Sharper estimates for Chebyshev's functions ϑ and ψ
aa r X i v : . [ m a t h . N T ] F e b Sharper estimates for Chebyshev’s functions ϑ and ψ Sadegh Nazardonyavi, Semyon Yakubovich
Departamento de Matem´atica, Faculdade de Ciˆencias , Universidade do Porto, 4169-007 Porto, Portugal
AbstractIn this article we present some improved results for Chebyshev’s functions ϑ and ψ using thenew zero-free region obtained by H. Kadiri and the calculated the first 10 zeros of the Riemannzeta function on the critical line by Xavier Gourdon. The methods in the proofs are similar tothose of Rosser-Shoenfeld papers on this subject. Definition 1.1.
For x > ψ -function by the formula ψ ( x ) = X n ≤ x Λ( n ) , where Λ( n ) = log p , n = p m for some m ;0 , otherwise.Since Λ( n ) = 0 unless n is a prime power, we can write the definition of ψ ( x ) as follows: ψ ( x ) = X n ≤ x Λ( n ) = ∞ X m =1 X p m ≤ x Λ( p m ) = ∞ X m =1 X p ≤ x /m log p. (1.1)The sum on m is actually a finite sum. In fact, the sum on p is empty if x /m <
2, that is, if(1 /m ) log x < log 2, or if m > log x log 2 = log x. Therefore, we have ψ ( x ) = X m ≤ log x X p ≤ x /m log p = X p ≤ x (cid:22) log x log p (cid:23) log p. This can be written in a slightly different form by introducing another function of Chebyshev.
Definition 1.2. If x >
0, we define Chebyshev’s ϑ -function by the equation ϑ ( x ) = X p ≤ x log p. ψ ( x ) can now be restated as follows: ψ ( x ) = X m ≤ log x ϑ ( x /m ) . (1.2)Using M¨obius inversion formula X k ≥ µ ( k ) ψ ( x /k ) = X k ≥ µ ( k ) X l ≥ ϑ ( x /kl ) = X n ≥ X k | n µ ( k ) ϑ ( x /n )= X n ≥ δ ,n ϑ ( x /n ) = ϑ ( x ) , where µ ( n ) = , n = 1;( − k , n = product of k distinct primes;0 , otherwise,and δ i,j is Kronecker’s delta function δ i,j = ( , i = j ;0 , i = j. Theorem 1.3 ([1], p. 76) . For x > we have ≤ ψ ( x ) x − ϑ ( x ) x ≤
12 log 2 log x √ x . Note that this inequality implies thatlim x →∞ (cid:18) ψ ( x ) x − ϑ ( x ) x (cid:19) = 0 . In other words, if one of ψ ( x ) /x or ϑ ( x ) /x tends to a limit then so does the other, and the twolimits are equal. ϑ ( x ) and π ( x ) In 1896 J. Hadamard and C. J. de la Vall´ee Poussin independently and almost simultaneouslysucceeded in proving that lim x →∞ π ( x ) log xx = 1 . This remarkable result is called the prime number theorem, and its proof was one of the crowningachievements of analytic number theory.In this section we give two formulas relating ϑ ( x ) and π ( x ). These can be used to show thatthe prime number theorem is equivalent to the limit relationlim x →∞ ϑ ( x ) x = 1 . Theorem 1.4 (Abel’s identity) . For any arithmetical function a ( n ) let A ( x ) = X n ≤ x a ( n ) , here A ( x ) = 0 if x < . Assume f has a continuous derivative on the interval [ y, x ] , where < y < x . Then we have X y
We have ψ ( x )log x < π ( x ) < Z x dψ ( t )log t . (1.10) Proof.
Since π ( x ) = X p ≤ x X p ≤ x ⌊ log x/ log p ⌋ log p (cid:22) log x log p (cid:23) log p = Z x ⌊ log x/ log t ⌋ log t dψ ( t ) . (1.11)and 1 ≤ ⌊ log x log t ⌋ ≤ log x log t Already Riemann, whose work was in many aspects decades beyond that of his contempo-raries, stated the elegant formula, which says that the weighted function ψ is in a certain sensemore natural than π and ϑ , since it possesses a (relatively simple) explicit expression, and relatesthe order of ψ ( x ) − x to a certain sum over non-trivial zeros of the zeta function; namely ψ ( x ) = x − X ρ x ρ ρ − ζ ′ (0) ζ (0) −
12 log(1 − x ) , ( x > , x = p m ) , (1.12)4here ρ = β + iγ is a non-trivial zero of ζ ( s ), and X ρ x ρ ρ = lim T →∞ X | γ |≤ T x ρ ρ . and when x = p m , then in the left-hand side of (1.12) put ψ ( x ) − Λ( x ). This explicit expressionfor ψ ( x ) was proved by H. von Mangoldt in 1895. The size of the error term in the prime number theorem depends on the location of zeros of theRiemann zeta function [10]. If { s = σ + it : σ > − c log | t | , | t | > t } is a zero free region, then an explicit error term in the prime number theorem is Theorem 1.8 ([5], p. 141) . There exists a constant a > such that for x tending to infinity,we have ψ ( x ) − x = O ( x exp( − a p log x )) ϑ ( x ) − x = O ( x exp( − a p log x ))Π( x ) − Li( x ) = O ( x exp( − a p log x )) π ( x ) − Li( x ) = O ( x exp( − a p log x )) One can choose a = 1 / and all constants “ O ” are effective. Theorem 1.9 ([5], p. 425) . There exist a positive constant α such that for x infinity we have π ( x ) − Li( x ) = O (cid:26) x exp( − α (log x ) / (log log x ) / ) (cid:27) The constant “ O ” is effective and can take α = 0 . . The corresponding asymptotic formulastake place for ϑ ( x ) , ψ ( x ) and Π( x ) . Cheng [2] gives an explicit zero-free region for the Riemann zeta-function derived from theVinogradov- Korobov method. He proves that the Riemann zeta-function does not vanish in theregion σ ≥ − . / | t | (log log | t | ) / , | t | ≥ x > | π ( x ) − li( x ) | ≤ . x (log x ) / exp( −
157 (log x ) / (log log x ) / )and for x ≥ e e . , there is a prime between x and ( x + 1) . Theorem 1.10 ([6]) . If | ζ ( σ + it ) | ≤ A | t | B (1 − σ ) / log / | t | , ( 12 ≤ σ ≤ , | t | ≥ , A = 76 . holds with a certain constant B , then for large | t | , ζ ( σ + it ) = 0 for σ ≥ − . B − / (log | t | ) / (log log | t | ) / Taking B = 4 . gives the zero-free region. .5 The results of Ingham Theorem 1.11 ([8], p. 66) . When x → ∞ , ψ ( x ) = x + O ( x exp( − a p log x log log x )) (1.13) π ( x ) = li( x ) + O ( x exp( − a p log x log log x )) (1.14) where a is a positive absolute constant. Let Θ be the upper bound of the real parts of the zeros of ζ ( s ). Clearly Θ ≤
1, since there isno zeros in σ >
1. And from the existence of the non-trivial zeros ρ and their symmetry aboutthe line σ = we infer that Θ ≥ . Thus ≤ Θ ≤
1, and this is the most that is knownabout Θ; but Θ = if (and only if) the Riemann hypothesis is true. We now have the followingtheorem, which is worthless if Θ = 1. Theorem 1.12 ([8], p. 83) . ψ ( x ) = x + O ( x Θ log x ) π ( x ) = li( x ) + O ( x Θ log x ) Theorem 1.13 ([8], p. 90) . If δ is any fixed positive number, then ψ ( x ) − x = Ω ± ( x Θ − δ )Π( x ) − li( x ) = Ω ± ( x Θ − δ ) Theorem 1.14 ([8], p. 100) . We have ψ ( x ) − x = Ω ± ( x / log log log x ) when x → ∞ . In fact, lim sup ψ ( x ) − xx / log log log x ≥ ψ ( x ) − xx / log log log x ≤ − Riemann Hypothesis verified until the 10 -th zero by Gourdon (October 12th 2004) [7].Recall N ( T ), F ( T ) and R ( T ) be defined as N ( T ) = { ρ : ζ ( ρ ) = 0 , < γ ≤ T } (2.1) F ( T ) = T π log T π − T π + 78 (2.2) R ( T ) =0 .
