In this document, we will show that
... might be a better approximation for the prime-counting function than
 x/lnx proposed by Bernhard Riemann

1. Better approximation for π(x)
Author:
Chris De Corte
KAIZY BVBA
Beekveldstraat 22 bus 1
9300 Aalst
Belgium
Tel: +32 495/75.16.40
E-mail: chrisdecorte@yahoo.com
Author : chrisdecorte@yahoo.com
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2. Abstract
In this document, we will show that ߨሺ‫ݔ‬ሻ = ߙ‫ ݔ‬ఉ might be a better approximation
for the prime-counting function than ߨሺ‫ݔ‬ሻ = ‫/ݔ‬lnሺ‫ݔ‬ሻ proposed by Bernhard
Riemann [1].
Key-words
prime number theorem (PNT), prime-counting function, asymptotic law of
distribution, Riemann hypothesis, Clay Mathematics.
Introduction
The following document originated during our study of primes and the reading
about the Riemann hypothesis [2,3].
We were baffled by the fact that the young Riemann had found such a complex
formula as a proposition for to the prime-counting function.
Moreover, we were struck by the complexity involved in extending this function
for values in the range 0 to 1.
We assumed that there must be very good reason to continue working with this
formula and that it increased the complexity significantly.
So, we wanted to test the accurateness of this formula with other ones on a set
of given primes.
We especially wanted to focus on the set of primes that were probably unknown
in the time of Riemann.
To our surprise, we found different formula’s that were also interesting and we
selected the one that gets our preference:
ߨሺ‫ݔ‬ሻ = ߙ‫ ݔ‬ఉ
One may want to have α and β dependent on the range of investigation but if we
have to choose values, we would assign ߙ = 0.2083666 ܽ݊݀ ߚ = 0.9294465.
Author : chrisdecorte@yahoo.com
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3. Methods & Techniques
We used Microsoft Excel to do our calculations.
We have split the calculations for 3 different prime ranges.
The first range goes from 2 to 13789 which we assumed was approximately the
highest prime known by Riemann. We therefore call this range “Riemann Time
(RT)”.
The second range goes from 2 to 999983. We call this range “Current Time 1
(CT1)”.
The third range goes from 2 to 49978001. We call this range “Current Time 2
(CT2)”.
For these 3 different ranges, we have set out the primes, counted them (π(x))
and also calculated x/ln(x).
Next, we drew a scatter chart in Excel with π(x) versus x.
We let Excel calculate for us the trend line that best fits the given set of data for
the 3 cases and return us the formula for the case of linear, polynomial, power
and logarithmic trend line.
Following, we use the formula’s given by Excel to calculate the values for all x.
We then calculated the correlations with π(x), the absolute errors at the
beginning and end of the range and compare them with x/ln(x).
Results
Riemann Time (till prime 13789):
Linear:
y = 0.11539x + 71.23392
Logarithmic:
y = 313.23041ln(x) - 1,802.25162
Polynomial :
y = -1.31204E-06x2 + 1.32917E-01x + 3.47937E+01
Power :
y = 5.11909E-01x8.43594E-01
correlations :
Author : chrisdecorte@yahoo.com
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4. Current Time 1 (till prime 999983):
Linear:
y = 7.73567E-02x + 2.24482E+03
Logarithmic:
y = 1.74981E+04ln(x) - 1.83312E+05
Polynomial :
y = -7.98849E-09x2 + 8.51679E-02x + 1.02140E+03
Power :
y = 2.99835E-01x9.02168E-01
correlations :
Current Time 2 (till prime 49978001):
Linear:
y = 0.0594x + 64307
Logarithmic:
y = 6.92061E+05ln(x) - 1.00281E+07
Polynomial :
y = -9.28361E-11x2 + 6.39209E-02x + 2.79717E+04
Power :
y = 0.2083666x0.9294465
correlations :
Author : chrisdecorte@yahoo.com
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5. Purely on correlations, Riemann’s approximation scores very well of course,
though the power formula is already a very serious (sometimes better)
competitor.
But there is more, like the general view of charts:
Here the Riemann approximation looks like the least good one, the power one
much better.
Also with regard to the absolute errors at the last prime of the range, the power
function scores much better:
The next graph is also very interesting. It plots
గሺ௫ሻ
ೣ
ౢ౤ሺೣሻ
ܽ݊݀ ߨሺ‫ݔ‬ሻ/ߙ‫ ݔ‬ఉ . This chart
clearly shows that the power approximation is the preferred one:
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6. Another advantage of the power formula is that it has a valid range between 0
and 1 which would make all the calculations done by Riemann with regard to
primes much easier.
We refer to our excel file to check the details.
Discussions:
We would be the last to question the genius of Bernhard Riemann but we have to
face it that at the time when he lived, there were no computers to check or
calculate results. So, we might have some competitive advantage here.
Suppose the international mathematical community would accept the change for
π(x) what would that then mean for the rest of the prime investigations and for
the Clay Mathematics prize?
Conclusion:
Advantages of power function for π(x):
1. Equally good (sometimes better) correlations
2. reaches asymptote better for large x in graph and lower absolute errors
3. Easier formula for calculations
4. Exists in range 0 to 1 and if necessary till -∞
Acknowledgements
I would like to thank this publisher, his professional staff and his volunteers for all
the effort they take in reading all the papers coming to them and especially I
would like to thank this reader for reading my paper till the end.
I would like to thank Matthew Mutch for providing me the raw file of primes up to
3 million.
I would like to thank my wife for having faith in my work.
Author : chrisdecorte@yahoo.com
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7. References
1. https://en.wikipedia.org/wiki/On_the_Number_of_Primes_Less_Than_a_Given_
Magnitude
2. http://en.wikipedia.org/wiki/Prime_number_theorem
3. https://en.wikipedia.org/wiki/Riemann_hypothesis
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