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Harold G. Diamond; Janos Pintz
Oscillation of Mertens’ product formula
Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), p. 523-533, doi: 10.5802/jtnb.687
Article PDF | Reviews MR 2605532 | Zbl 1214.11102
Class. Math.: 11N37, 34K11
Keywords: Mertens’ product formula, oscillation, Euler’s constant, Riemann hypothesis, zeta function
[1] R. J. Anderson and H. M. Stark, Oscillation theorems. In Analytic number theory (Philadelphia, Pa., 1980), pp. 79–106, Lecture Notes in Math. 899, Springer, 1981. MR0654520 (83h:10082). MR 654520 | Zbl 0472.10044
[2] T. M. Apostol, Introduction to analytic number theory. Undergraduate Texts in Mathematics, Springer, 1976. MR0434929 (55 #7892). MR 434929 | Zbl 0335.10001
[3] P. T. Bateman and H. G. Diamond, Analytic Number Theory: An Introductory Course. World Scientific Pub. Co., 2004. MR2111739 (2005h:11208). MR 2111739 | Zbl 1074.11001
[4] H. Cramér, Some theorems concerning prime numbers. Ark. f. Mat., Astron. och Fys. 15, No. 5 (1921), 1–33. JFM 47.0156.01
[5] H. G. Diamond, Changes of sign of $\pi (x) - {\rm {li \, }}x$. Enseign. Math. (2) 21 (1975), 1–14. MR0376566 (51 #12741). MR 376566 | Zbl 0304.10025
[6] A. Y. Fawaz, The explicit formula for $L_0(x)$, Proc. London Math. Soc. (3) 1 (1951), 86–103. MR0043841 (13 #327c). MR 43841 | Zbl 0042.27302
[7] A. Y. Fawaz, On an unsolved problem in the analytic theory of numbers, Quart. J. Math., Oxford Ser. (2) 3 (1952), 282–295. MR0051857 (14 #537a). MR 51857 | Zbl 0047.27901
[8] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed. Oxford Univ. Press, 1979. MR0568909 (81i:10002). MR 568909 | Zbl 0423.10001
[9] A. E. Ingham, Two conjectures in the theory of numbers. Am. J. Math. 64 (1942), 313–319. MR000202 (3 #271c). MR 6202 | Zbl 0063.02974
[10] J. E. Littlewood, Sur la distribution des nombres premiers. Comptes Rendus Acad. Sci. Paris 158 (1914), 1869–1872. JFM 45.0305.01
[11] F. Mertens, Ein Beitrag zur analytischen Zahlentheorie. J. reine angew. Math. 78 (1874), 46–62. JFM 06.0116.01
[12] H. L. Montgomery and R. C. Vaughan, Multiplicative number theory, I. Classical theory. Cambridge Studies in Adv. Math. 97. Cambridge Univ. Press, 2007. MR2378655. MR 2378655 | Zbl 1142.11001
[13] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), 64–94. MR0137689 (25 #1139). Article | MR 137689 | Zbl 0122.05001
[14] J. Sondow and E. W. Weisstein, Mertens’ Theorem. MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/MertensTheorem.html.
Harold G. Diamond; Janos Pintz
Oscillation of Mertens’ product formula
Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), p. 523-533, doi: 10.5802/jtnb.687
Article PDF | Reviews MR 2605532 | Zbl 1214.11102
Class. Math.: 11N37, 34K11
Keywords: Mertens’ product formula, oscillation, Euler’s constant, Riemann hypothesis, zeta function
Résumé - Abstract
Mertens’ product formula asserts that
$$ \prod _{p \le x} \Big ( 1 - \frac{1}{p} \Big )\, \log x \, \rightarrow \, e^{-\gamma } $$as $x \rightarrow \infty $. Calculation shows that the right side of the formula exceeds the left side for $2 \le x \le 10^8$. It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $\pi (x) - \textrm{li } x$, this and a complementary inequality might change their sense for sufficiently large values of $x$. We show this to be the case.
Bibliography
[2] T. M. Apostol, Introduction to analytic number theory. Undergraduate Texts in Mathematics, Springer, 1976. MR0434929 (55 #7892). MR 434929 | Zbl 0335.10001
[3] P. T. Bateman and H. G. Diamond, Analytic Number Theory: An Introductory Course. World Scientific Pub. Co., 2004. MR2111739 (2005h:11208). MR 2111739 | Zbl 1074.11001
[4] H. Cramér, Some theorems concerning prime numbers. Ark. f. Mat., Astron. och Fys. 15, No. 5 (1921), 1–33. JFM 47.0156.01
[5] H. G. Diamond, Changes of sign of $\pi (x) - {\rm {li \, }}x$. Enseign. Math. (2) 21 (1975), 1–14. MR0376566 (51 #12741). MR 376566 | Zbl 0304.10025
[6] A. Y. Fawaz, The explicit formula for $L_0(x)$, Proc. London Math. Soc. (3) 1 (1951), 86–103. MR0043841 (13 #327c). MR 43841 | Zbl 0042.27302
[7] A. Y. Fawaz, On an unsolved problem in the analytic theory of numbers, Quart. J. Math., Oxford Ser. (2) 3 (1952), 282–295. MR0051857 (14 #537a). MR 51857 | Zbl 0047.27901
[8] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed. Oxford Univ. Press, 1979. MR0568909 (81i:10002). MR 568909 | Zbl 0423.10001
[9] A. E. Ingham, Two conjectures in the theory of numbers. Am. J. Math. 64 (1942), 313–319. MR000202 (3 #271c). MR 6202 | Zbl 0063.02974
[10] J. E. Littlewood, Sur la distribution des nombres premiers. Comptes Rendus Acad. Sci. Paris 158 (1914), 1869–1872. JFM 45.0305.01
[11] F. Mertens, Ein Beitrag zur analytischen Zahlentheorie. J. reine angew. Math. 78 (1874), 46–62. JFM 06.0116.01
[12] H. L. Montgomery and R. C. Vaughan, Multiplicative number theory, I. Classical theory. Cambridge Studies in Adv. Math. 97. Cambridge Univ. Press, 2007. MR2378655. MR 2378655 | Zbl 1142.11001
[13] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), 64–94. MR0137689 (25 #1139). Article | MR 137689 | Zbl 0122.05001
[14] J. Sondow and E. W. Weisstein, Mertens’ Theorem. MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/MertensTheorem.html.
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