NEW ESTIMATES FOR SOME FUNCTIONS DEFINED OVER PRIMES

Transcription

1 #A52 INTEGERS 8 (208) NEW ESTIMATES FOR SOME FUNCTIONS DEFINED OVER PRIMES Christian Aler Institute of Mathematics, Heinrich Heine University Düsseldorf, Düsseldorf, Germany christian.aler@hhu.de Received: 5/6/7, Revised: 2/22/7, Acceted: 5/3/8, Published: 6/5/8 Abstract In this aer we give e ective estimates for some classical arithmetic functions defined over rime numbers. First we establish new elicit estimates for Chebyshev s #-function. Alying these new estimates, we derive new uer and lower bounds for some functions defined over rime numbers, for instance the rime counting function (), which imrove current best estimates. Furthermore, we use the obtained estimates for the rime counting function to give two new results concerning the eistence of rime numbers in short intervals.. Introduction Chebyshev s #-function is defined by #() = X log, ale where runs over all rimes not eceeding. Since there are infinitely many rimes, we have #()! as!. In 896, Hadamard [9] and, indeendently, de la Vallée-Poussin [37] roved a result concerning the asymtotic behavior for #(), namely #() (! ), (.) which is known as the Prime Number Theorem. In a later aer [38], where the eistence of a zero-free region for the Riemann zeta-function (s) to the left of the line Re(s) = was roved, de la Vallée-Poussin also estimated the error term in the Prime Number Theorem by showing that #() = + O( e( c 0 log )), (.2)

2 INTEGERS: 8 (208) 2 where c 0 is a ositive absolute constant. The currently best elicit version of (.2) is due to Dusart [, Corollary.2]. He used an elicit zero-free region for the Riemann zeta-function (s) due to Kadiri [20] to show that s 8 #() ale R (log )/4 e log /R (.3) for every 3, where R = The work of Korobov [23] and Vinogradov [39] imlies the currently asymtotically strongest error term in (.), namely that there is a ositive absolute constant c so that #() = + O e c log 3/5 (log log ) /5. Under the assumtion that the Riemann hyothesis is true, von Koch [22] deduced the asymtotic formula #() = + O( log 2 ). An elicit version of the von Koch result was given by Schoenfeld [33, Theorem 0]. Under the assumtion that the Riemann hyothesis is true, he has found that #() < (/(8 )) log 2 for every 599. In the current aer, we rove the following result concerning elicit bounds for Chebyshev s #-function which imroves several eisting bounds of similar shae. Theorem. For every and for every >, we have 9, 035, 709, 63, we have #() > 0.5 log 3, (.4) #() < log 3. (.5) For small values of, the estimates in Theorem for #() follow from Büthe [4, Theorem 2] and some direct comutations. For large values of, both estimates in Theorem follow from (.3) and some other recent results of Dusart [] which in turn are based on elicit zero-free regions for the Riemann zeta-function (s) due to Mossingho and Trudgian [26], an elicit zero-density estimate of Ramaré [30], and a numerical calculation of Gourdon [8] concerning the verification of the Riemann hyothesis for the first 0 3 nontrivial zeros of the Riemann zeta-function (s). The last comutation imlies that the Riemann hyothesis is true at least for every non-trivial zero + it with 0 < t < 2, 445, 999, 556, 030. Now let () denote the number of rimes not eceeding. Chebyshev s #-function and the rime counting function () are connected by the identity () = #() log + Z 2 #(t) t log 2 dt, (.6) t

3 INTEGERS: 8 (208) 3 which holds for every 2 (see, for instance, Aostol [, Theorem 4.3]). Using (.6), it is easy to see that the Prime Number Theorem (.) is equivalent to where li() denotes the logarithmic integral, namely li() = lim "!0 + () li() (! ), (.7) Z " 0 Z dt log t + dt +" log t The asymtotic formula (.7) was conjectured by Gauss [7] in 793. By [38], there is a ositive absolute constant c 2 such that () = li() + O( e( c 2 log )). The currently best elicit version of this result is due to Trudgian [36, Theorem 2]. He found that r! log () li() ale e (.8) (log ) 3/ for every 229. Again, the work of Korobov [23] and Vinogradov [39] imlies the currently asymtotically strongest error term for the di erence () li(), namely that there is a ositive absolute constant c 3 so that () = li() + O e c 3 (log ) 3/5 (log log ) /5. (.9) Ford [5,. 2] has found that the constant c 3 in (.9) can be chosen to be equal to Cheng used his own result concerning a zero-free region for the Riemann zeta-function (s) to show that the inequality () li() ale.88(log ) 3/5 e 57 (log )3/5 (log log ) /5 (.0) holds for every > 0 (see [5,. 2]). Panaitool [28,. 55] gave another comletely di erent asymtotic formula for the rime counting function by showing that for every ositive integer m, we have () = log k log k 2... log 2 k m log m + O. log m+2, (.) where the ositive integers k,..., k m are defined by the recurrence formula k m +!k m + 2!k m (m )!k = m m!. In the second art of this aer, we are interested in finding new elicit estimates for the rime counting function () which corresond to the first terms of the denominator in (.). In order to do this, we use (.6) to translate the inequalities obtained in Theorem into the following elicit uer bound.