137 log T + 0 .
443 log log T + 1 .
588 (2.3)6 heorem 2.1 ([12]) . For T ≥ , | N ( T ) − F ( T ) | < R ( T )Choose A such that F ( A ) = 10 . Then A =2 , , , , . , , A =28 . , , . Lemma 2.2 ([11]) . We have X ρ γ < . . Proposition 2.3. X ρ | γ | < . , X ρ γ < . · − , X ρ | γ | < . · − , X ρ γ < . · − , X ρ | γ | < . · − . Proof.
We use the same method as in [11], and letting r = 29 instead of 8. Theorem 2.4 ([9]) . The Riemann zeta-function ζ ( s ) doesn’t vanish in the region σ ≥ − R log | t | , ( | t | ≥ , R = 5 . ρ = β + iγ is a zero of Riemann zeta function, then β < − R log | t | , ( | t | ≥ , R = 5 . Let K ν ( z, x ) = 12 Z ∞ x t ν − H z ( t ) dt where z > x ≥ H z ( t ) = exp {− z ( t + 1 t ) } Lemma 2.5 ([14]) . K ν ( z, x ) + K − ν ( z, x ) = K ν ( z,
0) = K ν ( z ) K ( z, x ) < e − z z ( √ y ! e − zy + ( 38 + z ) √ Z ∞ y e − zw dw ) (2.4) K ( z, x ) < e − z z nh √ y + 2 y + ( 105128 z + 158 ) √ y + 2 + 2 z i e − zy + ( 105128 z + 158 ) √ Z ∞ y e − zw dw o (2.5)7 here y = ( √ x − / √ x ) / √ . If we let x go to 0, then K ( z ) ≤ r π z e − z (cid:18) z (cid:19) (2.6) K ( z ) ≤ r π z e − z (cid:18) z + 105128 z (cid:19) (2.7) ψ ( x ) − x for Large Values of x Lemma 2.6 ([14]) . Let < U ≤ V , and let Φ( y ) be non-negative and differentiable for U 137 + 0 . Y (cid:27) Z VU Φ( y ) y dy + E j ( U, V ) where the error term E j ( U, V ) is given by E j ( U, V ) = { − j } R ( Y )Φ( Y )+ { N ( V ) − F ( V ) − ( − j R ( V ) } Φ( V ) − { N ( U ) − F ( U ) + R ( U ) } Φ( U ) Corollary 2.7 ([14]) . If, in addition, U > π , then X U<γ ≤ V Φ( γ ) ≤ { π + ( − j q ( Y ) } Z VU Φ( y ) log y π dy + E j ( U, V ) where q ( y ) = 0 . 137 log y + 0 . y log y log( y/ π )Define for x ≥ X = r log xR where R = 5 . ν , positive integer m , and non-negative real T and T ,define 8 m ( ν ) = { (1 + ν ) m +1 + 1 } m (2.8) S ( m, ν ) =2 X β ≤ / <γ ≤ T mν | ρ | (2.9) S ( m, ν ) =2 X β ≤ / γ>T R m ( ν ) ν m | ρ ( ρ + 1) · · · ( ρ + m ) | (2.10) S ( m, ν ) =2 X β> / <γ ≤ T (2 + mν ) exp( − X / log γ )2 | ρ | (2.11) S ( m, ν ) =2 X β> / γ>T R m ( ν ) exp( − X / log γ ) ν m | ρ ( ρ + 1) · · · ( ρ + m ) | (2.12) Lemma 2.8 ([14]) . Let T and T be non-negative real numbers. Let m be a positive integer.Let x > and < δ < ( x − / ( xm ) . Then x (cid:12)(cid:12)(cid:12) ψ ( x ) − { x − log 2 π − 12 log (cid:18) − x (cid:19) } (cid:12)(cid:12)(cid:12) (2.13) ≤ √ x { S ( m, δ ) + S ( m, δ ) } + S ( m, δ ) + S ( m, δ ) + mδ | ρ ( ρ + 1) · · · ( ρ + m ) | ≤ γ m +1 we can use Lemma 2.6 to write bounds for S j ( m, δ ) in terms of integrals for suitable Φ( y ). Wenote that for m = 0 Z VU y − ( m +1) log y π dy = 1 + m log( U/ π ) m U m − m log( V / π ) m V m (2.15)In the below integral let y = exp( zt/ m ) Z VU y − ( m +1) e − X y log y π dy = Z V ′ U ′ e − zt m ( m +1) e − mX zt { zt m − log 2 π } z m e zt m dt = z m Z V ′ U ′ e − zt e − mX zt { zt m − log 2 π } dt = z m Z V ′ U ′ e − zt − mX zt tdt − z m log 2 π Z V ′ U ′ e − zt − mX zt dt = z m Z V ′ U ′ e − z ( t + mX z t ) tdt − z m log 2 π Z V ′ U ′ e − z ( t + mX z t ) dt, z = 2 X √ m , U ′ = (2 m/z ) log U , V ′ = (2 m/z ) log V . So z m Z V ′ U ′ e − z ( t + mX z t ) tdt − z m log 2 π Z V ′ U ′ e − z ( t + mX z t ) dt = z m { K ( z, U ′ ) − K ( z, V ′ ) } − zm log 2 π { K ( z, U ′ ) − K ( z, V ′ ) } Hence, Z VU y − ( m +1) e − X y log y π dy (2.16)= z m { K ( z, U ′ ) − K ( z, V ′ ) } − zm log 2 π { K ( z, U ′ ) − K ( z, V ′ ) } Also if we let y = exp( X /t ), we get Z VU y − e − X y log y π dy = Z V ′′ U ′′ e − X t e − t { X t − log 2 π } (cid:18) − X t e X t (cid:19) dt = − X Z V ′′ U ′′ t − e − t dt + X log 2 π Z V ′′ U ′′ t − e − t dt = X { Γ( − , V ′′ ) − Γ( − , U ′′ ) }− X log 2 π { Γ( − , V ′′ ) − Γ( − , U ′′ ) } , (2.17)where U ′′ = X / log U , V ′′ = X / log V andΓ( a, x ) = Z ∞ x t a − e − t dt is incomplete gamma function. Theorem 2.9. If log x > , then | ψ ( x ) − x | < xε ( x ) , | ϑ ( x ) − x | < xε ( x ) , where ε ( x ) = 1 . (cid:18) − . X (cid:19) X / e − X , (2.18) and X = r log xR , R = 5 . . Proof. Take m = 1 and T = T = 0 in (2.8) through (2.13). By Lemma 2.2, S (1 , δ ) + S (1 , δ ) < (0 . δ + δ δ . Also, as β = for | γ | ≤ A , and the zeros off the critical line occur in pairs which are symmetricalwith respect to this line, we have S (1 , δ ) + S (1 , δ ) ≤ δ + δ δ X γ>A φ ( γ ) , φ m ( y ) = e − X / log y y m +1 . We appeal to Corollary 2.7 with Φ( y ) = φ ( y ), j = 0, U = A , V = ∞ , and W = W , where for m > − W m = exp( X/ √ m + 1) . Not that q ( Y ) ≤ q ( A ). Also, as N ( A ) = F ( A ), we have E = 2 R ( Y ) φ ( Y ) − R ( A ) φ ( A ) . (2.19)Since K ν ( z, x ) + K − ν ( z, /x ) = K ν ( z ) and (2.16), we have Z ∞ A φ ( y ) log y π dy ≤ X { XK (2 X ) − log(2 π ) K (2 X ) } ≤ X K (2 X ) . Then, by (2.6) and (2.7), we conclude X γ>A φ ( γ ) ≤{ π + 0 . 137 log A + 0 . A log A log( A/ π ) } Z ∞ A φ ( y ) log y π dy + E < { π + 0 . 137 log A + 0 . A log A log( A/ π ) } (2 X ) { K (2 X ) } + E ≤{ π + 0 . 