# Explicit Estimate on Primes between Consecutive Cubes

@article{Cheng2008ExplicitEO,
title={Explicit Estimate on Primes between Consecutive Cubes},
author={Yuanyou Furui Cheng},
journal={arXiv: Number Theory},
year={2008}
}
• Y. Cheng
• Published 12 October 2008
• Mathematics
• arXiv: Number Theory
We give an explicit form of Ingham's Theorem on primes in the short intervals, and show that there is at least one prime between every two consecutive cubes $x\sp{3}$ and $(x+1)\sp{3}$ if $\log\log x\ge 15$.

### An Explicit Result for Primes Between Cubes

We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.217))$. Our new tool which we derive is a version of Landau's explicit formula for the Riemann zeta-function

### Explicit Estimates in the Theory of Prime Numbers

It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove that

### A zero density result for the Riemann zeta function

In this article, we prove an explicit bound for $N(\sigma,T)$, the number of zeros of the Riemann zeta function satisfying $\sigma < \Re s <1$ and $0 < \Im s < T$. This result provides a significant

### Prime Numbers in Short Intervals

The Riemann zeta function, ζ(s), is central to number theory and our understanding of the distribution of the prime numbers. This thesis presents some of the known results in this area before

### Discrete methods in geometric measure theory

The thesis addresses problems from the field of geometric measure theory. It turns out that discrete methods can be used efficiently to solve these problems. Let us summarize the main results of the

### Almost Ramanujan Expanders from Arbitrary Expanders via Operator Amplification

• Mathematics, Computer Science
• 2022
An efﬁcient algorithm that transforms any bounded degree expander graph into another that achieves almost optimal trade-off between spectral expansion λ and degree d and shows that any expanding ﬁnite group has almost Ramanujan expanding generators.

### Subclasses of presburger arithmetic and the weak EXP hierarchy

It is shown that for any fixed i > 0, the Σ i+1-fragment of Presburger arithmetic is complete for ΣiEXP, the i-th level of the weak EXP hierarchy, an analogue to the polynomial-time hierarchy residing between NEXP and EXPSPACE.

### A sharpened estimate on the pseudo-Gamma function

• Mathematics
• 2013
The pseudo-Gamma function is a key tool introduced recently by Cheng and Albeverio in the proof of \break the density hypothesis. This function is doubly symmetric, which means that it is

## References

SHOWING 1-10 OF 32 REFERENCES

### A note on the number of primes in short intervals

• Mathematics
• 1990
Let J(fl, T) = f, (ZX 1 , J(fl, T) < C0of log2 T/T, provided that T is sufficiently large. Here we prove a slightly stronger result from which we deduce a lower bound of the same order.

### Explicit Estimates for the Riemann Zeta Function

• Mathematics
• 2004
We apply van der Corput’s method of exponential sums to obtain explicit upper bounds for the Riemann zeta function on the line σ = 1/2. For example, we prove that if t ≥ e, then |ζ(1/2 + it)| ≤ 3t1/6

### On Vinogradov's Constant in Goldbach's Ternary Problem

Abstract This paper shows that under the assumption of the Generalized Riemann Hypothesis, every odd integer greater than 10 20 can be written as a sum of three primes. Using the computational

### The distribution of prime numbers

Preface Introduction 1. Elementary theorems 2. The prime number theorem 3. Further theory of ( ). applications 4. Explicit formulae 5. Irregularities of distribution Bibliography.

### Determining Mills' Constant and a Note on Honaker's Problem

• Mathematics
• 2005
In 1947 Mills proved that there exists a constant A such that bA 3 n c is a prime for every positive integer n. Determining A requires determining an efiective Hoheisel type result on the primes in

### Primes in Short Intervals

• Mathematics
• 2004
Contrary to what would be predicted on the basis of Cramér’s model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x+H)−ψ(x), for 0≤x≤N, is approximately

### A classical introduction to modern number theory

• Mathematics
• 1982
This second edition has been corrected and contains two new chapters which provide a complete proof of the Mordell-Weil theorem for elliptic curves over the rational numbers, and an overview of recent progress on the arithmetic of elliptic curve.

### A survey of results on primes in short intervals

Prime numbers have been a source of fascination for mathematicians since antiquity. The proof that there are infinitely many prime numbers is attributed to Euclid (fourth century B.C.). The basic

### On the zeros of the Riemann zeta function in the critical strip

We describe a computation which shows that the Riemann zeta function ζ(s) has exactly 75,000,000 zeros of the form σ+ it in the region 0 < t < 32,585,736.4; all these zeros are simple and lie on the

### The Distribution of Prime Numbers

THIS interesting “Cambridge Tract” is concerned mainly with the behaviour, for large values of x, of the function n(x), which denotes the number of primes not exceeding x. The first chapter gives