Bounds of the Mertens Functions

Darrell Coxa , Sourangshu Ghoshb, a Bachelor of Science, Department of Mathematics , University of Texas, Dallas, United States of America bUndergraduate Student ,Department of Civil Engineering , Indian Institute of Technology Kharagpur, West Bengal, India

ABSTRACT In this paper we derive new properties of and discuss about a likely upper bound of the absolute value of the Mertens function √log(푥!) > |푀(푥)| when 푥 > 1. Using this likely bound we show that we have a sufficient condition to prove the .

1. Introduction We define the Mobius Function 휇(푘). Depending on the factorization of n into prime factors the function can take various values in {−1, 0, 1}

• 휇(푛) = 1 if n has even number of prime factors and it is also square-free(divisible by no perfect square other than 1) • 휇(푛) = −1 if n has odd number of prime factors and it is also square-free • 휇(푛) = 0 if n is divisible by a perfect square.

푛 Mertens function is defined as 푀(푛) = ∑푘=1 휇(푘) where 휇(푘) is the Mobius function. It can be restated as the difference in the number of square-free integers up to 푥 that have even number of prime factors and the number of square-free integers up to 푥 that have odd number of prime factors. The Mertens function rather grows very slowly since the Mobius function takes only the value 0, ±1 in both the positive and negative directions and keeps oscillating in a chaotic manner. Mertens after verifying all the numbers up to 10,000 conjectured that the absolute value of 푀(푥) is always bounded by √푥. This was later disproved by Odlyzko and te Riele1. This conjecture is replaced by a weaker one by Stieltjes2 who conjectured that 푀(푥) = 1 푂(푥2). Littlewood3 proved that the Riemann hypothesis is equivalent to the statement that for every 휖 > 0 1 − −휖 the function 푀(푥)푥 2 approaches zero as x → ∞. This proves that the Riemann Hypothesis is equivalent to 1 +휖 conjecture that 푀(푥) = 푂(푥2 ) which gives a rather very strong upper bound to the growth of 푀(푥). Although there exists no analytic formula, Titchmarsh4 showed that if the Riemann Hypothesis is true and if there exist no multiple non-trivial zeros, then there must exist a sequence 푇푘 which satisfies 푘 ≤ 푇푘 ≤ 푘 + 1 such that the following result holds:

휌 ∞ 푛+1 푥 (−1) 2휋 2푛 푀0(푥) = lim ∑ − 2 + ∑ ( ) 푘→∞ 휌ζ′(휌) (2푛)! 푛ζ(2n + 1) 푥 휌 푛=1 |훾|<푇푘

Where ζ (z) is the Riemann zeta function, and 휌 are all the all nontrivial zeros of the Riemann zeta function 1 and 푀 (푥) is defined as 푀 (푥) = 푀(푥) − 휇(푥) 푖푓 푥 ∈ 푍+, 푀(푥) Otherwise (Odlyzko and te Riele1) 0 0 2 2. New Properties of Mertens Function

5 푥 푥 Lehman proved that ∑푖=1 푀(⌊푥/푖⌋)= 1. In general, ∑푖=1 푀(⌊푥/(푖푛)⌋)= 1, 푛 = 1, 2, 3, . . . , 푥(since⌊⌊푥/푛⌋/푖⌋ = (⌊푥/(푖푛)⌋). Let 푅′ denote a square where element (i, j) equals 1 if 푗 divides 푖 or 0 otherwise. (In a , element (푖, 푗) equals 1 if 푖 divides 푗 or if j = 1. Redheffer6 proved that the of the matrix equals the Mertens Function 푀(푥).) Let 푇 denote the matrix obtained from 푅′ by element-by-element 푥 푥 푥 푥 multiplication of the columns by 푀 (⌊ ⌋) , 푀 (⌊ ⌋) , 푀 (⌊ ⌋) … 푀(⌊ ⌋ ). For example, the T matrix for x = 12 is 1 2 3 푥

−2 0 0 0 0 0 0 0 0 0 0 0 −1 −1 0 0 0 0 0 0 0 0 0 0 −1 0 −1 0 0 0 0 0 0 0 0 0 −1 −1 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 푇 = [ ] 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 1 0 0 0 0 0 1

푥 (1) : ∑푖=1 푀(⌊푥/푖⌋) 푖 = 퐴(푥)

푥 Proof: Let us now take 퐴(푥) = ∑푖=1 φ(i) where φ is Euler’s totient function. Let 푈 denote the matrix obtained from 푇 by element-by-element multiplication of the columns by 휑(푗). The sum of the columns of 푈 then 푥 equals 퐴(푥). Now since 푖 = ∑푑|푖 φ(d) we can write ∑푖=1 푀(⌊푥/푖⌋) 푖 (the sum of the rows of U) equals 퐴(푥).

