An analytic method for bounding 𝜓(𝑥)

DOI10.1090/mcom/3264zbMath1450.11095arXiv1511.02032OpenAlexW2964018490MaRDI QIDQ4637585

Jan Büthe

Publication date: 24 April 2018

Published in: Mathematics of Computation (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1511.02032

zbMATH Keywords

Chebyshev functionprime counting functionanalytic algorithmSkewes number

Mathematics Subject Classification ID

Asymptotic results on arithmetic functions (11N37) Nonreal zeros of (zeta (s)) and (L(s, chi)); Riemann and other hypotheses (11M26) Distribution of primes (11N05)

Related Items (24)

On Ramanujan's prime counting inequality ⋮ On the average value of ⋮ The sum of divisors function and the Riemann hypothesis ⋮ Explicit estimates for the distribution of numbers free of large prime factors ⋮ An inequality related to the sieve of Eratosthenes ⋮ An explicit upper bound for \(L(1,\chi)\) when \(\chi\) is quadratic ⋮ Sharper bounds for the error term in the prime number theorem ⋮ New estimates for some integrals of functions defined over primes ⋮ Sharper bounds for the Chebyshev function \(\psi(x)\) ⋮ Some explicit estimates for the error term in the prime number theorem ⋮ New bounds for numbers of primes in element orders of finite groups ⋮ The reciprocal sum of the amicable numbers ⋮ The error term in the prime number theorem ⋮ Proof of the Sheldon Conjecture ⋮ New estimates for some functions defined over primes ⋮ Estimates for $\pi(x)$ for large values of $x$ and Ramanujan's prime counting inequality ⋮ The twin prime conjecture and other curiosities regarding prime numbers ⋮ The constant factor in the asymptotic for practical numbers ⋮ Primes in prime number races ⋮ Highly composite numbers and the Riemann hypothesis ⋮ Sharper bounds for the Chebyshev function 𝜃(𝑥) ⋮ Unnamed Item ⋮ The Riemann hypothesis is true up to 3·1012 ⋮ Improving bounds on prime counting functions by partial verification of the Riemann hypothesis

Cites Work

• Approximate formulas for some functions of prime numbers
• On the first sign change of $\theta (x) -x$
• Estimating $\pi (x)$ and related functions under partial RH assumptions
• An amortized-complexity method to compute the Riemann zeta function
• The impact of 𝜁(𝑠) complex zeros on 𝜋(𝑥) for 𝑥<10^{10¹³}
• Fast Algorithms for Multiple Evaluations of the Riemann Zeta Function
• Bounds for the Tails of Sharp-Cutoff Filter Kernels
• Sharper Bounds for the Chebyshev Functions θ(x) and ψ(x)
• The large sieve
• Sharper Bounds for the Chebyshev Functions θ(x) and ψ(x). II
• New bounds for $\psi (x)$
• A practical analytic method for calculating $\pi (x)$
• Explicit Bounds for Some Functions of Prime Numbers

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