4 INTEGERS: 8 (208) 4 Theorem 2. For every 49, we have () < log log 3.5 log log log log (.2) log 6 For all su ciently large values of, Theorem 2 is a consequence of (.8) and (.0). Again we aly Theorem to (.6) and get the following lower bound for the rime counting function which corresonds to the first terms of the denominator in (.). Theorem 3. For every () > log log 9, 033, 744, 403, we have 2.85 log log log log (.3) log 6 For all su ciently large values of, the lower bound obtained in Theorem 3 follows directly from (.8) and (.0). Those estimates for the rime counting function which corresond to the first terms of the asymtotic formula (.) are used to get e ective estimates for / () (see, for instance, [3]). In Section 4, we use the e ective estimates for Chebyshev s #-function obtained in Theorem to derive two new results concerning the eistence of rime numbers in short intervals. The origin of this roblem is Bertrand s ostulate which states that for each ositive integer n there is a rime number with n < ale 2n. We give the following two refinements. Theorem 4. For every 6, 034, 256 there is a rime number such that < ale log 3, and for every > there is a rime number such that < ale log 4. In Section 5 and Section 6, we use Theorem to derive some uer and lower bounds for the functions of rime numbers X, X log, and Y ale ale which imrove Dusart s estimates for these functions, see [2, Theorems 5.6, 5.7, and 5.9]. ale

5 INTEGERS: 8 (208) 5 2. Proof of Theorem The asymtotic formula (.2) imlies that for every ositive integer k there is a ositive real number k and a real (k) 2 so that #() < k / log k for every (k). In this direction, Dusart [2, Theorem 4.2] found the following remarkable e ective estimates. Lemma (Dusart). We have #() < k log k for every (k) with k k (k) 908, 994, 923 3, 594, 64 7, 73, 33, , 967, , 35, Now we give a roof of Theorem, where we find the corresonding value for the case k = 3 and 3 = 0.5. In order to do this, we use elicit estimates for Chebyshev s -function, which is defined by () = X log, m ale and some recent work of Dusart [] which is based on elicit zero-free regions for the Riemann zeta-function (s) due to Mossingho and Trudgian [26], an elicit zero-density estimate of Ramaré [30], and a numerical result of Gourdon [8] who announced to have verified that the first 0 3 nontrivial zeros of the Riemann zetafunction (s) lie on the line Re(s) = /2. Proof of Theorem. First, we check that the inequality #() < 0.5 log 3 (2.) holds for every e 35. We set g() = (log ) 3/4 e (log )/R, where R = Since g() is a decreasing function on the interval e 69R/4, we can use (.3) to get #() < 0.48/ log 3 for every e Alying [2, Corollary 4.5], we get ( )b 3 i #() < +.78b3 i e b i 3 + " i b 3 e 2b i+ i log 3 (2.2) for e bi ale ale e bi+, where b i and the corresonding " i are given in Table 5.2 of [2]. Substituting b 8 = 35, b = 50, b 24 = 400, b 32 = 2, 000, b 35 = 3, 500, and the

6 INTEGERS: 8 (208) 6 corresonding values of " i in (2.2), we see that the inequality (2.) also holds for every e 35 ale < e So to rove that (.4) holds for every 9, 035, 709, 63, it remains to deal with the case where 9, 035, 709, 63 ale < e 35. By Büthe [4, Theorem 2], we have #(t) t.95 t for every t such that, 423 ale t ale 0 9. Since 0.5 t >.95 log 3 t for every t 34, 485, 879, 392, the inequality (.4) holds for every such that 34, 485, 879, 392 ale < e 35. In addition, Büthe [4] found that 0.8 ale (t (t))/ t ale 0.8 for every t such that 00 ale t ale Using Lemma of [4], we get #(t) t.8 t 0.8t / (t /3 + t /5 + 2t /3 log t) for every t such that 0 4 ale t ale Since t /5 + 2t /3 log t ale t /3 for every t 783, 674, we get #(t) t.8 t 0.8t / t /3 (2.3) for every t with 783, 674 ale t ale Now we notice that 0.5t/ log 3 t.8 t + 0.8t / t /3 for every t 29, 946, 085, 320. Hence, by (2.3), the inequality (.4) is fulfilled for every such that 29, 946, 085, 320 ale < 34, 485, 879, 392. To rove that the inequality (.4) is also valid for every such that 9, 035, 709, 63 ale < 29, 946, 085, 320, we set f() = ( 0.5/ log 3 ). Let n denote the nth rime number. Since f is a strictly increasing function on (, ), it su ces to check with a comuter that #( n ) > f( n+ ) for every integer n such that (9, 035, 709, 63) ale n ale (29, 946, 085, 320). Now we show that (.5) holds for every >. Since (2.) is already roved, it su ces to show that (.5) holds for every such that < < e 35. In order to do this, we use another result of Büthe [4, Theorem 2]. He found that #() < for every such that ale ale 0 9, which clearly imlies that the inequality (.5) holds for every such that < < e 35. Remark. In [2, Proosition 3.2] it is shown that #() > 0.35/ log 3 for every e 30. We conclude from Theorem that this inequality also holds for every such that 9, 035, 709, 63 ale < e 30. A comuter check gives that the inequality #() > 0.35/ log 3 holds for every with, 332, 492, 593 ale < 9, 035, 709, 63. In the net roosition, we give a slight imrovement of Lemma for the case k = 4, which refines the inequality (.4) for every e 666+2/3. Proosition. For every 70,, we have #() < 00 log 4. (2.4)

