On the Shanks-Rényi race problem

Biases in the Shanks - Rényi Prime Number Race

1 N ov 1 99 9 BIASES IN THE SHANKS – RÉNYI PRIME NUMBERS RACE

Rubinstein and Sarnak investigated systems of inequalities of the form π(x; q, a 1) > · · · > π(x; q, a r), where π(x; q, a) denotes the number of primes up to x that are congruent to a mod q. They showed, under standard hypotheses on the zeros of Dirichlet L-functions mod q, that the set of positive real numbers x for which these inequalities hold has positive (logarithmic) density δ q;a 1 ,.....

Asymmetries in the Shanks–Rényi Prime Number Race

It has been well-observed that an inequality of the type π(x; q, a) > π(x; q, b) is more likely to hold if a is a non-square modulo q and b is a square modulo q (the so-called “Chebyshev Bias”). For instance, each of π(x; 8, 3), π(x; 8, 5), and π(x; 8, 7) tends to be somewhat larger than π(x; 8, 1). However, it has come to light that the tendencies of these three π(x; 8, a) to dominate π(x; 8, ...

the problem of divine hiddenness

این رساله به مساله احتجاب الهی و مشکلات برهان مبتنی بر این مساله میپردازد. مساله احتجاب الهی مساله ای به قدمت ادیان است که به طور خاصی در مورد ادیان ابراهیمی اهمیت پیدا میکند. در ادیان ابراهیمی با توجه به تعالی خداوند و در عین حال خالقیت و حضور او و سخن گفتن و ارتباط شهودی او با بعضی از انسانهای ساکن زمین مساله ای پدید میاید با پرسشهایی از قبیل اینکه چرا ارتباط مستقیم ویا حداقل ارتباط وافی به ب...

The hurdle-race problem

We consider the problem of how to determine the required level of the current provision in order to be able to meet a series of future deterministic payment obligations, in case the provision is invested according to a given random return process. Approximate solutions are derived, taking into account imposed minimum levels of the future random values of the reserve. The paper ends with numeric...