Related papers: Average Error of the Prime Number Theorem
Assuming a $q$-variant of the prime $k$-tuple conjecture uniformly, we compute mixed moments of the number of primes in disjoint short intervals and progressions, respectively. This involves estimating the mean of singular series along…
In this article we discuss a generalized Wirtinger inequality.
We prove an analogue of the prime number theorem for finite fields.
This is an article for a general mathematical audience on the author's work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. It is based on several one hour lectures, chiefly given at…
Understanding prediction errors and determining how to fix them is critical to building effective predictive systems. In this paper, we delineate four types of prediction errors and demonstrate that these four types characterize all…
We derived the sum identities for generalized harmonic and corresponding oscillatory numbers for which a sieve procedure can be applied. The obtained results enable us to understand better the properties of these numbers and their…
The aim of this paper is to prove Cotlar's ergodic theorem modeled on the set of primes.
This is an exposition of recent developments in the theory of bounded differences between primes. Readers are expected to be beginners of analytic number theory. The present text is a substantially improved and augmented version of the one…
On the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.
This paper gives an explicit version of Selberg's 1943 mean-value estimate for the prime number theorem in intervals under the Riemann hypothesis. Two applications are given: for primes in short intervals, and Goldbach numbers (sums of two…
Let $\sigma+i\gamma$ be a zero of the Riemann zeta function to the right of the line $\frac{1}{2}+it$. We show that this zero causes large oscillations of the error term of the prime number theorem. Our result is close to optimal both in…
Errors quoted on results are often given in asymmetric form. An account is given of the two ways these can arise in an analysis, and the combination of asymmetric errors is discussed. It is shown that the usual method has no basis and is…
We consider the general measurement scenario in which the ensemble average of an operator is determined via suitable data-processing of the outcomes of a quantum measurement described by a POVM. After reviewing the optimization of data…
We introduce extremely symmetric primes and provide some elementary properties of these.
We study the error bound for a smooth weighted prime number theorem, and its implication to the zero-free region for the Riemann zeta function using the method of Pintz. We also give an application to the average number of smooth weighted…
We will give a simple proof of the ambiguous class number formula.
Withdrawn by the authors due to an error in the proof of the finite field result (Thm. 1.5): The random primes used in the proof need NOT avoid the exceptional primes from Lemma 2.7, thus leaving Thm. 1.5 unproved.
We obtain error rates for large deviations of sums of i.i.d. random variables in, a particular case, of the domain of a non-symmetric infinite mean $\alpha=1$-stable law. The focus of this work is on the method of proof via analytic…
We investigate the probability that a random odd composite number passes a random Fermat primality test, improving on earlier estimates in moderate ranges. For example, with random numbers to $2^{200}$, our results improve on prior…
The common practice for displaying error bars on distributions of numbers of events is confusing and can lead to incorrect conclusions. A proposal is made for a different style of presentation that more directly indicates the level of…