Zeroes of the Mertens Function

I examined the Möbius and Mertens functions in a post titled The Möbius Function and Mertens Function on January 25th 2020. In number theory, we define the Mertens function as:$$M(n) = \sum_{1\le k \le n} \mu(k)$$where \( \mu (k)\) is the Möbius function. For any positive integer n, \(μ(n)\) has values in {−1, 0, 1} depending on the factorisation of \(n\) into prime factors:$$\mu(n) = \begin{cases} 1 & \quad \text{if } n \text{ is square-free + integer with even number of prime factors}\\ -1 & \quad \text{if } n \text{ is square-free + integer with odd number of prime factors}\\ 0 & \quad \text{if } n \text{ has a squared prime factor} \end{cases}$$

In that earlier post, I plotted the Mertens function for values up to one million but in this post I want to look at a smaller range and focus on the zeroes of the function in that range. Figure 1 shows a plot of the  Mertens function for values between 25500 and 26080.

Figure 1

We see that in this range, the zeroes occur in a run of three (25514, 25515, 25516) and later singly (26077).  I'm focusing on this range because these numbers are in the vicinity of my current diurnal age which is 26634 as of March 5th 2022. I'm afraid I've missed these previous zeroes in my daily number analysis because they don't register in a search of the OEIS unless you're looking at b-files.

The zeroes of the Mertens function comprise OEIS A028442:

A028442

Numbers \(k\) such that Mertens's function M(\(k\)) (A002321) is zero.

The initial values are:

2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, 329, 331, 332, 333, 353, 355, 356, 358, 362, 363, 364, 366, 393, 401, 403, 404, 405, 407, 408, 413, 414, 419, 420, 422, 423, 424, 425, 427

Figure 2 shows a further range from 26000 to 26750 where there a lot more zeroes:

Figure 2

Here the zeroes are:

26077, 26134, 26142, 26146, 26153, 26154, 26162, 26163, 26164, 26177, 26179, 26180, 26181, 26183, 26184, 26263, 26264, 26266, 26269, 26273, 26277, 26279, 26280, 26282, 26285, 26321, 26346, 26349, 26350, 26427, 26428, 26430, 26434, 26443, 26444, 26446, 26449, 26450, 26451, 26452, 26454, 26710

The zero upcoming for me is 26710 which is not that far off (Friday, May 20th 2022). After that there is not another zero until Tuesday, April 30th 2024 which is more than two years away. This distant zero will occur as the first in a closely spaced series: 27421, 27429, 27431 and 27432. After that, there is a large gap. See Figure 3.

Figure 3

The full list of zeroes from 27421 to 40000 is as follows:

27421, 27429, 27431, 27432, 27922, 27939, 27940, 27973, 27977, 28009, 28011, 28012, 28014, 28018, 28021, 28031, 28032, 28033, 28127, 28128, 28155, 28156, 28183, 28184, 28189, 28191, 28192, 28193, 28202, 28221, 28254, 28259, 28260, 28262, 28283, 28284, 28290, 28551, 28552, 28554, 28558, 28562, 28565, 28566, 28567, 28568, 29469, 30253, 30262, 30269, 30271, 30272, 33162, 33195, 33196, 33202, 33207, 33208, 33211, 33212, 33342, 33346, 33370, 33373, 33377, 33379, 33380, 33381, 33383, 33384, 33386, 33389, 33390, 33393, 33395, 33396, 33398, 33399, 33400, 33429, 33431, 33432, 33434, 33435, 33436, 33438, 33527, 33528, 33530, 33533, 33534, 35958, 35961, 35963, 35964, 35967, 35968, 35974, 35975, 35976, 35978, 35981, 35982, 36149, 36150, 36154, 36363, 36364, 36415, 36416, 36422, 36423, 36424, 36425, 36561, 36690, 39014, 39015, 39016, 39023, 39024, 39025, 39047, 39048, 39057, 39059, 39060, 39062, 39067, 39068, 39069, 39071, 39072, 39074, 39075, 39076, 39079, 39080, 39082, 39083, 39084, 39093, 39797, 39798, 39801, 39803, 39804, 39811, 39812, 39817

I find the Mertens function oddly fascinating and its graph certainly resembles the graph of cumulative random coin tosses where a tail counts as -1 and a head as +1. In the graph of the Mertens function however, the graph can run along the \(x\) axis for a bit because any number with repeated prime factors counts as 0.

