Conjecture 51 : Two
approximate relationships about prime numbers
Patrick Devlin sent the following two
approximate relationships & assertions about them, in a communication
entitled "ON THE DISTRIBUTION OF THE PRIMES"
1.
∑(p*Ln(p))
~ ∑(n);
p=2, 3, 5, 7, 11,...P; n=1,2,3,4,...P
2. ∑(Ln(p))
~ P; p=2, 3, 5, 7, 11,...P
Assertions:
1. The above relationships hold true for p (2 to Infinity) and n (1 to
Infinity).
2. They can be easily shown to be correct and equivalent to the Gauss
prime
number conjecture (which has now been proven) and so in some sense they
are
a restatement of that conjecture.
3. They provide an interesting tool for further investigation into the
precise distribution of the prime numbers.
4. Unlike
Euler's beautiful product identity for the zeta function which while
being
correct, is always out of phase (ie there is no specific point where
both
sides of the equation agree) these relationships hold true from the
outset
while being in phase with each other.
5. The error ratio diminishes to zero as p tends to infinity.
6. Relationship number 1, is very interesting because it suggests that
the
primes act as a moment balancing the moment exerted by the integers.
7. Relationship number 2 is interesting because it relates the nth prime
to
all of the preceding primes.
A
couple of comments [by CR] before the questions:
a) The
Devlin's first relationship may be written in a simpler way as:
1'.
∑(p*Ln(p))
~ P^{2}/2;
p=2, 3, 5, 7, 11,...P
b)
Regarding the 5th statement please see a pair of graphs that I have
produced: 1,
2.
Questions.
1. Please comment the Devlin's assertions
2. Do you devise a formal deduction of the
Devlin's approximate relationships?
3. Do you devise an interesting application
of the Devlin's approximate relationships?
Contributions came from
Luis Rodríguez and Anton Vrba
***
Rodríguez wrote:
The first conjecture seems to me
very interesting. I have not seen that before. I think it is better
to present the conjecture in this form:
Lim
Σ [p
log(p), p= 2 to P] = P(P + 1 ) / 2
Because the n is currently
reserved for the rank of a prime p. I myself have that confusion at
a first lecture.
The second "conjecture" is
[related to] a [known] theorem. Ross and Schoenfeld have fixed its
bounds:
If Theta(p) = Σ
[log(p), p= 2 to p], then:
p.(1  1/log(p)) < Theta(p) < p.(1 + 1/2log(p))
From this the Prime Numbers Theorem can be deducted. (See John
Williams, Les Nombres Premiers . Hermann Paris). Littlewood
demonstrated that the maximum oscillation that theta(p) can attain
is of order sqr(p). logloglog(p). That means that the Ross bounds
are too much ample.
Anton wrote:
Investigating above one can add the generalized case of Patrick’s
relation as
3.
∑
(p^r*Ln(p)) ~
∑
(n^r);
p=2, 3, 5, 7, 11,...P; n=1,2,3,4,...P
I have calculated with
r
as big as 500 and above relation holds as well as for
r<1
My
comment to Patrick’s assertion is (i) True for 1, 2, 5 and 7 (ii) False
for 3, (iii) No comment for 4 and 6 and finally (iv) Prime Number Theory
already states what is commented as True.
The above comment and Patrick’s relation can be simply demonstrated
as follows:
Considering that Prime Number Theory (PNT) tells us the average gap
of all primes less than
P
are
Ln(P),
therefore presume a prime
P_i
then the next prime by PNT is expected at
P_i+Ln(P_i).
Now lets define a recursive logarithmic series as follows:
a1.
q_1 = x,
for any x>1
a2.
q_i = q_(i1) + Log_a(q_(i1))
and the so obtained
q_i should
resemble the series of prime numbers if the base of the logarithm
a=e,
where
e
is the famous constant 2.718…
By calculation let
q_1=2
then
q_(10^6)=15479073
and
Prime[10^6]=15485863
that is a 0.044% error!
I have discovered or rediscovered that if we define
c
as follows
a3.
c=
∑
(q_i^r*Log_b(q_i) ) /
∑
(n^r) ; for
i=1,2,3,....k ; n=1,2,3,….Integer(q_k)
then
a4.
a
= b^c if
k
≈ infinity
for any
r, a,
b
and
x.
This leads me to believe the only logical reason for Patrick’s
relation
(1. and 2.) is because they are similar to the case
a=b=e
and
r=1
and
0
in
a1
to
a4
above. Hence, all above relationships, in my opinion, add no further
insight other than what is already known by the PNT, to the distribution
of primes as Patrick claims with assertion 3.
During the analysis when I started investigating the effect of the
bases of the log terms, I was utterly amazed to find the relation
a4
a=b^c,
which reinforces my believe that this relation is true, the error gets
extremely small very fast.
Now I pose the following questions:
Q1. Is the property of recursive logarithmic series
a1  a4
discovered or rediscovered? If rediscovered are there any references?
Q2. If discovered how to prove it other than by computational methods.
***
Werner Sand wrote
(30/11/06) a short
note on this issues:
Further approximate relationships see
www.primzahlen.de Theorie/Referenten: "Eine neue Methode zur
Berechnung der Anzahl der Primzahlen unter einer gegebenen Größe" by
Werner D. Sand.