137 log A + 0 . A log A log( A/ π ) } (2 X ) { X + 105512 X } r π X e − X + E < (0 . { X + 105512 X } X / e − X + E . (2.20)If W ≤ A , then Y = A . Then by (2.19) E = R ( A ) φ ( A ) = R ( A ) A { e − X / log A X / e X } X − / e − X . As the expression exp {− X log A + 12 log X + 2 X } takes its maximum at X = 12 log A + 12 q log A + log A we conclude that E < . · − X − / e − X . (2.21)If W > A , then Y = W and X > 40. As R ( y ) / log y is decreasing for y > e e , (2.19) gives E < R ( Y ) φ ( Y ) =2 R ( Y )log Y φ ( Y ) log Y< R ( A )log A φ ( W ) log W =2 R ( A )log A X √ e − √ X = √ R ( A )log A Xe − √ X = √ R ( A )log A X / e X − √ X X − / e − X < (3 . · − ) X − / e − X . 11o that we conclude (2.21) for this case also. Then by (2.20) X γ>A φ ( γ ) < (0 . { X + 105512 X } X / e − X . (2.22)As log x ≥ 110 0 . √ x = (0 . e − R X < − X − / e − X . (2.23)Choose δ = 2(0 . / { X } X / e − X . (2.24)So S (1 , δ ) + S (1 , δ ) √ x + S (1 , δ ) + S (1 , δ ) < δ + δ δ . √ x + X γ>A φ ( γ ) < δ + δ δ (cid:18) − X − / e − X + (0 . { X + 105512 X } X / e − X (cid:19) < δ + δ δ (0 . { X + 105512 X } X / e − X < δ (0 . (cid:18) X (cid:19) X / e − X . Let ε ( x ) = S (1 , δ ) + S (1 , δ ) √ x + S (1 , δ ) + S (1 , δ ) + δ . Then ε ( x ) < δ (0 . (cid:18) X (cid:19) X / e − X + δ . / { X } X / e − X + (0 . / { X } X / e − X =2(0 . / { X } X / e − X < (1 . { X } X / e − X . So | ψ ( x ) − { x − log 2 π − 12 log(1 − x ) }| < xε ( x ) , where ε ( x ) = (1 . { X } X / e − X . By Theorem 13 of [13], | ψ ( x ) − ϑ ( x ) | < . √ x Thus, it would appear that for ϑ ( x ) we should increase ε ( x ) by 1 . / √ x . However, we can treatit as in (2.23) to show it is absorbed when we round up some of the coefficients. Theorem 2.10. If log x ≥ | ψ ( x ) − x | < xε ∗ ( x ) , here ε ∗ ( x ) = ε ( x ) √ ( X p π (4 X − X ) + 32 √ πX ) and r ( x ) = 1 + 1532 X . Proof. Take δ = 1 √ . { X } X / e − X . (2.25)We may assume X ≥ . x ≥ e ), since if X ≤ . 36 or (more accurate x ≤ e ), ε ∗ ( x ) > ε ( x ). We take m = 1, T = 0, and T = X − / e X . (2.26)As X > 32 we have A < T < e X = W and W < T .We can treat { S (1 , δ ) + S (1 , δ ) } / √ x and the error terms E j ( U, V ) arising from the use ofCorollary 2.7, as we did in the proof of the previous theorem. Thus we can proceed as though S (1 , δ ) = 2 + δ X <γ ≤ T φ ( γ ) < δ (cid:18) π − q ( T ) (cid:19) Z T A φ ( y ) log y π dy. (2.27)If ν ≤ x > 0, thenΓ( ν, x ) = Z ∞ x t ν − e − t dt ≤ x ν − Z ∞ x e − t dt = x ν − e − x and Γ( ν, x ) = Z ∞ x t ν − e − t dt ≥ (1 + x ) ν − e − x . Hence, by (2.17), we have in effect S (1 , δ ) < δ π (cid:8) X { Γ( − , V ′′ ) − Γ( − , U ′′ ) } − X log 2 π { Γ( − , V ′′ ) − Γ( − , U ′′ ) } (cid:9) ≤ δ π (cid:8) X Γ( − , V ′′ ) − X log 2 π Γ( − , V ′′ ) (cid:9) < δ π (cid:8) X ( V ′′ ) − − X log 2 π (1 + V ′′ ) − (cid:9) e − V ′′ , (2.28)where V ′′ = X log T = 4 X X − X . (2.29)Then V ′′ > X + 34 log X, e − V ′′ < X − / e − X . (2.30)Also X ( V ′′ ) − − X log 2 π ( V ′′ ) − = (cid:18) − X X (cid:19) (cid:18) X − 34 log X − log 2 π (1 + 1 /X − X ) / (4 X )) (cid:19) < X. So, effectively S (1 , δ ) < δ π X / e − X . (2.31)13imilarly, we can proceed as though S (1 , δ ) = 2 + 2 δ + δ δ X γ>T φ ( γ ) < δ + δ δ (cid:18) π + q ( T ) (cid:19) Z ∞ T φ ( y ) log y π dy. (2.32)By (2.16) Z ∞ T φ ( y ) log y π dy = 2 { X K (2 X, U ′ ) − X log(2 π ) K (2 X, U ′ ) } ≤ X K (2 X, U ′ ) , (2.33)where U ′ = 1 X log T = 1 − X X . (2.34)Write temporarily q = 3 log X X , y = 1 √ (cid:18) √ U ′ − √ U ′ (cid:19) . (2.35)Then y is negative, and y = 12 (cid:18) U ′ + 1 U ′ (cid:19) − q − q ) . So, by splitting the integral in Lemma 2.5 at w = 0, we get √ Z ∞ y e − Xw dw = √ Z y e − Xw dw + √ Z ∞ e − Xw dw = √ Z y e − Xw dw + √ π √ X = √ Z y e − Xw dw + √ π √ X< √ Z y dw + √ π √ X = q √ − q + √ π √ X . (2.36)Hence, by (2.4) we get X log(2 π ) K (2 X, U ′ ) < X log(2 π ) e − X X ( √ y ! e − Xy + (cid:18) 38 + 2 X (cid:19) (cid:18) √ π √ X + q √ − q (cid:19)) ≤ 14 log(2 π ) e − X (cid:26) (cid:18) 38 + 2 X (cid:19) (cid:18) √ π √ X + q √ − q (cid:19)(cid:27) = √ π π ) X / e − X ( √ πX / + (cid:18) X + 1 X (cid:19) q √ X p π (1 − q ) !) (2.37)As 1 + zy < e zy , we have (2 y + 2 /z ) e − zy < /z = 1 /X . Hence, by (2.5) we get X K (2 X, U ′ ) With T and D given by (2.42) and (2.43), if T ≥ D , δ > , and m is a positiveinteger, then S ( m, δ ) + S ( m, δ ) < mδ π ((cid:18) log T π + 1 m (cid:19) + 1 m − . − . m ( m + 1) T ) . (2.45)16 roof. By (2.9) and (2.44) S ( m, δ ) = (2 + mδ ) X β ≤ / <γ ≤ T | ρ | < (2 + mδ ) S + X D<γ ≤ T γ . Taking Φ( y ) = y − , j = 0, U = D , V = T , and W = 0 in Lemma 2.6 gives X D<γ ≤ T γ ≤ π Z T D y log y π dy + (cid:26) . 137 + 0 . D (cid:27) Z T D y dy + E ≤ π (cid:26) log T π − log D π (cid:27) + (0 . 137 + 0 . D ) (cid:26) − T + 1 D (cid:27) + E . Then (2.44) together with N ( D ) = 620 gives S ( m, δ )2 + mδ 137 + 0 . D ) (cid:26) T − D (cid:27) + 1 T ( N ( T ) − F ( T ) − R ( T )) − D ( N ( D ) − F ( D ) − R ( D )) < − π log D π + 1 D (cid:18) . 137 + 0 . D − N ( D ) + F ( D ) + R ( D ) (cid:19) + 14 π log T π − T (cid:18) . 