory, Attitudes towards job questionnaires, School Collective efficacy. Results: Cluster analysis approach showed two distinct teacher’s profiles named at-risk and non at-risk teachers. Main differences were due to burnout levels, attitudes toward job and extra-mansions at work. No differences were found related to teachers’ socio-demographic characteristics and their years of experience. Conclusions: The two teachers’ profiles resulting from the cluster analysis show several similarities, including collective efficacy and job satisfaction levels. Results are discussed in relation as to how teachers’ positive emotions towards their job can work as protective factors against the risk of burnou

Low-lying zeros of quadratic Dirichlet LL-functions: A transition in the Ratios Conjecture

We study the $1$-level density of low-lying zeros of quadratic Dirichlet $L$-functions by applying the $L$-functions Ratios Conjecture. We observe a transition in the main term as was predicted by the Katz-Sarnak heuristic as well as in the lower order terms when the support of the Fourier transform of the corresponding test function reaches the point $1$. Our results are consistent with those obtained in previous work under GRH and are furthermore analogous to results of Rudnick in the function field case.Comment: 15 page

Low-lying zeros of elliptic curve L-functions: Beyond the ratios conjecture

We study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over $\mathbb Q$. We are primarily interested in the family of all twists coprime to the conductor of E and compute a very precise expression for the corresponding 1-level density. In particular, for test functions whose Fourier transforms have sufficiently restricted support, we are able to compute the 1-level density up to an error term that is significantly sharper than the square-root error term predicted by the L-functions Ratios Conjecture.Comment: 33 page