Estimates of Some Functions Over Primes without R.H.

Abstract

Some computations made about the Riemann Hypothesis and in particular, the verification that zeroes of zeta belong on the critical line and the extension of zero-free region are useful to get better effective estimates of number theory classical functions which are closely linked to zeta zeroes like psi(x), theta(x), pi(x) or the k-th prime number.

Cited by in corpus (46)

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• Some heuristics on the gaps between consecutive primes
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• Updating the error term in the prime number theorem
• On the Interval [n,2n]: Primes, Composites and Perfect Powers
• On the Density of Ranges of Generalized Divisor Functions
• Higher Mertens constants for almost primes
• On The Prime Numbers In Intervals
• Explicit Chen's theorem
• On Legendre's, Brocard's, Andrica's, and Oppermann's Conjectures
• Every odd number greater than 1 is the sum of at most five primes
• On the Density of Ranges of Generalized Divisor Functions with Restricted Domains
• Sharper estimates for Chebyshev's functions $\vartheta$ and $\psi$
• Linear Algebraic Number Theory, Part I: Foundations
• On the Distribution of Twin Primes
• Le plus grand facteur premier de la fonction de Landau
• A Fowler-Nordheim Integrator can Track the Density of Prime Numbers
• A Totient Function Inequality
• A lower bound on the number of rough numbers
• Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap
• Inequalities For The Primes Counting Function
• Improved estimate for the prime counting function $\pi (x)$
• Revisiting Generalized Bertand's Postulate and Prime Gaps
• On the constant factor in several related asymptotic estimates
• An analytical proof for Lehmer's totient conjecture using Mertens' theorems
• On the arithmetic and geometric means of the prime numbers
• New bounds for the sum of the first $n$ prime numbers
• On the geometric mean of the first n primes
• Finite order elements in the integral symplectic group
• Diophantine Inequalities with Primes, Auxiliary Inequalities, Evaluations of the Difference between Consecutive Primes
• On a conjecture of Pomerance
• An asymptotic upper bound on prime gaps
• An Equivalent Statement to Nicolas' Criterion
• When $\pi (n)$ does not divide $n$
• On Sparsely Schemmel Totient Numbers
• An explicit upper bound for the first $k$-Ramanujan prime
• Derived Ramanujan Primes : R'_{n}
• Squarefree Integers And Extreme Values Of Some Arithmetic Functions
• Oscillations in Mertens Theorems and Other Finite Sums and Products
• Finding Large Primes
• New upper bounds for Ramanujan primes
• An upper bound for the prime gap
• Weighted Prime Powers Truncation of the Asymptotic Expansion for the Logarithmic Integral: Properties and Applications
• On the number of primes up to the $n$th Ramanujan prime