137 + 0 . D − N ( T ) + F ( T ) + R ( T ) (cid:19) < − . π log T π − T (cid:18) . 137 + 0 . D − N ( T ) + F ( T ) + R ( T ) (cid:19) . (2.46)Since S ( m, δ ) = 2 X β ≤ / γ>T R m ( δ ) δ m | ρ ( ρ + 1) · · · ( ρ + m ) | . Taking Φ( y ) = y − ( m +1) , j = 0, U = T , V = ∞ , and W = 0 in Lemma 2.6 and using (2.15) gives δ m S ( m, δ )2 R m ( δ ) = X β ≤ / γ>T | ρ ( ρ + 1) · · · ( ρ + m ) | < X γ>T γ m +1 ≤ π Z ∞ T y − ( m +1) log y π dy + (cid:18) . 137 + 0 . T (cid:19) Z ∞ T y − ( m +1) y dy + E ∗ = 12 π (cid:18) m T m + 1 mT m log T π (cid:19) + (cid:18) . 137 + 0 . T (cid:19) m + 1) T m +11 + E ∗ = 1 T m (cid:26) πm log T π + 12 πm + (cid:18) . 137 + 0 . T (cid:19) m + 1) T (cid:27) + E ∗ = 1 T m (cid:26) πm log T π + 12 πm + (cid:18) . 137 + 0 . T (cid:19) m + 1) T + + E ∗ T m (cid:27) . S ( m, δ ) + S ( m, δ ) < (2 + mδ ) (cid:20) − . π log T π − T (cid:18) . 137 + 0 . D − N ( T ) + F ( T ) + R ( T ) (cid:19)(cid:21) + 2 R m ( δ ) δ m T m (cid:20) πm log T π + 12 πm + (cid:18) . 137 + 0 . T (cid:19) m + 1) T + E ∗ T m (cid:21) =(2 + mδ ) (cid:20) − . π log T π − T (cid:18) . 137 + 0 . D − N ( T ) + F ( T ) + R ( T ) (cid:19)(cid:21) + (2 + mδ ) (cid:20) πm log T π + 12 πm + (cid:18) . 137 + 0 . T (cid:19) m + 1) T + E ∗ T m (cid:21) = 2 + mδ π (cid:20) log T π + 4 π (cid:26) − . − T (cid:18) . 137 + 0 . D − N ( T ) + F ( T ) + R ( T ) (cid:19)(cid:27)(cid:21) + 2 + mδ π (cid:20) m log T π + 2 m + 4 π (cid:18) . 137 + 0 . T (cid:19) m + 1) T + 4 πE ∗ T m (cid:21) = 2 + mδ π ((cid:18) log T π + 1 m (cid:19) + 1 m + J ) , where J =4 π (cid:26) − . − T (cid:18) . 137 + 0 . D − N ( T ) + F ( T ) + R ( T ) (cid:19)(cid:27) + 4 π (cid:18) . 137 + 0 . T (cid:19) m + 1) T + 4 πE ∗ T m =4 π (cid:26) − . − T (cid:18) . 137 + 0 . D (cid:19) + (cid:18) . 137 + 0 . T (cid:19) m + 1) T (cid:27) < π (cid:26) − . − T (cid:18) . 137 + 0 . D (cid:19) (cid:18) − m + 1 (cid:19)(cid:27) < − . − . m ( m + 1) T . Theorem 2.12. Let T ≥ D . Let m be a positive integer, let Ω denote the right side of (2.45)and let Ω =(0 . R m ( δ ) z m δ m { zK ( z, A ′ ) − m log(2 π ) K ( z, A ′ ) } + R m ( δ ) δ m { R ( Y ) φ m ( Y ) − R ( A ) φ m ( A ) } , (2.47) where z = 2 X √ m = 2 p mb/R , A ′ = (2 m/z ) log A , Y = max { A, exp p b/ ( m + 1) R } . If b > / and < δ < (1 − e − b ) /m , then | ψ ( x ) − x | < εx, ( x ≥ e b ) , where ε = Ω e − b/ + Ω + mδ e − b log 2 π. roof. Take T = 0, then S ( m, δ ) = 0 and by Corollary 2.7 and 2.16, δ m R m ( δ ) S ( m, δ ) = X β> / γ> e − X / log γ | ρ ( ρ + 1) · · · ( ρ + m ) | < X γ>A e − X / log γ γ m +1 = X γ>A φ m ( γ ) ≤ { q ( Y ) } Z ∞ A φ m ( y ) log y π dy + E < . (cid:18) z m K ( z, A ′ ) − zm log(2 π ) K ( z, A ′ ) (cid:19) + { R ( Y ) φ m ( Y ) − R ( A ) φ m ( A ) } . So S ( m, δ ) + S ( m, δ ) < Ω . Since1 x | ψ ( x ) − x + log 2 π + 12 log(1 − x ) | < S ( m, δ ) + S ( m, δ ) √ x + S ( m, δ ) + S ( m, δ ) + mδ . So 1 x | ψ ( x ) − x | < S ( m, δ ) + S ( m, δ ) √ x + S ( m, δ ) + S ( m, δ ) + mδ πx< Ω √ x + Ω + mδ πx ≤ Ω e − b/ + Ω + mδ e − b log 2 π. Theorem 2.13. Let T ≥ D and A ≤ T ≤ exp p b/R . Let m be a positive integer and let Ω = 2 + mδ π (cid:2) X { Γ( − , T ′′ ) − Γ( − , A ′′ ) } − X log(2 π ) { Γ( − , T ′′ ) − Γ( − , A ′′ ) } (cid:3) + 2 + mδ { R ( T ) φ ( T ) − R ( A ) φ ( A ) } + Ω ∗ , (2.48) where A ′′ = b/ ( R log A ) , T ′′ = b/ ( R log T ) , and Ω ∗ is obtained from Ω by deleting the term − R ( A ) φ m ( A ) in (2.47) and then replacing A by T in the definition of A ′ and Y . If b > / and < δ < (1 − e − b ) /m , then (2.41) holds for all x ≥ e b , where ε = Ω e − b/ + Ω + mδ e − b log 2 π. (2.49)If we use the following bounds for Γ( ν, x ) and ν < 1, for large b we get a better bounds thanthose given in three theorems before the last one; x ν e − x x + 1 − ν < Γ( ν, x ) < x ν − e − x { x + ( ν − x + ( ν − ν − } , ( x > , ν < . .4 Bounds for ϑ ( x ) − x for Large Values of x Theorem 2.14 ([14]) . We have ϑ ( x ) < . , x, ( x > , . , x <ϑ ( x ) , ( x ≥ , , ,ψ ( x ) − ϑ ( x ) < . , √ x + 3 √ x, ( x > , . , √ x <ψ ( x ) − ϑ ( x ) , ( x ≥ . Corollary 2.15 ([14]) . We have ϑ ( x ) > . x, ( x ≥ , ,ϑ ( x ) > . x, ( x ≥ , ,ϑ ( x ) > . x, ( x ≥ , ,ϑ ( x ) > . x, ( x ≥ , . Theorem 2.16 ([14]) . If x ≥ , then | ψ ( x ) − x | < . x log x , | ϑ ( x ) − x | < . x log x . Corollary 2.17 ([14]) . If x ≥ , , then ϑ ( x ) − x ≤ ψ ( x ) − x < . , x log x . Corollary 2.18 ([14]) . We have | ϑ ( x ) − x | < . , x log x , ( x ≥ , , | ϑ ( x ) − x | < x log x , ( x ≥ , . Theorem 2.19 ([14]) . If x > , then | ψ ( x ) − x | <η k x log k x , | ϑ ( x ) − x | <η k x log k x , where η = 8 . , η = 11 , , η = 1 . · . heorem 2.20. If ε ( x ) is defined as (2.18), then ϑ ( x ) − x ≤ ψ ( x ) − x We have ϑ ( x ) < . x, ( x > ,ϑ ( x ) > . x, ( x ≥ ) . Proof. If 8 · ≤ x < e , then ϑ ( x ) < ψ ( x ) − √ x − √ x < (cid:26) . − √ x − 67 1 √ x (cid:27) x< (cid:26) . − e − / − e − / (cid:27) x< . x. By handling the intervals [ e , e ), etc., similarly, we derive the same inequality. And for x ≥ e we use the table and ϑ ( x ) < ψ ( x ). This proves for all x ≥ · . For x < · , it followsfrom (4.5) of [13] and Dusart[4] which says ϑ ( x ) < x in this domain.If 8 · ≤ x < , then ϑ ( x ) > ψ ( x ) − √ x − √ x > (cid:26) − . − √ x − 65 1 √ x (cid:27) x> (cid:26) . − (8 · ) − / − 65 (8 · ) − (2 / (cid:27) x> . x. x ≥ ϑ ( x ) > ψ ( x ) − . √ x − . √ x > (cid:26) − . · − − . 