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Sign Changes in the Sums of the Liouville Function

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Sign Changes in the Sums of the Liouville Function SIGN CHANGES IN SUMS OF THE LIOUVILLE FUNCTION PETER BORWEIN, RON FERGUSON, AND MICHAEL J. MOSSINGHOFF Abstract. The Liouville function λ(n) is the completely multiplicative func- tion whose value is −1 at each prime. We develop some algorithms for com- Pn puting the sum T (n) = k=1 λ(k)/k, and use these methods to determine the smallest positive integer n where T (n) < 0. This answers a question originat- ing in some work of Tur´an, who linked the behavior of T (n) to questions about the Riemann zeta function. We also study the problem of evaluating P´olya’s Pn sum L(n) = k=1 λ(k), and we determine some new local extrema for this function, including some new positive values. 1. Introduction The Liouville function λ(n) is the completely multiplicative function defined by λ(p) = −1 for each prime p. Let ζ(s) denote the Riemann zeta function, defined for complex s with <(s) > 1 by −1 X 1 Y 1 ζ(s) = = 1 − . ns ps n≥1 p It follows then that −1 ζ(2s) Y 1 X λ(n) (1) = 1 + = ζ(s) ps ns p n≥1 for <(s) > 1. Let L(n) denote the sum of the values of the Liouville function up to n, n X L(n) := λ(k), k=1 so that L(n) records the difference between the number of positive integers up to n with an even number of prime factors and those with an odd number (counting multiplicity). P´olya noted in 1919 [15] that the Riemann hypothesis follows if L(n) does not change sign for sufficiently large n. This may be established by applying parital summation in (1) and then employing a well-known theorem of Landau on the convergence of Dirichlet series with terms of constant sign; see for instance [14]. It is also known that the zeros of the zeta function are all simple if L(n) is eventually of constant sign. P´olya proved in [15] that L((p − 3)/4) = 0 for any prime p > 7 for which p ≡ 3 mod 4 and the number of classes of positive definite quadratic forms of discriminant −p is 1, so L((p−3)/4) = 0 for p = 11, 19, 43, 67, and 163. He verified that L(n) ≤ 0 1991 Mathematics Subject Classification. Primary: 11Y35; Secondary: 11M26. Key words and phrases. Liouville function, P´olya’s sum, Tur´an’s sum, Riemann hypothesis. Research of P. Borwein supported in part by NSERC of Canada and MITACS. 1 2 PETER BORWEIN, RON FERGUSON, AND MICHAEL J. MOSSINGHOFF for all n between 2 and approximately 1500, and found that in fact L(n) = 0 for several values of n beyond those obtained from the quadratic imaginary fields with class number 1; for example, L(586) = 0. In 1940, Gupta [6] verified that L(n) ≤ 0 for 2 ≤ n ≤ 20000. Also, D. H. Lehmer’s review of Gupta’s paper mentions that L(48512) = −2, so it seems possi- ble that Lehmer may have performed some extensive computations on this problem around this time, too. Haselgrove [7] later verified that L(n) remains negative up to 250 000, and in his article [8] he remarks that Lehmer had in fact checked this condition up to n = 600 000. In 1942, Ingham [10] noted that the Riemann hypothesis,√ and the simplicity√ of the zeros of ζ(s), follows more generally if either L(n) < c n or L(n) > −c n, for some positive constant c. Further, it would also follow from either of these conditions that the imaginary parts of the zeros of ζ(s) lying in the upper half plane would satisfy infinitely many nontrivial linear relations over the field of rationals. Similar statements hold for a particular weighted sum of the Liouville function. Landau proved in his thesis that the convergence of the sum X λ(n) = 0 n n≥1 is in fact equivalent to the prime number theorem. (Landau also considered the analogous statement involving the M¨obius function, µ(n).) Define the function T (n) by n X λ(k) T (n) := , k k=1 and let Un(s) denote the sum of the first n terms of the series for the zeta function, n X 1 U (s) := . n ks k=1 In 1948, Tur´an[18] explored connections between the values of T (n) and Un(s), and their relation to the Riemann hypothesis. He proved that the Riemann hypothesis follows if Un(s) does not vanish in the half-plane <(s) > 1 for sufficiently large n, or more generally in the half-plane c (2) <(s) ≥ 1 + √ n for some positive constant c. Somewhat