푥 2 By the Schwarz inequality, 퐴(푥)/√푥(푥 + 1)(2푥 + 1)/6 is a lower bound of √∑푖=1 푀(⌊푥/푖⌋) . 퐴(푥) =

푥 7 8 ∑푖=1 φ(i) Is further simplified by Walfisz and Hardy and Wright as

1 푥 푥 3 퐴(푥) = ∑푥 φ(i) = ∑푥 휇(푘) ⌊ ⌋ (1 + ⌊ ⌋) = 푥2 + 푂(푛(log 푛)2/3(log log 푛)4/3) 푖=1 2 푘=1 푘 푘 휋2

3 2 퐴(푥) 푥 3√3 푥2 √∑푥 푀(⌊푥/푖⌋)2 > > 휋2 = 푖=1 ( )( ) ( )( ) ( )( ) √푥 푥 + 1 2푥 + 1 √푥 푥 + 1 2푥 + 1 휋 √푥 푥 + 1 푥 + 1/2 6 6

This can be further simplified to

3√3 1 √∑푥 푀(⌊푥/푖⌋)2 /푥 > 푖=1 휋 √(1+1/푥 )(1 + 1/2푥)

Taking limit of infinity on both the sides, we get

3√3 lim √∑푥 푀(⌊푥/푖⌋)2 /푥 > 푥→∞ 푖=1 휋

27 This shows that ∑푥 푀(⌊푥/푖⌋)2 at large values of x is greater than 푥2. 푖=1 휋2 Let 훬(푖) denote the Mangoldt function (훬(푖) equals 푙표푔(푝) if 푖 = 푝푚 for some prime 푝 and some 푚 ≥ 1 or 9 푥 0 otherwise). Mertens proved that ∑푖=1 푀(⌊푥/푖⌋)푙표푔 푖 = 휓(푥) where 휓(푥) denotes the second (ψ(x) =∑푖≤푥 훬(푖) ).

푥 Theorem (2) : ∑푥 푀 (⌊ ⌋) log(푖) 휎 (푖)/2 = 푙표푔(푥!) 푖=1 푖 0

푥 Proof: Let 휎푥(푖) denote the sum of positive divisors function (휎푥(푖) = ∑푑|푖 푑 ). Replacing 휑(푗) with 푙표푔(푗) in the U matrix gives a similar result.

푎 푎1 푘 푎1+..+푎푘 Let 휆(푛) denote the (휆(1) = 1 or if = 푝1 . . 푝푘 , 휆(푛) = (−1) ). ∑푑|푛 휆(푑) Equals 1 if n is a perfect square or 0 otherwise. Let 퐿(푥) = ∑n≤x 휆(푑) . Let us also assume 퐻(푥) = ∑n≤x 휇(푛)log (푛) . 퐻(푥)/(푥 푙표푔(푥)) → 0 as 푥 → ∞ and lim 푀(푥)/푥 − 퐻(푥)/(푥 푙표푔(푥))) = 0. The statement lim 푀(푥)/푥 = 0 푥→∞ 푥→∞ 10 is equivalent to the prime number theorem. Also, 훬(푛) = − ∑d|n 휇(푑)log (푑) . (Apostol ).

The generalization of the Euler’s totient function is Jordan totient function. Let it be denoted as 퐽푘(푥) which is defined as number of set of k positive integers which are all less than or equal to n that will form a co-prime 푥 set of (푘 + 1) positive integers together with 푛. Let us define (푥) = ∑푖=1 퐽푘(푖) . It is known that ∑푑|푛 퐽푘(푑) = 푛푘. Then we get the following theorem.