001 1 √ x − . √ x (cid:27) x> n . − (1 . − − (1 . − / o x> . x Theorem 2.28. If x ≥ · | ψ ( x ) − x | < . x log x , | ϑ ( x ) − x | < . x log x . Proof. If 8 · ≤ x < e , then ϑ ( x ) − x >ψ ( x ) − x − √ x − √ x> − (cid:26)(cid:18) . √ x + 65 1 √ x (cid:19) log x (cid:27) x log x> − (cid:26)(cid:18) . e − / + 65 e − / (cid:19) (28) (cid:27) x log x> − . x log x . We continue to use the table in this way until e . If 10 ≤ x < e ϑ ( x ) − x >ψ ( x ) − x − . √ x − . √ x> − (cid:26)(cid:18) . · − + 1 . 001 1 √ x + 1 . √ x (cid:19)(cid:27) x log x> − n(cid:16) . · − + (1 . e − + (1 . e − / (cid:17) o x> − . x log x . We continue again until e . For x ≥ e , we apply Theorem 2.9 and note that ε ( x ) log x < . ϑ ( x ) − x > −{ ε ( x ) log x } x log x > − . x log x . Theorem 2.29. If x > · , then | ψ ( x ) − x | < η k x log k x , whereProof. If x ≥ · we proceed as previous theorem. For x ≥ · from table we get.If 1 < x < · , since ϑ ( x ) < x , we have ψ ( x ) − x <ϑ ( x ) − x + √ x + 43 √ x < ( log k x √ x + 43 log k x √ x ) x log k x ≤ (cid:26) (2 k ) k e k + 43 (3 k/ k e k (cid:27) x log k x , ( k = 1 , , , x > · k 1 2 3 4 η k k = 1 , , , 4; and ψ ( x ) − x >ϑ ( x ) − x + √ x + 23 √ x > − . √ x + √ x + 23 √ x = ( − . 06 log k x √ x + 23 log k x √ x ) x log k x> − c k x log k x where c = 0 . 445 and c = 1 . 592 and c = 8 . 887 and c = 66 . Theorem 2.30. For x ≥ · | ϑ ( x ) − x | < η k x log k x where Table 2: η k 1 2 3 4 η k Proof. For 1 < x < ϑ ( x ) − x > − . √ x = − . 06 log k x √ x . x log k x ≥ − . 06 (2 k ) k e k x log k x for k = 1 , , , Theorem 2.31. If ε ( x ) is defined as in Theorem 2.9, then ϑ ( x ) − x ≤ ψ ( x ) − x < xε ( x ) , ( x > ψ ( x ) − x ≥ ϑ ( x ) − x > − xε ( x ) , ( x ≥ Proof. We need to verify them for x < e . As ε ( x ) increases for 1 < x < x > . < ε ( x ) < . ≤ x < e . From the table we deducethem for 10 ≤ x < e . For 132 ≤ x < , we have ε ( x ) > . ψ ( x ) < . x < (1 + ε ( x )) x < x < ≤ x < ϑ ( x ) > . x > (1 − ε ( x )) x For 71 ≤ x < Second method. By Theorem 9 of [14] ψ ( x ) − x < xε ( x ) , ( x > ε is defined in (3.9) of [14]. On the other hand ε ( x ) < ε ( x ) for 408 < x < e . Bycomputation for smaller values. The same hold for ϑ ( x ) − x > − xε ( x ). | ψ ( x ) − x | and | ϑ ( x ) − x | Lemma 2.32 ([14]) . If ν ≤ , z > , and x > , we have K ν ( z, x ) < Q ν ( z, x ) where Q ν ( z, x ) = x ν +1 z ( x − H z ( x ) , H ( t ) = e − ( t +1 /t ) Lemma 2.33 ([14]) . If z > and x > , then ( x − Q ( z, x )+(1 + 2 z − z (1 + x ) ) K ( z, x ) Let ε ( x ) = p /πX / e − X Then | ψ ( x ) − x | < xε ( x ) , ( x ≥ and | ϑ ( x ) − x | < xε ( x ) , ( x ≥ Proof. The main part of the proof is concerned with large x in which case the proof is similar toTheorem 2.10, but we ultimately take m = 2 rather than m = 1. In place of (2.26), we let T = e νx , (2.50)where ν will be specified later. We assume that ν , m , X are such that T ≥ A, √ m + 1 ≤ ν ≤ X ≥ νX = log T ≥ log A W m = e X/ √ m +1 ≤ T = e νX ≤ e X = W . In place of (2.27), we get S ( m, δ ) ≤ mδ (cid:26) π − q ( T ) (cid:27) Z T A φ ( y ) log y π dy + E ! (2.52)where E = { N ( T ) − F ( T ) + R ( T ) } + φ ( T ) − { N ( A ) − F ( A ) + R ( A ) } φ ( A ) < R ( T ) φ ( T ) (2.53)and R ( T ) = 0 . 137 log T + 0 , 443 log log T + 1 . V ′′ = X log T we have V ′′ = X νX = Xν = X { (1 − ν ) ν + 2 − ν } = Y + 2 X − νX, (2.54)where Y = X (1 − ν ) ν (2.55)Proceeding as in (2.28) and (2.31) and using (2.53), we find S ( m, δ ) < mδ π e − V ′′ { X ( V ′′ ) − − (log 2 π ) X ( V ′′ ) − } + 2 + mδ E < mδ π e − Y − X T XG + (2 + mδ ) R ( T ) φ ( T ) (2.56)where G = ν (cid:18) ν − log 2 πX (cid:19) (2.57)As R ( y ) / log y decrease for y > e e , we have R ( T ) φ ( T ) = R ( T )log T φ ( T ) log T ≤ R ( A )log A φ ( T ) log T ≤ R ( A )log A e − V ′′ T log T = R ( A )log A e − Y − X + νX T log T = R ( A )log A e − Y − X log T . by (2.54). Then (2.50) and (2.51) yieldlog T = νX ≤ X hence R ( T ) φ ( T ) < . Xe − Y − X . (2.58)We have S ( m, δ ) ≤ R m ( δ ) δ m (cid:18)(cid:26) π + q ( T ) (cid:27) Z ∞ T φ m ( y ) log y π dy + E (cid:19) (2.59)26here E = { R ( T ) + F ( T ) − N ( T ) } φ m ( T ) < R ( T ) φ m ( T ) = 2 R ( T ) φ ( T ) T − m (2.60)Also Z ∞ T φ m ( y ) log y π dy = z m { K ( z, U ′ ) − m log 2 πz K ( z, U ′ ) } (2.61)where z = 2 X √ m and U ′ = 2 mz log T = 2 m X √ m log T = √ m log T X = √ m νXX = ν √ m By assuming ν > √ m (2.62)we have U ′ > 1; also m ≥ ν ≤ K ( z, U ′ ) − m log 2 πz K ( z, U ′ ) 1) ( m − ν ( mν − X − T − ( m − e − Y − X = G m m − X − T − ( m − e − Y − X (2.63)where G = ( m − ν ( mν − (cid:18) ν + 1 mX (cid:19) (2.64)Then Z ∞ T φ m ( y ) log y π