7 INTEGERS: 8 (208) 7 Proof. Let R = We use (.3) to get #() < 00/ log 4 for every e Similarly to the roof of Theorem, we use Table 5.2 of [2] to check that the inequality (2.4) also holds for every such that e 00 ale < e Since / log 3 t < 00/ log 4 t for every t satisfying < t ale e 00, Lemma imlies the validity of the desired inequality for every such that 89, 967, 803 ale < e 23. To rove that the inequality (2.4) is also fulfilled for every such that 70, ale < 89, 967, 803, we set f() = ( 00/ log 4 ). Since f is a strictly increasing function for every >, it su ces to check with a comuter that #( n ) > f( n+ ) for every integer n such that (70, ) ale n ale (89, 967, 803). 3. New Estimates for the Prime Counting Function Let k be a ositive integer and let k and (k) be ositive real numbers so that the inequality #() < k log k (3.) holds for every (k). A classical way to derive new elicit estimates for the rime counting function is to consider the function Z J k, k, (k)() = ( (k)) log 2 t + #( (k)) log (k) + log + k log k+ + (k) k log k+2 t dt. This function was introduced by Rosser and Schoenfeld for the case k = [32,. 8] and by Dusart in general [0,. 9]. Using (.6) and (3.), we get J k, k, (k)() ale () ale J k, k, (k)() (3.2) for every (k). In this section, we use (3.2) and aly the estimates for Chebyshev s #-function obtained in the revious section to establish new elicit estimates for the rime counting function () which corresond to the first terms of the asymtotic formula (.). 3.. Proof of Theorem 2 Now we give a roof of Theorem 2. For this urose, we use (3.2), Theorem, and a recent result concerning the sign of li() (), see [4, Theorem 2]. Proof of Theorem 2. Let = 0 5, let f() be given by the right-hand side of (.2), and let r() be the denominator of f(). By (3.2) and Theorem, we get () ale J 3,0.5, () for every. In the first ste of the roof, we comare f() with J 3,0.5, (). In order to rove that the function g() = f() J 3,0.5, () is ositive for every, it su ces to show that g( ) > 0 and that the derivative

8 INTEGERS: 8 (208) 8 of g is ositive for every. By Dusart [2, Table 5.], we have #( ) 999, 999, 965, 752, 660. Further, ( ) = 29, 844, 570, 422, 669 and so we comute g( ) To show that the derivative of g is ositive for every, we set h (y) = y 3823y y y y y y y y y y and comute that h (y) > 0 for every y Therefore, we get g 0 ()r 2 () log 7 h (log ) > 0 for every. Hence f() J 3,0.5, () > 0 for every, and we conclude from (3.2) that the inequality (.2) holds for every. In the second ste, we check (.2) for every such that, 095, 698 ale < 0 5 by comaring f() with the logarithmic integral li(). In order to do this, we set h 2 (y) = 0.5y 0.75y y y y y y y y y y Then it is easy to see that h 2 (y) 0 for every y Hence, for every 23, 502, we have f 0 () li 0 () = h 2 (log )/(r 2 () log 3 ) 0. In addition, we have f(, 095, 698) li(, 095, 698) > 0. Hence f() > li() for every, 095, 698. Now we use a result of Büthe [4, Theorem 2], namely that () < li() (3.3) for every such that 2 ale ale 0 9, to show that the desired inequality holds for such that, 095, 698 ale < 0 5. Finally, to deal with the case where 0 ale <, 095, 698, we notice that f() is strictly increasing for every such that 0 ale <, 095, 698. So we check with a comuter that f( n ) > ( n ) for every integer n such that (0) ale n ale (, 095, 698) +. A comuter check for smaller values of comletes the roof. Under the assumtion that the Riemann hyothesis is true, von Koch [22] deduced that () = li() + O( log ). An elicit version of von Koch s result is due to Schoenfeld [33, Corollary ]. Under the assumtion that the Riemann hyothesis is true, he found that the inequality holds for every () li() < log (3.4) 8 2, 657. Büthe [5,. 2495] roved the following result. Lemma 2 (Büthe). The inequality (3.4) holds unconditionally for every with 2, 657 ale ale