푥 Theorem (3): ∑푥 푀 (⌊ ⌋) 푖푘 = 퐵(푥) 푖=1 푖

퐵(푥) is expanded by McCarthy11 to be

푛푟+1 퐵(푥) = ∑푥 퐽 (푖) = + 푂(푛푟) 푖=1 푘 (푟+1)ζ(r+1)

푥 푛푘+1 We therefore get ∑푥 푀 (⌊ ⌋) 푖푘 = 퐵(푥) = + 푂(푛푘) . 푖=1 푖 (푘+1)ζ(k+1)

Likewise we can derive some other similar relationships using the 푇 matrix that are as listed below:

푥 Theorem (4): ∑푥 푀 (⌊ ⌋) 휎 (푖) = ∑푥 푖푘 for 푘 ∈ 푍+ 푖=1 푖 푘 푖=1

푥 Theorem (5): ∑푥 푀 (⌊ ⌋) where the summation is over those 푖 values that are perfect squares equals 퐿(푥) 푖=1 푖

푥 Theorem (6): ∑푥 푀 (⌊ ⌋) 훬(푖) = −퐻(푥) 푖=1 푖

푥 6 Theorem (7): ∑푥 푀 (⌊ ⌋) 2휔(푛) = ∑푥 |휇(푖)| ~ 푥2 + 푂(√푛) = 푁표. 표푓 푆푞푢푎푟푒 퐹푟푒푒 퐼푛푡푒푔푒푟푠 푖=1 푖 푖=1 휋2

푥 Theorem (8): ∑푥 푀 (⌊ ⌋) 푑(푛2) = ∑푥 2휔(푖) where 푑(푥) is the sum of all the divisors of 푥 푖=1 푖 푖=1

푥 Theorem (9): ∑푥 푀 (⌊ ⌋) 푑2(푛) = ∑푥 푑(푖2) 푖=1 푖 푖=1

푥 푖 휇2(푖) Theorem (10): ∑푥 푀 (⌊ ⌋) ( ) = ∑푥 푖=1 푖 φ(i) 푖=1 φ(i)

Similarly many other relationships can be found between various arithmetic functions and the Mertens Functions. 3. A Likely Upper Bound of |M(x)| The following conjecture is based on data collected for x ≤ 500, 000.

푥 2 Conjecture (1): 푙표푔(푥!) > ∑푖=1 푀(⌊푥/푖⌋) > 휓(푥) when 푥 > 7

By Stirling’s formula, 푙표푔(푥!) = 푥 푙표푔(푥) − 푥 + 푂(푙표푔(푥)), since 푙표푔(푥) increases more slowly than any 푥 2 1+휀 positive power of log(푥), this is a better upper bound of ∑푖=1 푀(⌊푥/푖⌋) than 푥 for any 휀 > 0. This likely 푥 bound can be used to prove the Riemann Hypothesis since ∑푖=1 푀(⌊푥/푖⌋) > |푀(푥)| and therefore we can 푥 2 1+휀 write √log(푥!) > |푀(푥)| . Since the growth of ∑푖=1 푀(⌊푥/푖⌋) is lesser than 푥 for any 휀 > 0. We can say

1 − −휖 푀(푥)푥 2 → 0 푎푠 푥 → ∞.

푥 2 Figure 1 for a plot of 표푔(푥!), ∑푖=1 푀(⌊푥/푖⌋) , and 휓(푥) for 푥 = 1, 2, 3, . . . , 1000.

푥 2 Fig 1: Plot of 푙표푔(푥!), ∑푖=1 푀(⌊푥/푖⌋) , and 휓(푥) for 푥 = 1, 2, 3, . . . , 1000

푥 2 Let 푗(푥) = ∑푖=1 푀(⌊푥/푖⌋) where the summation is over 푖 values where 푖|푥. Let 푙1, 푙2, 푙3 denote the x values where 푗(푥) is a local maximum (that is, greater than all preceding 푗(푥) values) and let 푚1, 푚2, 푚3..... denote the values of the local maxima. The local maxima occur at x values that equal products of powers of small primes (Lagarias12 discussed colossally abundant numbers and their relationship to the Riemann hypothesis). See Figure 2 for a plot of 푙푖/(푙표푔(푙푖) 푚푖), 푚푖/푙푖 , and 1/ 푙표푔(푙푖) for 푖 = 1, 2, 3, . . . , 772 (corresponding to the local maxima for x ≤ 15, 000, 000, 000). (푀(푥) Values for large x were computed using Del´eglise and Rivat’s13 algorithm.) The first two curves cross frequently, so there are 푖 values where mi is approximately equal to 푙푖/√푙표푔(푙푖) .