dy < z m G m m − X − T − ( m − e − Y − X = 1 m − G XT − ( m − e − Y − X We define G = R m ( δ )2 m { πq ( T ) } = { πq ( T ) } (cid:26) (1 + δ ) m +1 + 12 (cid:27) m (2.65)27hen S ( m, δ ) ≤ R m ( δ )2 m (cid:18) δ (cid:19) m π { πq ( T ) } Z ∞ T φ m ( y ) log y π dy + R m ( δ )2 m (cid:18) δ (cid:19) m E = (cid:18) δ (cid:19) m π G Z ∞ T φ m ( y ) log y π dy + R m ( δ )2 m (cid:18) δ (cid:19) m E < (cid:18) δ (cid:19) m π G Z ∞ T φ m ( y ) log y π dy + G (cid:18) δ (cid:19) m E < (cid:18) δ (cid:19) m π ( m − G G XT − ( m − e − Y − X + G (cid:18) δ (cid:19) m E Now 1 + mδ < R m ( δ ) / m < G . We obtain S ( m, δ ) + S ( m, δ ) < π G Xe − Y − X (cid:26) G T + 1 m − G (cid:18) δT (cid:19) m T (cid:27) + 2 G R ( T ) φ ( T ) (cid:26) (cid:18) δT (cid:19) m (cid:27) If G and G were independent of ν , and hence of T , then the expression inside the first braceswould be minimized by choosing T = 2 δ (cid:18) G G (cid:19) /m (2.66)Postponing the reconciliation of this with the previous definition of T , we obtain S ( m, δ ) + S ( m, δ ) + 12 mδ < π G Xe − Y − X (cid:26) G T + 1 m − G (cid:18) δ (cid:19) m T − m (cid:27) + 2 G R ( T ) φ ( T ) (cid:26) (cid:18) δT (cid:19) m (cid:27) + 12 mG δ = 12 π G Xe − Y − X (cid:26) mm − G − /m G /m δ (cid:27) + 2 G R ( T ) φ ( T ) (cid:18) G G (cid:19) + 12 mG δ = 12 mG (cid:26) π ( m − G − /m G /m Xe − Y − X δ + δ (cid:27) + 2 G R ( T ) φ ( T ) (cid:18) G G (cid:19) The expression inside the last braces is minimized by choosing δ = (cid:26) π ( m − G − /m G /m e − Y (cid:27) / X / e − X (2.67)so that (2.66) becomes T = 2 δ (cid:18) G G (cid:19) /m = (cid:18) G G (cid:19) / m (cid:26) π ( m − G e Y (cid:27) / X − / e X (2.68)28oreover, since R ( T ) φ ( T ) < . Xe − Y − X , S ( m, δ ) + S ( m, δ ) + 12 mδ 2, it is not hard to see that G decreases as ν increases, G is alsoincreasing function of ν . Hence, k ( ν ) is strictly increasing for increasing ν ∈ (1 / √ , k ( ν ) → ν → / √ k (1) > X ≥ ν ∈ (1 / √ , 1) such that k ( ν ) = 1. Henceforth, let ν be this number sothat ν depends on X ; then G , G , Y and T are defined in terms of ν by (7.12), (7.25), (7.10)and (7.5), (7.24b). Of course, (7.17) holds since m = 2. Hence (7.26) will be fully establishedonce it is shown that T ≥ A . We have, for 1 / √ < ν ≤ H ( ν ) ≡ G G = ν (2 ν − 1) ( ν − log(2 π ) /X ) ν + 1 / (2 X ) < ( ν − log(2 π ) /X ) , all X; > . ν (2 ν − , X ≥ 10. (2.74)If we define for j = 0 and 1, ν j = 1 − X log X (2 + 3 j ) π (2.75)29e see that H ( ν ) < , ( X ≥ π ) , H ( ν ) > . , ( X > k ( ν j ) = 12 π XH ( ν j ) / e − X (1 − ν ) e − X (1 − ν ) /ν = 12 π XH ( ν j ) / exp (cid:26) − log X (2 + 3 j ) π (cid:27) exp (cid:26) − ν j X log X (2 + 3 j ) π (cid:27) = 2 + 3 j H ( ν j ) / exp (cid:26) − ν j X log X (2 + 3 j ) π (cid:27) we see that k ( ν ) < k ( ν ) , ( X ≥ π ) , k ( ν ) > k ( ν ) , ( X > ν < ν, (log x ≥ . , ν < ν , (log x > 0) (2.76)Of course ν < ν in all cases. For log x ≥ T > e ν X > A Hence, S (2 , δ ) + S (2 , δ ) + δ 16. It is a simple matter to verifythat G G < ν − < , Y < X (1 − ν ) /ν < . 025 (2.77) G G = ν ν − (cid:18) − log 2 πνX (cid:19) (cid:18) νX (cid:19) < (1 − νX )(1 + νX ) < , X ≥ > ν / (2 ν − ≥ . , X ≥ 11. (2.78)Then S (2 , δ ) + S (2 , δ ) + δ 137 + 0 . / log T T log( T / π ) < πT . 137 + 0 . / log A log( A/ π ) < . πT =1 + 0 . √ π (cid:18) G G (cid:19) / e − Y/ X / e − X < . X / e − X < exp(0 . X / e − X )31urther, R ( δ )2 = (cid:26) (1 + δ ) + 12 (cid:27) = (cid:26) δ (3 + 3 δ + δ ) (cid:27) < (cid:18) . δ (cid:19) < (cid:26) exp (cid:18) . δ (cid:19)(cid:27) = exp(3 . δ ) < exp(2 . X / e − X )Then G 4. If log x ≥ . ν > ν > / √ 2. Hence, (cid:18) (cid:19) / < L ( ν ) < , ( x ≥ e . ) (2.86)In addition, M ( ν ) < exp 14 (cid:26) − νX + 12 νX − Xν (1 − ν ) (cid:27) < exp 14 (cid:26) − νX + 12 νX − Xν (1 − ν ) (cid:27) = exp 14 (cid:26) − νX (cid:18) X π (cid:19)(cid:27) < E ( x ) (2.87)32here E ( x ) = exp 14 (cid:26) − ν X (cid:18) X π (cid:19)(cid:27) = exp 14 ν (cid:26) − X − X (1 − ν ) (cid:27) (2.88)It is clear from the first part of (2.88) that E ( x ) < x . So for log x ≥ | ψ ( x ) − x | , | ϑ ( x ) − x | < xε ( x ) . For smaller x we use the table and relation 0 ≤ ψ ( x ) − ϑ ( x ) < . √ x . x Let T = 1 δ (cid:18) R m ( δ )2 + mδ (cid:19) /m and leave T unspecified for the moment. We showed that by letting T = e νX , S ( m, δ ) < mδ (cid:26) π − q ( T ) (cid:27) Z T A φ ( y ) log y π dy + 2 R ( T ) φ ( T ) ! S ( m, δ ) < R m ( δ ) δ m (cid:18)(cid:26) π + q ( T ) (cid:27) Z ∞ T φ m ( y ) log y π dy + 2 R ( T ) φ ( T ) T − m (cid:19) so that S ( m, δ ) + S ( m, δ ) < π h ( T ) + e ( T )where h ( T ) = 2 + mδ Z TA φ ( y ) log y π dy + R m ( δ ) δ m Z ∞ T φ m ( y ) log y π dy and e ( T ) = q ( T ) ( − mδ Z TA φ ( y ) log y π dy + R m ( δ ) δ m Z ∞ T φ m ( y ) log y π dy ) + R ( T ) φ ( T ) { mδ + 2 R m ( δ )( δT ) m } The situation for S ( m, δ ) + S ( m, δ ) is entirely similar. If we leave T and D unspecified butsubject to 2 ≤ D ≤ A and T ≥ D , then we get S ( m, δ ) + S ( m, δ ) < π h ( T ) + e ( T )where h ( T ) = 2 + mδ Z TD y log y π dy + R m ( δ ) δ m Z ∞ T y m +1 log y π dy + (2 + mδ ) π (cid:26) G ( D ) + 14 π log D π (cid:27) G ( D ) = X <γ ≤ D γ + 1 / / − π ((cid:18) log D π − (cid:19) + 1 ) + 1 D (cid:26) . 