Experimental Results

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The following table shows the quantity of square-free numbers composed from the product of primes up to Nmax. For example there are 135 square-free numbers less than 1000 that are constructed from 3 primes. The entries that are in bold italics indicate the peaks of the distributions which can be seen to require more primes as Nmax increases. The shift of the peaks and the 'spreading out' of the distribution is the beginning of a trend described by the Erdős-Kac theorem.

Nmax Nmax Nmax Nmax Nmax Nmax Nmax Nmax Nmax
Number of

primes in n

10 100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000 1,000,000,000
1 . 4 25 168 1,229 9,592 78,498 664,579 5,761,456 50,847,535
2 . 2 30 288 2,600 23,313 209,867 1,903,878 17,426,029 160,785,135
3 . 5 135 1,800 19,900 206,964 2,086,746 20,710,806 203,834,084
4 . 16 429 7,039 92,966 1,103,888 12,364,826 133,702,610
5 . 24 910 18,387 286,758 3,884,936 48,396,263
6 . 20 1,235 32,396 605,939 9,446,284
7 . 8 1,044 38,186 885,674
8 . 1 516 29,421
9 . 110
Total

(odd N°of primes)

4 30 303 3,053 30,421 303,857 3,039,127 30,395,384 303,963,666
Total

(even N° of primes)

2 30 304 3,029 30,372 304,068 3,040,163 30,0397,310 303,963,450
Total quantity of

squarefree numbers

7 61 608 6,083 60,794 607,926 6,079,291 60,792,695 607,927,117

We can also see from the table that:

  • The quantity of square-free numbers built from an even number of primes is approximately the same as those built from an odd number of primes.
  • The total quantity of square-free numbers (which includes 1) rapidly approaches the predicted quantity of which is approximately N x 0.6079271018...

If we select a random square-free number then it seems there is a 50-50 chance that it will have either an odd or an even number of primes. If you can prove that the parity of the primes in a square-free number can be modelled like the toss of a fair coin - heads for even, tails for odd - then you have proved the Riemann Hypothesis.

Illuminated switch

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A illuminated light switch has an internal light source either a neon or a LED which allows the user to locate the switch in the dark. Most European illuminated switches are two pole requiring the live and neutral wires to pass into the switch which enables the neon to be powered directly from the mains via a resistor. The internal light source in a single pole illuminated switch derives its power when the switch is OFF from current passing through the external light bulb.


Just a line break or two


The spreading Gaussian distribution of distinct primes


This definition can be extended to Real numbers



Representations

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As an integral

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Using the Euler product one finds that

 

where   is the Riemann zeta function and the product is taken over primes. Then, using this Dirichlet series with Perron's formula, one obtains:

 

where C is a closed curve encircling all of the roots of  

Conversely, one has the Mellin transform

 

which holds for  .

A curious relation given by Mertens himself involving the second Chebyshev function is:

 

A good evaluation, at least asymptotically, would be to obtain, by the method of steepest descent, an inequality:

 

assuming that there are not multiple non-trivial roots of   you have the "exact formula" by residue theorem:

 

Weyl conjectured that Mertens function satisfied the approximate functional-differential equation

 

where H(x) is the Heaviside step function, B are Bernoulli numbers and all derivatives with respect to t are evaluated at t = 0.

Titchmarsh (1960) provided a Trace formula involving a sum over mobius function and zeros of Riemann Zeta in the form

 

where 't' sums over the imaginary parts of nontrivial zeros, and (g, h) are related by a Fourier transform so

 

As a sum over Farey sequences

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Another formula for the Mertens function is

    where       is the Farey sequence of order n.

This formula is used in the proof of the Franel–Landau theorem.[1]

As a determinant

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M(n) is the determinant of the n × n Redheffer matrix, a (0,1) matrix in which aij is 1 if either j is 1 or i divides j.

Calculation

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  1. ^ Edwards, Ch. 12.2