Fig 2: Plot of 푙푖/(푙표푔(푙푖) 푚푖), 푚푖/푙푖 , and 1/ 푙표푔(푙푖) for 푖 = 1, 2, 3, . . . , 772

푥 2 See Figure 3 for a plot of 푗(푥) and ∑푖=1 푀(⌊푥/푖⌋) for x = 1, 2, 3, ..., 10,000. See Figure 4 for a plot of 2 푙표푔(푙푖), 푙표푔(푚푖), 푙표푔(푀(푙푖) ), and 푙표푔(푚푖/휎0(푙푖)) for i = 1, 2, 3, ..., 772 (when 푀(푙푖) = 0, 푙표푔(푀(푙푖) 2 ) is set to −1). See Figure 5 for a plot of |푀(푙푖)|/ √푙푖 for 푖 = 1, 2, 3, . . . , 772. The largest known value of |푀(푥)|/ √푥 (computed by Kotnik and van de Lune14 for x ≤ 1014) is 0.570591 (for 푀(7, 766, 842, 813) = 50, 286). The largest |푀(푙푖)|/ √푙푖 value for x ≤ 15, 000, 000, 000 is 0.568887 (for 푙푖 = 7, 766, 892, 000). The largest known value of |푀(푥)|/ √푥 (computed by Kuznetsov15 is 0.585767684 (for 푀(11, 609, 864, 264, 058, 592, 345) = −1, 995, 900, 927).)

푥 2 Fig 3: Plot of 푗(푥) and ∑푖=1 푀(⌊푥/푖⌋) for x = 1, 2, 3, ..., 10,000

2 Fig 4: Plot of 푙표푔(푙푖), 푙표푔(푚푖), 푙표푔(푀(푙푖) ), and 푙표푔(푚푖/휎0(푙푖)) for i = 1, 2, 3, ..., 772

Fig 5: Plot of |푀(푙푖)|/ √푙푖 for 푖 = 1, 2, 3, . . . , 772

Let 푙푖 and 푚푖 be similarly defined for the function 휎0(푥). (푙푖 , 푖 = 1, 2, 3, . .. are known as “highly composite” numbers. Ramanujan16 initiated the study of such numbers. Robin17 computed the first 5000 highly ′ ′ ′ composite numbers.) Let 푚푖 denote 푗(푙푖). See Figure 6 for a plot of 푙푖/(푙표푔(푙푖)푚푖), 푚푖/푙푖 , 푎푛푑 1/ 푙표푔(푙푖) for 푖 = 2, 3, 4, . . . , 160 (corresponding to the local maxima for 푥 ≤ 2, 244, 031, 211, 966, 544, 000). (M(x) values for large x were computed using an algorithm similar to that used by Kuznetsov. The computations were done ′ on an Intel i7-6700K CPU with 64 GB of RAM.) Although the first two curves cross frequently, 푚푖 does not appear to converge to 푙푖/√푙표푔(푙푖).

′ ′ Fig 6: Plot of 푙푖/(푙표푔(푙푖)푚푖), 푚푖/푙푖 , 푎푛푑 1/ 푙표푔(푙푖) for 푖 = 2, 3, 4, . . . , 160

′ 2 Fig 7: Plot of 푙표푔(푙푖) + 푙표푔(푙표푔(푙푖)), 푙표푔(푙푖), 푙표푔(푚푖), and 푙표푔(푀(푙푖) ), for 푖 = 2, 3, 4, . . . , 160