137 log D + 0 . (cid:18) log log D + 1log D (cid:19) + 2 . − N ( D ) (cid:27) ( T ) = − πT (cid:26) mδ (cid:18) . 137 + 0 . D (cid:19) − R m ( δ )( m + 1)( δT ) m (cid:18) . 137 + 0 . T (cid:19)(cid:27) + 2 πT (cid:26) mδ − R m ( δ )( δT ) m (cid:27) { N ( T ) − F ( T ) − R ( T ) } Let C ( D ) = 4 π (cid:18) . 137 + 0 . D (cid:19) , S ( D ) = X <γ ≤ D γ + 1 / / Theorem 2.36. Let T be defined as above and satisfy T ≥ D , where ≤≤ A . Let m be apositive integer and let δ > . Then S ( m, δ ) + S ( m, δ ) < Ω ∗ where Ω ∗ = 2 + mδ π ((cid:18) log T π + 1 m (cid:19) + 4 πG ( D ) + 1 m − mC ( D )( m + 1) T ) where G ( D ) and C ( D ) are defined above. Moreover, if Ω ∗ = 12 π h ( T ) + e ( T ) then | ψ ( x ) − x | < ε ∗ x, ( x ≥ e b ) where ε ∗ = Ω ∗ e − b/ + Ω ∗ + m δ + e − b log 2 π Remark . After doing this article we realized that a similar theorem to Theorem 2.35 for | ψ ( x ) − x | was done by P. Dusart. But as you may see in this article we give the proofs withmore details. Acknowledgments The work of the second author is supported by Center of Mathematics of the University ofPorto. The work of the first author is supported by the Calouste Gulbenkian Foundation, underPh.D. grant number CB/C02/2009/32. Research partially funded by the European RegionalDevelopment Fund through the programme COMPETE and by the Portuguese Governmentthrough the FCT under the project PEst-C/MAT/UI0144/2011. Sincere thanks to ´Elio Coutinhofrom Informatics Center of Faculty of Science of the University of Porto for providing accessesto fast computers for necessary computations. References [1] Tom M. Apostol. Introduction to analytic number theory . Springer-Verlag, New York, 1976.Undergraduate Texts in Mathematics.[2] Yuanyou Cheng. An explicit zero-free region for the Riemann zeta-function. Rocky MountainJ. Math. , 30(1):135–148, 2000. 343] N. Costa Pereira. Estimates for the Chebyshev function ψ ( x ) − θ ( x ). Math. Comp. ,44(169):211–221, 1985.[4] P. Dusart. Estimates of some functions over primes without R.H. arXiv:1002.0442.[5] William John Ellison. Les nombres premiers . Hermann, Paris, 1975. En collaborationavec Michel Mend`es France, Publications de l’Institut de Math´ematique de l’Universit´e deNancago, No. IX, Actualit´es Scientifiques et Industrielles, No. 1366.[6] Kevin Ford. Zero-free regions for the Riemann zeta function. In Number theory for themillennium, II (Urbana, IL, 2000) , pages 25–56. A K Peters, Natick, MA, 2002.[7] X. Gourdon. The 10 first zeros of the Riemannzeta function, and zeros computation at very large height. http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf .[8] A. E. Ingham. The distribution of prime numbers . Cambridge Mathematical Library. Cam-bridge University Press, Cambridge, 1990. Reprint of the 1932 original, With a foreword byR. C. Vaughan.[9] Habiba Kadiri. Une r´egion explicite sans z´eros pour la fonction ζ de Riemann. Acta Arith. ,117(4):303–339, 2005.[10] Kevin S. McCurley. Explicit estimates for the error term in the prime number theorem forarithmetic progressions. Math. Comp. , 42(165):265–285, 1984.[11] J. Barkley Rosser. The n -th prime is greater than n log n . Proc. London Math. Soc. (2) ,45:21–44, 1939.[12] J. Barkley Rosser. Explicit bounds for some functions of prime numbers. Amer. J. Math. ,63:211–232, 1941.[13] J. Barkley Rosser and Lowell Schoenfeld. Approximate formulas for some functions of primenumbers. Illinois J. Math. , 6:64–94, 1962.[14] J. Barkley Rosser and Lowell Schoenfeld. Sharper bounds for the Chebyshev functions θ ( x )and ψ ( x ). Math. Comp. , 29:243–269, 1975. Collection of articles dedicated to Derrick HenryLehmer on the occasion of his seventieth birthday.[15] Lowell Schoenfeld. Sharper bounds for the Chebyshev functions θ ( x ) and ψ ( x ). II. Math.Comp. , 30:337–360, 1976.Email: [email protected]: [email protected] 35n this case for b ≥ ε < Ω e − b/ + Ω + md/ π ) e − b For b = log(8 · ) ≈ . m = 1, δ = 9( − 6) and ε = 2 . − b = log 10 ≈ . m = 2, δ = 5 . − 8) and ε = 4 . − | ψ ( x ) − x | < xε, ( x ≥ e b ), ε = CX / e − X b m δ ε b m δ ε η for the case ε = CX / e − X b η η η η b ≥ ε < Ω ∗ e − b/ + Ω ∗ + mδ/ π ) e − b so that instead of Ω ∗ we use general theorem ε = p /π.... ). D=2500, minimize all:39able 5: η for the case ε = CX / e − X b η η η η 950 2.36304(-8) 0.0000236304 0.0236304 23.63041000 2.44643(-8) 0.0000256875 0.0269719 28.32041050 2.52703(-8) 0.0000277973 0.030577 33.63471100 2.60413(-8) 0.0000299475 0.0344396 39.60551150 2.67764(-8) 0.0000321317 0.038558 46.26961200 2.74833(-8) 0.0000343541 0.0429427 53.67831250 2.81629(-8) 0.0000366118 0.0475953 61.87391300 2.87972(-8) 0.0000388762 0.0524828 70.85181350 2.94051(-8) 0.0000411671 0.057634 80.68761400 2.99885(-8) 0.0000434834 