9 INTEGERS: 8 (208) 9 Now we use Lemma 2 to obtain the following weaker but more comact uer bounds for (). Corollary. We have () < for every 0, where log a log a 2 log 2 a 3 log 3 a 4 log 4 a 5 log 5 a.5 a a a a , 03, 975, , 284, 442, 297. Proof. We only show that the inequality () < log log 3.5 log log log log 5 (3.5) holds for every 32. The roofs of the remaining inequalities are similar and we leave the details to the reader. For every , Theorem imlies the validity of (3.5). Denoting the right-hand side of (3.5) by f(), we set g() = f() li() log /(8 ). We comute that g(0 4 ) > 0 6 and g 0 () 0 for every 0 4. Hence f() li() + log /(8 ) for every 0 4. Alying Lemma 2, we see that the inequality (3.5) also holds for every with 0 4 ale ale A comarison with li() shows that f() > li() for every 4, 54, 694. From (3.3), it follows that the inequality (3.5) holds for every such that 4, 54, 694 ale < 0 4. Since f() is strictly increasing for every 67, in order to verify that f() > () holds for every such that 67 ale < 4, 54, 694, it su ces to check that f( n ) > ( n ) for every integer n such that (67) ale n ale (4, 54, 694) +. We conclude by direct comutation. Remark. In [2, Theorem.3], the resent author claimed that the inequality () < log log 3.35 log log log log log 6 (3.6) held for every e Fortunately, this is really the case. However, there was a mistake in the first art of the roof, where it was claimed that the inequality (3.6) was valid for every 0 4. In the current aer, we filled the ga by roving Theorem 2. Using Proosition, we get the following uer bound for the rime counting function which imroves the inequality (.2) for all su ciently large values of.

10 INTEGERS: 8 (208) 0 Proosition 2. For every 4, we have () < log log 3 log 2 3. (3.7) log 3 Proof. The roof is similar to the roof of Theorem 2 and we leave the details to the reader. We denote the right-hand side of (3.7) by f() and let = 0 5. Comaring f() with J 4,00, () and using (3.2) and Proosition, we get f() > () for every 0 5. Net we comare f() with li() and conclude that the desired inequality also holds for every such that e 7 ale < 0 5. A direct comutation for smaller values of comletes the roof. Integration by arts in (.3) imlies that for every ositive integer m, we have () = log + log log log (m log )! log m +O log m+. (3.8) In this direction, we get the following uer bound for the rime counting function. Proosition 3. For every >, we have () < log + log log log log log log log 8. (3.9) Proof. We set = 0 5. Further, let f() be the right-hand side of (3.9). A comarison with J 3,0.5, () shows that f() > J 3,0.5, () for every 0 5. By (3.2) and Theorem, we get f() > () for every. Net we comare f() with li() and get f() > li() for every, 509, 42. We can use (3.3) to obtain f() > () for every such that, 509, 42 ale ale 0 5. It remains to deal with the cases where < ale, 509, 42. Since f() is a strictly increasing function for every 47, it su ces to check that f( n ) > ( n ) for every integer n such that (47) ale n ale (, 509, 42) +. For smaller values of, we conclude by direct comutation. Remark. Note that the inequality () < log + log log log log 5 holds for every >. The roof is similar to that of Proosition 3 if we use Proosition instead of Theorem. We get the following weaker but more comact uer bound for the rime counting function. Corollary 2. For every () < 27, 777, 762, 89, we have log + log log 3.

11 INTEGERS: 8 (208) Proof. From Proosition 3, it follows that the required inequality holds for every Denoting the right-hand side of the desired inequality by f(), we get f() > li() for every 33, 272, 003, 003. We use (3.3) to finish the roof for every 33, 272, 003, 003. Now we check that f( n ) ( n ) for every integer n satisfying (27, 777, 762, 97) ale n ale (33, 272, 003, 003). Since f is an increasing function for every 7, we get f() > () for every such that 27, 777, 762, 97 ale < 33, 272, 003, 003. A direct comuter check for small values of comletes the roof New lower bounds for the rime counting function Here we give a roof of Theorem 3. In order to do this, we use (3.2), Theorem, and a numerical calculation that verifies the desired inequality for smaller values of. Proof of Theorem 3. Let = 0 0. Further, let f() be the right-hand side of (.3) and let r() be the denominator of f(). To rove that the function g() = J 3, 0.5, () f() is ositive for every, it su ces to show that g( ) > 0 and that the derivative of g is ositive for every. By Dusart [2, Table 5.], we have #( ) ale 9, 999, 939, 83. We combine this with ( ) = 455, 052, 5 to comute g( ) > 28.. To show that the derivative of g is ositive for every, we set h(y) = 28930y y y y y y y y y y Clearly, we have h(y) > 0 for every y log( ). Hence g 0 ()r 2 () log 7 h(log ) 0 for every. Therefore, J 3, 0.5, () f() = g() > 0 for every. Using (3.2) and Theorem, we get the required inequality for every 9, 035, 709, 63. To deal with the remaining cases where 9, 033, 744, 403 ale ale 9, 035, 709, 63, we note that f() is increasing for every 9. So we check with a comuter that ( n ) > f( n+ ) for every integer n such that (9, 033, 744, 403) ale n ale (9, 035, 709, 63). Remark. Theorem 3 imroves the lower bound for () obtained in [2, Theorem.4]. In the net corollary, we establish some weaker lower bounds for the rime counting function. Corollary 3. We have () > log log a 2 log 2 a 3 log 3 a 4 log 4 a 5 log 5 (3.0) for every 0, where