′ 2 See Figure 7 for a plot of 푙표푔(푙푖) + 푙표푔(푙표푔(푙푖)), 푙표푔(푙푖), 푙표푔(푚푖), and 푙표푔(푀(푙푖) ), for 푖 = 2, 3, 4, . . . , 160 2 (when 푀(푙푖) = 0, 푙표푔(푀(푙푖) ), is set to −1). The vertical distance between the first and third curves appears ′ to become roughly constant. See Figure 8 for a plot of (푙표푔(푙푖) + 푙표푔(푙표푔(푙푖)) − 푙표푔(푚푖),for i = 2, 3, 4... 160. 1 2 See Figure 9 for a plot of 푙표푔(푙 ) + 푙표푔(푙표푔(푙 )), 푙표푔(∑푙푖 푀(⌊푙 /푖⌋) ), and 푙표푔(푙 ) for 푖 = 2, 3, 4, . . . , 160. 푖 2 푖 푖=1 푖 푖 1 2 2 푙표푔(푙 ) + 푙표푔(푙표푔(푙 )) Is greater than 푙표푔(∑푙푖 푀(⌊푙 /푛⌋) ) and 푙표푔(∑푙푖 푀(⌊푙 /푛⌋) ) is greater than 푙표푔(푙 ) 푖 2 푖 푛=1 푖 푛=1 푖 푖 1 for i > 4. This is evidence in support of Conjecture 1. See Figure 10 for a plot of 푙표푔(푙 ) + 푙표푔(푙표푔(푙 )) 푖 2 푖 2 푙푖 − 푙표푔(∑푛=1 푀(⌊푙푖/푛⌋) ) 푓표푟 푖 = 2, 3, 4, . . . , 160.

′ Fig: 8 for a plot of (푙표푔(푙푖) + 푙표푔(푙표푔(푙푖)) − 푙표푔(푚푖),for i = 2, 3, 4... 160.

1 2 Fig 9: Plot of 푙표푔(푙 ) + 푙표푔(푙표푔(푙 )), 푙표푔(∑푙푖 푀(⌊푙 /푖⌋) ), and 푙표푔(푙 ) for 푖 = 2, 3, 4, . . . , 160 푖 2 푖 푖=1 푖 푖

1 2 Fig 10: Plot of 푙표푔(푙 ) + 푙표푔(푙표푔(푙 )) − 푙표푔(∑푙푖 푀(⌊푙 /푛⌋) ) 푓표푟 푖 = 2, 3, 4, . . . , 160. 푖 2 푖 푛=1 푖

4. Conclusion In this paper we derived new relations between Mertens function with a different arithmetic functions and also discussed about a likely upper bound of the absolute value of the Mertens function √(log (x!) )>|M(x)| when 푥 > 1 with sufficient numerical evidence. Using this likely upper bound we showed that we have a sufficient condition to prove the Riemann Hypothesis using the Littlewood condition.

5. References 1. Odlyzko, A. M. and te Riele, H. J. J. "Disproof of the ." J. reine angew. Math. 357, 138-160, 1985. 2. Stieltjes, T. C. R. A. S. 1885. 3. J. E. Littlewood, Quelques cons_equences de l'hypoth_ese que la fonction_(s) n'a pas de z_eros dans le demi-plan Re(s) > 1=2. C. R. Acad. Sci. Paris 154, 263-266 (1912) 4. Titchmarsh, E. C. The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, 1960. 5. R. S. Lehman, On Liouville's Function, Math. Comput. 14:311-320 (1960) 6. R. M. Redheffer, Eine explizit losbare Optimierungsaufgabe, Internat. Schriftenreine Numer. Math., 36 (1977) 7. Walfisz, A. Ch. 5 in Weyl's exponential sums in modern . Berlin: Deutscher Verlag der Wissenschaften, 1963. 8. Hardy, G.H. and Wright, E.M. "The Average Order of phi (n)." §18.5 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 268-269, 1979 9. F. Mertens, Uber eine zahlentheoretische Funktion, Akademie Wis- senschaftlicher Wien Mathematik-Naturlich Kleine Sitzungsber, 106 (1897) 761-830 10. T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976 11. McCarthy, P.J.: Introduction to Arithmetical Functions. Springer, New York (1986) 12. J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, American Mathematical Monthly 109, 534-543 (2002) 13. M. Del_eglise and J. Rivat, Computing the Summation of the Mobius Function, Experimental Mathematics 5, 291-295 (1996) 14. T. Kotnik and J. van de Lune, On the order of the Mertens function, Experimental Mathematics 13, pp. 473-481 (2004) 15. E. Kuznetsov, Computing the Mertens function on a GPU, arXiv:1108.0135v1 [math.NT], (2011) 16. S. Ramanujan, Highly composite numbers, Proc. London Math. Soc. 14 (1915), 347-407 17. G. Robin, M_ethodes d'optimalisation pour un probl_eme de th_eories des nombres., RAIRO Inform. Th_eor. 17, 239-247, 1983