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Updating the error term in the prime number theorem
An improved estimate is given for $$|\theta (x) -x|$$|θ(x)-x|, where $$\theta (x) = \sum _{p\le x} \log p$$θ(x)=∑p≤xlogp. Four applications are given: the first to arithmetic progressions that haveExpand
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An improved upper bound for the argument of the Riemann zeta-function on the critical line II
Abstract Text This paper concerns the function S ( T ) , where π S ( T ) is the argument of the Riemann zeta-function along the critical line. The main result is that | S ( T ) | ⩽ 0.112 log T +Expand
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Bounds on the number of Diophantine quintuples
We consider Diophantine quintuples $\{a, b, c, d, e\}$. These are sets of distinct positive integers, the product of any two elements of which is one less than a perfect square. It is conjecturedExpand
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Linnik's approximation to Goldbach's conjecture, and other problems
We examine the problem of writing every sufficiently large even number as the sum of two primes and at most $K$ powers of 2. We outline an approach that only just falls short of improving the currentExpand
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An improved explicit bound on $|ζ(1/2 + it)|$
Abstract This article proves the bound | ζ ( 1 2 + i t ) | ≤ 0.732 t 1 6 log ⁡ t for t ≥ 2 , which improves on a result by Cheng and Graham. We also show that | ζ ( 1 2 + i t ) | ≤ 0.732 | 4.678 + iExpand
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On a conjecture of Shanks
Abstract The conjecture in question concerns the function ϕ n related to the distribution of the zeroes of the Riemann zeta-function, γ n , over the Gram points g n . It is the purpose of thisExpand
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There Are No Socialist Primes Less Than 109
  • T. Trudgian
  • Mathematics, Computer Science
  • Integers
  • 23 October 2013
There are no primes $p$ with $5<p<10^{9}$ for which $2!, 3!, \ldots, (p-1)!$ are all distinct modulo $p$; it is conjectured that there are no such primes.
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Searching for Diophantine quintuples
We consider Diophantine quintuples $\{a, b, c, d, e\}$. These are sets of distinct positive integers, the product of any two elements of which is one less than a perfect square. It is conjecturedExpand
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The error term in the prime number theorem
TLDR
We make explicit a theorem of Pintz concerning the error term in the prime number theorem. Expand
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On the success and failure of Gram's Law and the Rosser Rule
This article contains a survey on results relating to Gram’s Law and the distribution of the zeroes of the Riemann zeta-function. It is shown that Gram’s Law and the Rosser Rule fail a positiveExpand
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