12 INTEGERS: 8 (208) 2 a a a a , 532, 44, 449 7, 822, 207, 95, 33, 532, , 099, 53. Proof. By comaring each right-hand side of (3.0) with the right-hand side of (.3), we see that each inequality holds for every 9, 033, 744, 403. For smaller values of we use a comuter. Now we aly Proosition to obtain the following result which refines Theorem 3 for all su ciently large values of. Proosition 4. For every 9, 423, we have () > log log 3 log (3.) log 3 Proof. Let = 0 6 and let f() denote the right-hand side of (3.). A comarison with J 4, 00, () gives that J 4, 00, () > f() for every 0 6. Now we use (3.2) and Proosition to see that () > f() for every 0 6. To rove that the inequality (3.) is also valid for every such that 9, 423 ale < 0 6, it su ces to check with a comuter that ( n ) > f( n+ ) for every integer n such that (9, 423) ale n ale (0 6 ), because f is a strictly increasing function on the interval (, ). The asymtotic eansion (3.8) imlies that the inequality () > log + log log log log (n )! log n holds for all su ciently large values of. The best elicit result in this direction was found in [2, Theorem.2]. The following refinement of it is a consequence of Theorem 3. Proosition 5. For every () > log + log 2 + 9, 027, 490, 297, we have 2 log log log log log log 8. Proof. Let U() denote the right-hand side of the required inequality and let R(y) = U(y) log y/y. Further, we set S(y) = (y 7 y 6 y y 4 3.5y y y )/y 6. Then S(y) > 0 for every y > Moreover, y 3 R(y)S(y) = y 4 T (y),

13 INTEGERS: 8 (208) 3 where T (y) = y y y y y y By Theorem (3), () > S(log ) > T (log ) S(log ) log 4 = U() for every 9, 033, 744, 403. So it remains to deal with the case where 9, 027, 490, 297 ale < 9, 033, 744, 403. Since U() is a strictly increasing function for every 44, it su ces to check with a comuter that ( n ) > U( n+ ) for every integer n such that (9, 027, 490, 297) ale n ale (9, 033, 744, 403). 4. On the Eistence of Prime Numbers in Short Intervals Bertrand s ostulate states that for each ositive integer n there is a rime number with n < ale 2n; it was roved, for instance, by Chebyshev [7] and by Erdős [3]. In the following, we note some of the remarkable imrovements of Bertrand s ostulate. The first result is due to Schoenfeld [33, Theorem 2]. He discovered that for every 2, 00, there is a rime number with < < ( + /6, 597). Ramaré and Saouter [3, Theorem 3] roved that for every 0, 726, 905, 04 there is a rime number so that < ale ( + /28, 33, 999). Further, they gave a table of sharer results which hold for large, see [3, Table ]. Kadiri and Lumley [2, Table 2] obtained a series of imrovements. For instance, they showed that for every e 50 there is a rime number such that < < (+/2, 442, 59, 73). Dusart [9, Théorème ] roved that for every 3, 275 there eists a rime number such that < ale ( + /(2 log 2 )) and then reduced the interval himself [0, Proosition 6.8] by showing that for every 396, 738 there is a rime number satisfying < ale ( + /(25 log 2 )). Trudgian [36, Corollary 2] roved that for every 2, 898, 242 there eists a rime number with < ale + log 2. (4.) Recently, Dusart [2, Corollary 5.5] imroved Trudgian s result by showing that for every 468, 99, 632 there eists a rime number such that < ale + 5, 000 log 2. (4.2) In [2, Theorem.5], it is shown that for every 58, 837 there is a rime number such that < ale ( +.87/ log 3 ). In [2, Proosition 5.4], Dusart refined the last result by showing that for every 89, 693 there eists a rime number such that < ale + log 3. (4.3)

