# Computing π(x) analytically

@article{Platt2015ComputingA, title={Computing $\pi$(x) analytically}, author={Dave Platt}, journal={Math. Comput.}, year={2015}, volume={84}, pages={1521-1535} }

We describe a rigorous implementation of the Lagarias and Odlyzko Analytic Method to evaluate the prime counting function and its use to compute unconditionally the number of primes less than 10.

#### 28 Citations

A practical analytic method for calculating 휋(x)

- Mathematics, Computer Science
- Math. Comput.
- 2017

A practical analytic method for the computation of pi(x), the number of prime numbers <= x, which is similar to the one proposed by Lagarias and Odlyzko but uses the Weil-Barner explicit formula instead of curve integrals. Expand

New bounds for π(x)

- Computer Science, Mathematics
- Math. Comput.
- 2015

The proof relies on two new arguments: smoothing the prime counting function which allows to generalize the previous approaches, and a new explicit zero density estimate for the zeros of the Riemann zeta function. Expand

A computational history of prime numbers and Riemann zeros

- Mathematics
- 2018

We give an informal survey of the historical development of computations related to prime number distribution and zeros of the Riemann zeta function.

Estimates of ψ,θ for large values of x without the Riemann hypothesis

- Mathematics, Computer Science
- Math. Comput.
- 2016

The quintessence of the method of Rosser and Schoenfeld on the upper bounds for the usual Chebyshev prime and prime power counting functions is drawn to find an upper bound function directly linked to a zero-free region. Expand

Improvements to Turing's method II

- Mathematics
- 2016

This article improves the estimate of the size of the definite inte- gral of S(t), the argument of the Riemann zeta-function. The primary appli- cation of this improvement is Turing's Method for the… Expand

Estimating π(x) and related functions under partial RH assumptions

- Computer Science, Mathematics
- Math. Comput.
- 2016

The aim of this paper is to give a direct interpretation of the validity of the Riemann hypothesis up to a certain height $T$ in terms of the prime-counting function $\pi(x)$. This is done by proving… Expand

Will a physicist prove the Riemann Hypothesis?

- Physics, Medicine
- Reports on progress in physics. Physical Society
- 2019

The Riemann Hypothesis is formulated and some physical problems related to this hypothesis are reviewed: the Polya--Hilbert conjecture, the links with Random Matrix Theory, relation with the Lee--Yang theorem on the zeros of the partition function and phase transitions, random walks, billiards etc. Expand

Updating the error term in the prime number theorem

- Mathematics
- 2014

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 have… Expand

Prime Numbers in Short Intervals

- Mathematics
- 2017

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… Expand

An improved analytic method for calculating $$\pi (x)$$π(x)

- Mathematics
- 2014

We provide an improved version of the analytic method of Franke et al. for calculating the prime-counting function $$\pi (x)$$π(x), which is more flexible and, for calculations not assuming the… Expand

#### References

SHOWING 1-10 OF 33 REFERENCES

Isolating some non-trivial zeros of zeta

- Computer Science, Mathematics
- Math. Comput.
- 2017

A rigorous algorithm to compute Riemann’s zeta function on the half line and its use to isolate the non-trivial zeros of zeta with imaginary part to an absolute precision of ±2−102 is described. Expand

Prime sieves using binary quadratic forms

- Mathematics, Computer Science
- Math. Comput.
- 2004

An algorithm that computes the prime numbers up to N using O(N/log log N) additions and N 1/2+o(1) bits of memory is introduced. Expand

Computing pi(x): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method

- Mathematics, Computer Science
- Math. Comput.
- 1996

It is shown that it is possible to compute π(x) in O( x 2/3 /log 2 x) time and O(x 1/3l log 3 x log log x) space. Expand

Computing n(X) the meissel-lehmer method

- Mathematics
- 1985

E. D. F. Meissel, a German astronomer, found in the 1870’s a method for computing individual values of π(x), the counting function for the number of primes ≤ x. His method was based on recurrences… Expand

Artin's Conjecture, Turing's Method, and the Riemann Hypothesis

- Mathematics, Computer Science
- Exp. Math.
- 2006

A group-theoretic criterion under which one may verify the Artin conjecture for some Galois representations, up to finite height in the complex plane is presented and a rigorous algorithm for computing general L-functions on the critical line via the fast Fourier transform is developed. Expand

Multiplicative Number Theory

- Mathematics
- 1967

From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The… Expand

Introduction to analytic number theory

- Mathematics
- 1976

This is the first volume of a two-volume textbook which evolved from a course (Mathematics 160) offered at the California Institute of Technology during the last 25 years. It provides an introduction… Expand

Analytic computation of the prime-counting function

- Mathematics
- 2004

The main topic of this dissertation is the Lagarias-Odlyzko analytic algorithm for computing π(x)—the number of primes up to x. This algorithm is asymptotically the fastest known algorithm for the… Expand

Computing π(x): the Meissel

- Lehmer method, Math. Comp
- 1985

Computing π(x): the Meissel-Lehmer

- method, Math. Comp
- 1985