For example, consider the run of zeroes 26449, 26450, 26451, 26452 where we have:

• \(26449 = 26449\) which has \( \mu \) = -1 which brings the graph to the \(x\) axis
• \(26450 = 2 \times 5^2 \times 23^2\) which has \( \mu \) = 0 so graph stays on \(x\) axis
• \(26451 = 3^2 \times 2939\) which has \( \mu \) = 0 so graph stays on \(x\) axis
• \(26452 = 2^2 \times 17 \times 389\) which has \( \mu \) = 0 so graph stays on \(x\) axis

Once we reach \(6453 = 7 \times 3779\) we have \( \mu \) = 1 and we leave the \(x\) axis.

OEIS A319520 records increasing runs of zero in the Mertens function:

A319520

Starts of strictly increasing runs of 0's in Mertens's function A002321.

The initial members of the sequence are 2, 39, 331, 422, 45371, 22898822, 871469945 ... where we have:

• 2 is a term because M(2) = 0 for a run of one zero
• 39 is a term because M(39) = M(40) = 0 for a run of two zeroes
• 331 is a term because M(331) = M(332) = M(333) = 0 for a run of three zeroes
• 422 is a term because M(422) = ... = M(425) = 0 for a run of four of four zeroes
• 45371 is a term because M(45371) = ... = M(45376) = 0 for a run of six zeroes

Figure 4 shows the graph of the Mertens function in the vicinity of 45371 to 45376:

Figure 4: permalink

The Möbius Function and Mertens Function

I had come across the Möbius function before but didn't see it as all that important. However, I was reminded of it recently when I encountered a Numberphile video on Mertens conjecture (uploaded 23rd January 2020).

In number theory, we define the Mertens function as:$$M(n) = \sum_{1\le k \le n} \mu(k)$$where \( \mu (k)\) is the Möbius function. The Mertens conjecture is that for all \(n > 1\): $$\left| M(n) \right| < \sqrt { n }$$To quote from Wikipedia:

In mathematics, the Mertens conjecture is the disproven statement that the Mertens function \(M(n)\) is bounded by \( \sqrt {n}\), which implies the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes in an 1885 letter to Charles Hermite (reprinted in Stieltjes (1905)) and Franz Mertens (1897), and disproved by Andrew Odlyzko and Herman te Riele (1985). It is a striking example of a mathematical proof contradicting a large amount of computational evidence in favour of a conjecture.

I set about writing a program in SageMath to plot the first million values of the Mertens function. I managed to execute it in SageMathCell but the code was clunky as I had to get the program to determine the values of -1, 0 and 1 based on the factorisation. Later I realised that the Möbius function would replace the need for this. In SageMath, the spelling is moebius and moebius(1) --> 1 etc. So let's define the Möbius function:

For any positive integer n, \(μ(n)\) has values in {−1, 0, 1} depending on the factorisation of \(n\) into prime factors:$$\mu(n) =
  \begin{cases}
    1       & \quad \text{if } n \text{ is a square-free positive integer with an even number of prime factors}\\
   -1  & \quad \text{if } n \text{ is a square-free positive integer with an odd number of prime factors}\\
0  & \quad \text{if } n \text{ has a squared prime factor}
  \end{cases}$$Figure 1 shows the results of plotting the first one million values of the Möbius function. Here is the permalink to the SageMathCell calculation.

Figure 1

The square root of \( \pm \) 1,000,000 is \( \pm \) 1,000 and as can be seen the function is well within those bounds. However, as mentioned earlier, the bounds are eventually exceeded. Sometimes, instead of \(M(n) \), the function \(m(n) \) is used where:$$m(n)=\frac{M(n)}{\sqrt{n}}$$For the Mertens conjecture to hold, the following condition would be necessary: \(-1 <m(n)<1\). However, for enormously large values of \(n\), it has been shown that \(m(n)<-1.837625 \) and \(m(n)>1.826054\) are possible.