14 INTEGERS: 8 (208) 4 Theorem 4 of the current aer gives two imrovements of (4.3): by decreasing the coe cient of the term / log 3 and, on the other hand, by increasing the eonent of log in it. We rove Theorem 4 by using some elicit estimates for the Chebyshev #-function. Proof of Theorem 4. Similarly to the roof of Theorem, we get #() < log 3 (4.4) for every e 40. Setting f() = 0.087/ log 3, we use (4.4) to get #( + f()) #() > log log 3 0 for every e 40, which imlies that for every e 40 there is a rime number satisfying < ale / log 3. From (4.2) it is clear that the claim follows for every with 468, 99, 632 ale < e 40. To deal with the cases where 56, 007 ale < 468, 99, 632, we check with a comuter that the inequality n ( / log 3 n ) > n+ holds for every integer n such that (6, 034, 393) ale n ale (468, 99, 632). Finally, we notice that (( / log 3 )) > () for every such that 6, 034, 256 ale < 6, 034, 393, which comletes the roof of the first art. We define g() = 98.2/ log 4. To rove the second art, we first notice that #() < log 4 (4.5) for every e 25. The roof of this inequality is quite similar to the roof of Proosition and we leave the details to the reader. Using (4.5), we obtain the inequality #( + g()) #() > log log 4 0 for every e 25. Analogously to the roof of the first art, we check with a comuter that for every < < e 25 there is a rime so that < ale ( / log 4 ). Using (4.), Dudek [8, Theorem 3.4] claims that for every integer m there eists a rime number between n m and (n + ) m for all n. Actually, he gave a roof for a slightly weaker bound m 4, 97, 69, 788. Alying Theorem 4 to Dudek s roof, we get the following refinement. Proosition 6. Let m (n + ) m for all n. 3, 239, 773, 03. Then there is a rime between n m and

15 INTEGERS: 8 (208) 5 Proof. Let m M 0, where M 0 = 3, 239, 773, 03. First, we set = n m in Theorem 4 to see that there is a rime number satisfying n m ale < n m log 4 (4.6) (n m ) for every n 2. We have n m log 4 (n m ) ale n m + mn m (4.7) if and only if 98.2n/ log 4 n ale m 5. Setting n 0 (t) = ma{k 2 N 98.2k/ log 4 k ale t 5 }, we get n 0 (m) n 0 (M 0 ) Now we aly (4.7) to (4.6) to find that there is a rime with n m ale < n m + mn m (4.8) for every integer n satisfying 2 ale n ale n 0 (m). By the binomial theorem, we have n m + mn m ale (n + ) m. Hence (4.8) imlies that there is a rime between n m and (n + ) m for every 2 ale n ale n 0 (m). On the other hand, Dudek [8,. 42] showed that for every integer t 000 there is a rime between n t and (n + ) t for every n n (t), where n (t) = e(000 e(9.807)/t). Therefore, 000 e(9.807) 000 e(9.807) n (m) = e ale e ale m Since n (m) ale n 0 (m), we finish the roof for all n 2. The remaining case n = is clear. M 0 5. On Estimates of Two Sums Over Primes In this section, we give some refined estimates for the sums X ale and where runs over rimes not eceeding. X ale log, 5.. On the Sum of the Recirocals of All Prime Numbers Not Eceeding In 737, Euler [4] roved that the sum of the recirocals of all rime numbers diverges. In articular, this result imlies that there are infinitely many rimes. Further, Euler [4, Theorema 9] and later Gauss [6] stated that the sum of the

16 INTEGERS: 8 (208) 6 recirocals of all rime numbers not eceeding grows like log log. In 874, Mertens [25,. 52] used several results of Chebyshev s aers (see [6] and [7]) to find that log log is the right order of magnitude for this sum by showing X ale = log log + B + O log. (5.) Here B denotes the Mertens constant (see [34]) and is defined by B = + X log + = , (5.2) where = denotes the Euler- Mascheroni constant. In 962, Rosser and Schoenfeld [32,. 74] derived a remarkable identity which connects the sum of the recirocals of all rime numbers not eceeding with Chebyshev s #-function, namely X ale #() = log log + B + log Z (#(y) y)( + log y) y 2 log 2 dy. (5.3) y Alying (.2) to (5.3), they [32,. 68] refined the error term in Mertens result (5.). Using (5.3) and elicit estimates for the Chebyshev #-function, Rosser and Schoenfeld [32, Theorem 5] showed that log log + B 2 log 2 < X ale < log log + B + 2 log 2, where the left-hand side inequality is valid for every > and the right-hand side inequality holds for every 286. After some remarkable imrovements, the currently best known estimates for the sum of the recirocals of all rime numbers not eceeding are due to Dusart [2, Theorem 5.6]. He used (5.3) to show that (3.) imlies X ale log log B ale k k log k + (k + 2) k (k + ) log k+ (5.4) for every 0 (k). Then he alied Lemma with k = 3 and 3 = 0.5 to get 5 log 3 ale X ale log log B ale 5 log 3 (5.5) for every 2, 278, 383, see [2, Theorem 5.6]. Following Dusart s roof of (5.5), we obtain the following slight refinements of these estimates by using Theorem.

