Zeta function zeros, powers of primes, and quantum chaos
نویسندگان
چکیده
منابع مشابه
Primes, Quantum Chaos, and Computers
The primes, which are the fundamental building blocks of the multiplicative structure of the integers, have fascinated mathematicians at least since the time of the ancient Greeks. Euclid already knew that there are infinitely many primes, and his elegant proof is still commonly used today. Further progress in studying primes was made over the succeeding centuries. However, many very basic ques...
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Assuming a special version of the Montgomery-Odlyzko law on the pair correlation of zeros of the Riemann zeta function conjectured by Rudnick and Sarnak and assuming the Riemann Hypothesis, we prove new results on the prime number theorem, difference of consecutive primes, and the twin prime conjecture. 1. Introduction. Assuming the Riemann Hypothesis (RH), let us denote by 1=2 ig a nontrivi...
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Assuming the Riemann Hypothesis, Montgomery and Taylor showed that at least 67.25% of the zeros of the Riemann zeta-function are simple. Using Montgomery and Taylor's argument together with an elementary combinatorial argument, we prove that assuming the Riemann Hypothesis at least 67.275% of the zeros are simple.
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The Riemann zeta function is given by: [equation, see published text]. Zeta(s) may be analytically continued to the entire s-plane, except for a simple pole at s = 0. Of great interest are the complex zeros of zeta(s). The Riemann hypothesis states that the complex zeros all have real part 1/2. According to the prime number theorem, pn approximately n logn, where pn is the nth prime. Suppose th...
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Abstract We study the relations between the distribution of the zeros of the Riemann zeta-function and the distribution of primes in “almost all” short intervals. It is well known that a relation like ψ(x)−ψ(x−y) ∼ y holds for almost all x ∈ [N, 2N ] in a range for y that depends on the width of the available zero-free regions for the Riemann zeta-function, and also on the strength of density b...
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ژورنال
عنوان ژورنال: Physical Review E
سال: 2003
ISSN: 1063-651X,1095-3787
DOI: 10.1103/physreve.68.026206