Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis

en · official · sha256:5f1373a3a196d5f74f2a8960dbb8f24d2f6aa830186406166aeeb427fb4976dd

Abstract. In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that |psi(x) - x| and |theta(x) - x| are bounded from above by sqrt(x) log x (log x - log log x) / (8 pi) for all x >= 101 and x >= 2657 respectively, where psi(x) and theta(x) are the Chebyshev psi and theta functions. Using the extra precision offered by these results, we also prove new explicit descriptions for the error in each of Mertens' theorems which improve earlier bounds by Schoenfeld.

en · official · sha256:e499bdd060acae04f4ef8331961bc46ea3db1348316e92e6be67ca480264b080

Corollary 1.3. If the RH is true, then | sum_{p<=x} (log p)/p - log x - E | <= 3(log x)^2 / (8 pi sqrt(x)) for all x >= 43.1, | sum_{p<=x} 1/p - log log x - B | <= 3 log x / (8 pi sqrt(x)) for all x >= 24.4, | e^C log x prod_{p<=x} (1 - 1/p) - 1 | <= 3 log x / (8 pi sqrt(x)) for all x >= 23.8, | (e^(-C)/log x) prod_{p<=x} (1 - 1/p)^(-1) - 1 | <= 3 log x / (8 pi sqrt(x)) for all x >= 24.2.

Referencing norms: MertensErrorCorollary13, cor13-attributed, cor13-depends-on-theorem12, cor13-depends-transitively

en · official · sha256:a08d309ce88ad96157caebc04d7983f8c6145a90a53a5d3f3ff3a39a663992ef

we can collect observations from Schoenfeld [16] and Grenie-Molteni [9] to see that |psi(x) - x| <= ((log x)/(8 pi) + 2) sqrt(x) log x for all x >= 2, |psi(x) - x| <= sqrt(x) (log x)^2 / (8 pi) for all x >= 73.2, |psi(x) - x| <= sqrt(x) log x (log x - 2) / (8 pi) for all x >= 2.3 * 10^9. (1)

Referencing norms: UnconditionalBoundApplies, UnconditionalPsiBoundLargeX, attributed_to, candidate_x, eq1-attributed, eq1-attribution, unc-hi-1e10, unc-lo-1e10, unconditional_bound_applies, unconditional_bound_value_lower, unconditional_bound_value_upper

en · official · sha256:7d9c2c83c17dee4bccb018e2d943fbb2c97428e0ae3841fb520263502b606986

Methodology. For x > 10^19, we prove Theorem 1.1 by making a method of Goldston [8] explicit, and Theorem 1.2 by utilising a smoothed explicit formula for psi(x) from Rosser and Schoenfeld [15]. Computations by Buthe [4] are used to verify Theorems 1.1 and 1.2 for each x <= 10^19. Once Theorems 1.1 and 1.2 are proved, we apply them in standard arguments to establish each result in Corollary 1.3.

en · official · sha256:db66e27f56ee68799c9c000b9d5a2c35cd71d4a980871cb93023d040707ba37a

Theorem 1.1. If the RH is true and x >= 11, then |psi(x) - x| <= sqrt(x) log x ( (log x)/(8 pi) - (1/(2 pi) + 1.465/log x) log log x + 1.2325 ).

en · official · sha256:5976cc77b452dc182cbb69b32a761f2aa0b57720e2ab11a3e24358d3e3c3fd23

Theorem 1.2. If the RH is true, then |psi(x) - x| <= sqrt(x) log x (log x - log log x) / (8 pi) for all x >= 101, (2) |theta(x) - x| <= sqrt(x) log x (log x - log log x) / (8 pi) for all x >= 2657. (3)

Referencing norms: ConditionalPsiBoundApplies, ConditionalPsiThetaBoundTheorem12, ConditionalThetaBoundApplies, conditional_psi_bound_applies, conditional_theta_bound_applies, psi-hi-101, psi-hi-1e6, psi-hi-2657, psi-hi-3e12, psi-lo-101, psi-lo-1e6, psi-lo-2657, psi-lo-3e12, psi_bound_value_lower, psi_bound_value_upper, thm12-attributed, thm12-conditional-on-rh, thm12-depends-directly

en · official · sha256:d42ce7e0466160b3cb30f4216ba9b208dc232720fdae77335f3e0a7dbe3e3c79

SHARPER BOUNDS FOR THE ERROR IN THE PRIME NUMBER THEOREM ASSUMING THE RIEMANN HYPOTHESIS ETHAN SIMPSON LEE AND PAWEL NOSAL

Referencing norms: LeeNosalSource