17 INTEGERS: 8 (208) 7 Proosition 7. We have 20 log log 4 ale X ale log log B ale 20 log log 4, where the left-hand side inequality holds for every > and the right-hand side inequality is valid for every 46, 909, 074. Proof. We use (5.4) and Theorem to see that these inequalities hold for every 9, 035, 709, 63. To verify that the left-hand side inequality also holds for every such that 2 ale < 9, 035, 709, 63, we check with a comuter that for every ositive integer n ale (9, 035, 709, 63), X k kalen log log n+ + B 3 20 log 3 n+ 6 log 4. n+ Clearly, the left-hand side inequality holds for every such that < < 2. A similar calculation shows that the right-hand side inequality also holds for every such that 46, 909, 074 ale < 9, 035, 709, On Another Sum Over All Prime Numbers Nnot Eceeding In 857, de Polignac [29, art 3] stated without roof that log is the right asymtotic behavior for X log, (5.6) ale where runs over rimes not eceeding. A rigorous roof of this statement was given by Mertens [25] in 874. He showed that X ale log In 909, Landau [24, 55] imroved (5.7) by finding X ale log = log + O(). (5.7) = log + E + O(e( 4 log )), where E is a constant defined by X log E = ( ) = Rosser and Schoenfeld [32,. 74] connected the sum in (5.6) with Chebyshev s #- function by showing X ale log = log + E + #() Z #(y) y y 2 dy. (5.8)

18 INTEGERS: 8 (208) 8 Using their own elicit estimates for the Chebyshev #-function, they roved that log + E 2 log < X ale log < log + E + 2 log, where the left-hand side inequality is valid for every > and the right-hand side inequality holds for every 39, see [32, Theorem 6]. In [0, Theorem 6.], Dusart utilized (5.8) to show that (3.) imlies X ale log log E ale k (k ) log k + k log k (5.9) for every 0 (k). Then he alied Lemma with k = 3 and 3 = 0.5 to (5.9) and obtained the currently best estimates for the sum given in (5.6), namely 0.3 log 2 < X ale log log E < 0.3 log 2 for every 92, 560, see [, Theorem 5.7]. Now (5.9) and Theorem imly the following refinement. Proosition 8. We have 3 40 log log 3 ale X ale log log E ale 3 40 log log 3, where the left-hand side inequality is valid for every > and the right-hand side inequality holds for every 30, 972, 320. Proof. From (5.9) and Theorem, we may conclude that the desired inequalities hold for every 9, 035, 709, 63. Similarly to the roof of Proosition 7, we use a comuter to check the desired inequalities for smaller values of. 6. Refined Estimates for a Product Over Primes The asymtotic formula (5.) imlies that Y ale = e log + O log 2. In [32, Theorem 7], Rosser and Schoenfeld found that e log 2 log 2 < Y < e + log ale 2 log 2, (6.)

19 INTEGERS: 8 (208) 9 where the left-hand side inequality is valid for every 285 and the right-hand side inequality holds for every >. After several imrovements, the sharest known estimates for this roduct are due to Dusart [, Theorem 5.9]. Following Rosser s and Schoenfeld s roof of (6.), Dusart used (5.4) and Lemma with k = 3 and k = 0.5 to find e 0.2 log log 3 < Y < e log log 3 ale for every 2, 278, 382. We use the same method combined with Proosition 7 to obtain the following Proosition 9. For every Y ale ale and for every >, we have Y Proof. First, let < e log 46, 909, 038, we have > e log 46, 909, 074 and let 20 log 3 S = X >(log( /) + /) = 3 6 log 4, (6.2) 20 log log (6.3) ( ) log X X /k k. k=2 > Using the right-hand side inequality in Proosition 7 and the definition (5.2) of B, we get Y > e log e 3 S 20 log 3 6 log 4. ale Now we use the inequality e t + t, which holds for every real t, and the fact that S < 0 to obtain the inequality (6.2) for every 46, 909, 074. A comuter check comletes the roof of the first art. Analogously, we use the left-hand side inequality of Proosition 7 to get Y < e log e S + 20 log log 4 ale for every >. By Rosser and Schoenfeld [32,. 87], we have ) log ) for every > and we arrive at the end of the roof. S <.02/(( Acknowledgement. I would like to eress my great areciation to Jan Büthe and Marc Deléglise for the comutation of several secial values of Chebyshev s #- function. Furthermore, I would like to thank the anonymous reviewer for the useful comments and suggestions to imrove the quality of this aer.