0.0630509 91.42381450 3.05317(-8) 0.0000457975 0.0686963 103.0441500 3.10407(-8) 0.0000481131 0.0745754 115.5921550 3.15265(-8) 0.0000504425 0.080708 129.1331600 3.19825(-8) 0.0000527712 0.0870724 143.6691650 3.23916(-8) 0.0000550657 0.0936117 159.141700 3.27774(-8) 0.0000573605 0.100381 175.6671750 3.31425(-8) 0.0000596565 0.107382 193.2871800 3.346(-8) 0.0000619011 0.114517 211.8561850 3.37469(-8) 0.000064119 0.121826 231.471900 3.40132(-8) 0.0000663258 0.129335 252.2041950 3.42502(-8) 0.0000685004 0.137001 274.0022000 3.52772(-8) 0.0000740822 0.155573 326.7032100 3.55621(-8) 0.0000782365 0.17212 378.6652200 3.56974(-8) 0.0000821041 0.188839 434.3312300 3.57438(-8) 0.0000857851 0.205884 494.1222400 3.56337(-8) 0.0000890842 0.22271 556.7762500 3.54302(-8) 0.0000921186 0.239508 622.7222600 3.50936(-8) 0.0000947527 0.255832 690.7472700 3.71195(-8) 0.000111358 0.334075 1002.233000 3.36966(-8) 0.000107829 0.345053 1104.173200 3.26308(-8) 0.000114208 0.399727 1399.043500 2.78888(-8) 0.000103189 0.381798 1412.653700 2.55845(-8) 0.000102338 0.409352 1637.414000 2.01043(-8) 0.0000844381 0.35464 1489.494200 1.69966(-8) 0.0000764846 0.344181 1548.814500 1.15857(-8) 0.0000544527 0.255928 1202.864700 8.85926(-9) 0.0000442963 0.221481 1107.415000 5.07767(-9) 0.0000264039 0.1373 713.9625200 3.612(-9) 0.000019866 0.109263 600.9465500 2.06644(-9) 0.0000117787 0.0671385 382.6895700 1.40731(-9) 8.11312(-6) 0.0467721 269.641 | ψ ( x ) − x | < xε, ( x ≥ e b ), ε = C ′ X / e − X b m δ ε b m δ ε η for the case ε = C ′ X / e − X b η η η η η for the case ε = C ′ X / e − X b η η η η 900 2.27887(-8) 0.0000216493 0.0205668 19.5385950 2.36469(-8) 0.0000236469 0.0236469 23.64691000 2.4477(-8) 0.0000257009 0.0269859 28.33521050 2.52801(-8) 0.0000278081 0.0305889 33.64781100 2.60488(-8) 0.0000299561 0.0344495 39.6171150 2.67822(-8) 0.0000321386 0.0385663 46.27961200 2.74877(-8) 0.0000343597 0.0429496 53.6871250 2.81663(-8) 0.0000366162 0.0476011 61.88141300 2.87997(-8) 0.0000388796 0.0524875 70.85821350 2.94071(-8) 0.0000411699 0.0576378 80.6931400 2.999(-8) 0.0000434855 0.063054 91.42831450 3.05328(-8) 0.0000457993 0.0686989 103.0481500 3.10416(-8) 0.0000481145 0.0745774 115.5951550 3.15272(-8) 0.0000504435 0.0807096 129.1351600 3.1983(-8) 0.000052772 0.0870738 143.6721650 3.2392(-8) 0.0000550663 0.0936128 159.1421700 3.27777(-8) 0.000057361 0.100382 175.6681750 3.31427(-8) 0.0000596569 0.107382 193.2881800 3.34602(-8) 0.0000619014 0.114518 211.8571850 3.3747(-8) 0.0000641193 0.121827 231.4711900 3.40133(-8) 0.0000663259 0.129336 252.2041950 3.42503(-8) 0.0000685005 0.137001 274.0022000 3.52773(-8) 0.0000740823 0.155573 326.7032100 3.55621(-8) 0.0000782366 0.172121 378.6652200 3.56974(-8) 0.0000821041 0.188839 434.3312300 3.57465(-8) 0.0000857917 0.2059 494.162400 3.56337(-8) 0.0000890842 0.22271 556.7762500 3.54303(-8) 0.0000921187 0.239509 622.7222600 3.50936(-8) 0.0000947527 0.255832 690.7472700 3.71195(-8) 0.000111358 0.334075 1002.233000 3.3697(-8) 0.00010783 0.345057 1104.183200 3.26308(-8) 0.000114208 0.399727 1399.043500 2.8266(-8) 0.000105998 0.397491 1490.593750 2.4491(-8) 0.0000979639 0.391855 1567.424000 2.01043(-8) 0.0000844381 0.35464 1489.494200 1.69978(-8) 0.0000764902 0.344206 1548.934500 1.15857(-8) 0.0000544527 0.255928 1202.864700 8.86144(-9) 0.0000443072 0.221536 1107.685000 4.98003(-9) 0.0000253981 0.12953 660.6065100 4.16392(-9) 0.0000216524 0.112592 585.485200 3.42352(-9) 0.0000178468 0.0930354 484.993 η for the case ε = C ′ X / e − X b m δ ε η η η η3800 6 1.67(-12) 5.86122(-12) 2.23312(-8) 0.000085082 0.324163 1235.063810 5 1.94(-12) 5.80739(-12) 2.21842(-8) 0.0000847437 0.323721 1236.613820 5 1.92(-12) 5.74859(-12) 2.20171(-8) 0.0000843255 0.322967 1236.963830 5 1.90(-12) 5.69039(-12) 2.18511(-8) 0.0000839082 0.322207 1237.283840 5 1.88(-12) 5.63277(-12) 2.16862(-8) 0.0000834918 0.321443 1237.563850 5 1.86(-12) 5.57575(-12) 2.15224(-8) 0.0000830764 0.320675 1237.83860 5 1.84(-12) 5.51930(-12) 2.13597(-8) 0.000082662 0.319902 1238.023870 5 1.82(-12) 5.46344(-12) 2.11982(-8) 0.0000822488 0.319126 1238.213880 5 1.80(-12) 5.40817(-12) 2.10378(-8) 0.0000818369 0.318346 1238.363890 5 1.78(-12) 5.35348(-12) 2.08786(-8) 0.0000814265 0.317563 1238.53900 5 1.77(-12) 5.29927(-12) 2.07201(-8) 0.0000810158 0.316772 1238.583910 5 1.75(-12) 5.24559(-12) 2.05627(-8) 0.0000806058 0.315975 1238.623920 5 1.73(-12) 5.19249(-12) 2.04065(-8) 0.0000801975 0.315176 1238.643930 5 1.71(-12) 5.13998(-12) 2.02515(-8) 0.000079791 0.314376 1238.643940 5 1.70(-12) 5.08798(-12) 2.00975(-8) 0.0000793852 0.313572 1238.613950 5 1.68(-12) 5.03642(-12) 1.99442(-8) 0.0000789791 0.312757 1238.523960 5 1.66(-12) 4.98545(-12) 1.97922(-8) 0.0000785752 0.311944 1238.423970 5 1.64(-12) 4.93509(-12) 1.96417(-8) 0.0000781739 0.311132 1238.313980 5 1.63(-12) 4.88504(-12) 1.94913(-8) 0.0000777704 0.310304 1238.113990 5 1.61(-12) 4.83561(-12) 1.93424(-8) 0.0000773697 0.309479 1237.924000 5 1.60(-12) 4.78674(-12) 1.91948(-8) 0.0000769713 0.308655 1237.71