weaker conditions suffice as well (see [18] and [19]). Since ζ(s) has no zeros in <(s) > 1, these conditions may seem plausible. In fact, Tur´anproved that if Un(s) does not vanish in the half-plane (2), then c T (n) > −√1 n for a positive constant c1, and that this condition implies the Riemann hypothesis. In particular, therefore, the Riemann hypothesis follows if T (n) remains positive for large n. We remark that in 1983 Montgomery [13] established that Un(s) does in fact 4 have zeros with large real part, proving that for each positive number c < π − 1 and every n > n0(c), the function Un(s) has zeros in the half-plane <(s) > 1 + c(log log n)/ log n. SIGN CHANGES IN SUMS OF THE LIOUVILLE FUNCTION 3 In [18], Tur´anreported that the positivity of T (n) was checked for n ≤ 1000 by five Danish mathematicians: Eilertsen, Kristensen, Petersen, Poulsen, and Winther. In 1952, Larsen [11] corrected some minor errors in the previously reported values for T (293) and T (1000), and checked that T (n) > 0 for n ≤ 4528. The stopping point here was selected no doubt because L(4528) = −74, and L(n) does not achieve a smaller value until n = 6317. We mention however that Larsen’s reported value for T (4528) is not correct (its actual value is .0035514 ...), and that his reported minimum over this range at n = 2837 is erroneous as well. In fact, Larsen remarked that his value for T (2837) is something of a near miss, since he calculated it to be less than 1/2837. However, only n = 3, 8, 13, and 32 have the property that T (n) < 1/n, at least until shortly before the first sign change of T (n), described later in this article. In 1953, Bateman and Chowla [2] remarked that the condition T (n) > 0 was checked up to n = 100 000 at the Institute for Numerical Analysis, a laboratory affiliated with the National Bureau of Standards that existed at UCLA between 1947 and 1954. It is not mentioned who performed these calculations, but it seems likely that it was again D. H. Lehmer, who was involved with computations at the Institute around this time. (Lehmer’s review of the article of Bateman and Chowla in fact includes some additional information on the calculation to 100 000, namely, that the computations were performed on the National Bureau of Standards Western Automatic Computer, known as the SWAC [9].1) Haselgrove proved that both L(n) and T (n) change sign in 1958 [8]. We briefly describe his proof, which builds on the work of Ingham [10]. Assume that the Riemann hypothesis is valid, and that the zeros of the zeta function are all simple. Let {ρn} denote the sequence of zeros of ζ(s) on the critical line in the upper half 1 plane, and write ρn = 2 + iγn, with γn < γn+1 for n ≥ 1. Define the function A(x) by (3) A(x) := e−x/2L(ex), ∗ and for a positive real number m, define the function Am(x) by ! 1 γ ζ(2ρ ) ∗ X n n iγnx (4) Am(x) := + 2< 1 − 0 e . ζ(1/2) m ρnζ (ρn) 0<γn<m Then, for any fixed positive real numbers m and y, Ingham proved that ∗ ∗ ∗ (5) lim inf A(x) ≤ lim inf Am(x) ≤ Am(y) ≤ lim sup Am(x) ≤ lim sup A(x). x→∞ x→∞ x→∞ x→∞ ∗ Therefore, if one can find values for m and y for which Am(y) > 0, then it follows that L(n) > 0 for infinitely many integers n. (Naturally, one obtains the same conclusion if the Riemann hypothesis is false, or if its zeros are not all simple.) ∗ Haselgrove found that selecting m = 1000 and y = 831.847 produces Am(y) ≈ .00495. This computation employed the first 649 zeros of the Riemann zeta function. Tur´an’s problem was investigated in the same way. Define (6) B(x) := ex/2T (ex) 1The SWAC was employed in a number of computations in number theory around this time, including Robinson’s determination of the first five Mersenne primes found with the aid of a computer [16]. 4 PETER BORWEIN, RON FERGUSON, AND MICHAEL J. MOSSINGHOFF and let ! 1 γ ζ(2ρ ) ∗ X n n iγnx (7) Bm(x) := − + 2< 1 − 0 e . ζ(1/2) m (ρn − 1)ζ (ρn) 0<γn<m ∗ Then inequalities analogous to those in (5) hold for the functions B(x) and Bm(x), and Haselgrove found that selecting m = 1000 and y = 853.853 or y = 996.980 ∗ produces a negative value of Bm(y). It follows that T (n) < 0 for infinitely many integers n. Haselgrove’s results, however, do not identify a specific value of n where L(n) > 0 or T (n) < 0.
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