20 INTEGERS: 8 (208) 20 References [] T. Aostol, Introduction to Analytic Number Theory, Sringer-Verlag, New York-Heidelberg, 976. [2] C. Aler, New bounds for the rime counting function, Integers 6 (206), Paer No. A22, 5. [3] D. Berkane and P. Dusart, On a constant related to the rime counting function, Mediterr. J. Math. 3 (206), no. 3, [4] J. Büthe, An analytic method for bounding (), Math. Com. 87 (208), no. 32, [5] J. Büthe, Estimating () and related functions under artial RH assumtions, Math. Com. 85 (206), no. 30, [6] P. L. Chebyshev, Sur la fonction qui détermine la totalité des nombres remiers inférieurs à une limite donnée, Mémoires des savants étrangers de l Acad. Sci. St.Pétersbourg 6 (848), -9. [Also in J. math. ures ali. 7 (852), ] [7] P. L. Chebyshev, Mémoire sur les nombres remiers, Mémoires des savants étrangers de l Acad. Sci. St.Pétersbourg 7 (850), [Also in J. math. ures al. 7 (852), ] [8] A. Dudek, An elicit result for rimes between cubes, Funct. Aro. Comment. Math. 55 (206), no. 2, [9] P. Dusart, Inégalités elicites our (X), (X), (X) les nombres remiers, C. R. Math. Acad. Sci. Soc. R. Can. 2 (999), no. 2, [0] P. Dusart, Estimates of some functions over rimes without R.H., rerint, 200. Available at ariv.org/ [] P. Dusart, Estimates of, for large values of without the Riemann hyothesis, Math. Com. 85 (206), no. 298, [2] P. Dusart, Elicit estimates of some functions over rimes, Ramanujan J. 45 (208), no., [3] P. Erdős, Beweis eines Satzes von Tschebyschef, Acta Litt. Sci. Szeged 5 (932), [4] L. Euler, Variae observationes circa series infinitas, Comment. Acad. Sci. Petrool. 9 (744), [5] K. Ford, Vinogradov s integral and bounds for the Riemann zeta function, Proc. London Math. Soc. 85 (2002), no. 3, [6] C. F. Gauss, Werke 0., Teubner, Leizig, 97, -6. [7] C. F. Gauss, Werke 2, Königliche Gesellschaft der Wissenschaften, Göttingen, 863, [8] X. Gourdon, The 0 3 first zeros of the Riemann zeta function, and zeros comutation at very large height. numbers.comutation.free.fr/constants/miscellaneous/ zetazerose3-e24.df. [9] J. Hadamard, Sur la distribution des zéros de la fonction (s) et ses conséquences arithmétiques, Bull. Soc. Math. France 24 (896),

21 INTEGERS: 8 (208) 2 [20] H. Kadiri, Une région elicite sans zéros our la fonction de Riemann, Acta Arith. 7 (2005), no. 4, [2] H. Kadiri and A. Lumley, Short e ective intervals containing rimes, Integers 4 (204), Paer No. A6, 8. [22] H. von Koch, Sur la distribution des nombres remiers, Acta Math. 24 (90), no., [23] N. M. Korobov, Estimates of trigonometric sums and their alications, Usehi Mat. Nauk 3 (958), no. 4 (82), [24] E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 2 vols., Leizig, Teubner, 909. Rerinted in 953 by Chelsea Publishing Co., New York. [25] F. Mertens, Ein Beitrag zur analytischen Zahlentheorie, J. Reine Angew. Math. 78 (874), [26] M. J. Mossingho and T. S. Trudgian, Nonnegative trigonometric olynomials and a zero-free region for the Riemann zeta-function, J. Number Theory 57 (205), [27] W. Narkiewicz, The Develoment of Prime Number Theory. From Euclid to Hardy and Littlewood, Sringer Monograhs in Mathematics, Sringer-Verlag, Berlin, 2000, ii+448. [28] L. Panaitool, A formula for () alied to a result of Koninck-Ivić, Nieuw Arch. Wiskd. (5) (2000), no., [29] A. de Polignac, Recherches sur les nombres remiers, Comtes Rendus Acad. Sci. Paris 45, , , , [30] O. Ramaré, An elicit density estimate for Dirichlet L-series, Math. Com. 85 (206), no. 297, [3] O. Ramaré and Y. Saouter, Short e ective intervals containing rimes, J. Number Theory 98 (2003), no., [32] J. B. Rosser and L. Schoenfeld, Aroimate formulas for some functions of rime numbers, Illinois J. Math. 6 (962), [33] L. Schoenfeld, Sharer bounds for the Chebyshev functions () and () II, Math. Com. 30 (976), no. 34, [34] N. J. A. Sloane, Sequence A07776, The on-line encycloedia of integer sequences. Available at oeis.org/a [35] T. S. Trudgian, An imroved uer bound for the argument of the Riemann zeta-function on the critical line II, J. Number Theory 34 (204), [36] T. S. Trudgian, Udating the error term in the rime number theorem, Ramanujan J. 39 (206), no. 2, [37] C.-J. de la Vallée Poussin, Recherches analytiques la théorie des nombres remiers, Ann. Soc. scient. Bruelles 20 (896), [38] C.-J. de la Vallée Poussin, Sur la fonction (s) de Riemann et le nombre des nombres remiers inférieurs à une limite donnée, Mem. Couronnés de l Acad. Roy. Sci. Bruelles 59 (899), -74. [39] I. M. Vinogradov, A new estimate of the function (+it), Izv. Akad. Nauk SSSR. Ser. Mat. 22 (958), 6-64.