I have asked to Mr. Sand for an English version of his
pdf article, published in German. Hopefully soon we may have it available... Two days latter he wrote and sent the following summary
of his article:
Assertion:
Σ(p, p=2 to
x) ~ π(x²)
or
Σ(p, p=2 to sqrt(x)) ~
π(x),
where p =
prime and x = real,
π = number of primes
Generalization:
Σ(p^r, p=2
to x) ~ π(x^(r+1))
or
Σ(p^r, p=2 to x^(1/(r+1))) ~
π(x)
where r =
real >= 0 (r=0 means counting the primes)
Proof for r =
m = integer:
General
formula for the sum of powers of natural numbers:
Σ(k^m, k=1
to n) = (n^(m+1))/(m+1) + (n^m)/2 + O(n^(m1),
where the
second and third summand can be omitted without effect on the order:
Σ(k^m, k=1
to n) ~ (n^(m+1))/(m+1)
Instead of n
natural numbers we choose the
π(n)
~ n/ln n prime numbers, which means we divide the equation by ln n:
Σ(p^m, p=2
to n) ~ ((n^(m+1))/(m+1)) * (1/ln n) = (n^(m+1))/ln(n^(m+1)) =
π(n^(m+1)),
q.e.d.
Proof for r =
real:
Instead of
sum choose integral: int(x^r) = (x^(r+1))/(r+1). Then proof like
above.
Let
S1 be
π(x)
~ Σ(p, p=2 to sqrt(x)) and
S2 be
π(x)
~ Σ(p², p=2 to x^(1/3)).
Then the
quality of S1 or S2 is comparable to Legendre:
π(x)
~ x/(ln x1), in some cases better.
I didn't
check yet Si with i >=3.
See
this new article from W. Sand (April, 09)
***
As you all can observe,
the approximate equations published by Sand are distinct to the
submitted by Patrick Devlin, because the argument of the summing
operator is distinct.
But share with them that they admit  conjecturally 
a generalization to powers of the terms added, equal to the
generalization that Anton made to the Devlin's approximate equations.
In the Devlin's generalized equations the arguments are
p^r.Ln(p) and Ln(p) while in the Sand's equations the argument is p^r.
Let's give a comparison for the three equations without
generalizations, in the following manner:
Author 
Argument of the
Σ operator 
Limits of the sum 
Value approximated by Σ

Devlin 
p.ln(p) 
2, P 
P^{2}/2 
Devlin 
ln(p) 
2, P 
P 
Sand 
p 
2, P 
π(P^{2}) 
Do you like these results? Have they been
suggested before? Are they useful?
***
Sebastián Martín Ruiz sent (Dec. 7,
2006) the following link related:
***
Anton Vrba wrote (Dec. 8, 06):
Author 
Argument of the
Σ operator 
Limits of the sum 
Value approximated by Σ 
Deviation at
π(P)=10^{5} 
Devlin 
p ln(p) 
2, P 
D_{1} = P^{2}/2 
0.060% 
Devlin 
1 ln(p) 
2, P 
D_{0} = P 
0.094% 
Sand 
p 
2, P 
S_{1 }
=
π(P^{2}) 
0.061% 
Sand 
1 
2,P 
S_{0} =
π(P) 
exact, counting 
Vrba 
p/ln(p) 
2, P 
V_{1}=
π(P)^{2}/2 
8.01% 
Vrba 
1/ln(p) 
2, P 
V_{0} =
π(P)^{2}/P 
1.07% 
After seeing the
first three entries in above table the last three entries had to follow.
Note that is the number of primes squared in last two lines
Note the
following relationship D_{1 }/D_{0} = V_{1 }
/V_{0 }= P/2 and extended over Sands relationships then
(2 D_{1
})/(P D_{0} ) = (2 S_{1 })/(P S_{0} ) =
(2 V_{1 })/(P V_{0} )_{ }= 1
The easy proof
for S_{1}/S_{0} was forwarded by Werner Sand: The Prime
Number Theory states that number of primes equals x / ln x hence
S_{1 }
/S_{0} = π(P^{2})/ π(P) ~ (P²/ln P²) / (P/ln P) = (P²*ln
P)/(2ln P*P) = P/2.
Using the above,
the following relationship between primes and prime counting function
has now been identified
(2 π(P^{2}))
/ ( π(P) P) = 1
Example:
Calculating for the first 10^{5} primes above evaluates to 0.958
and substituting known values for P^{2}=10^{22},
then P=10^{11} and calculating π(P) with
Mathematica the above evaluates to 0.978.
Concerning the
high deviation of V_{1 }it is to be noted that V_{1}
sums the counting function as defined in the PNT and we know that x/ln(x)
consistently gives a lower value for the number of primes than the
actual value (see the
graph on Wolfram’s site) and thus this consistent lower value
integrates into the larger deviation.
***
Werner Sand contributes with another approximate
relationship, in this case based on the definition of
Merten's
constant.
Complement of
Anton Vrba's table:
Author

Argument
of the
Σ operator 
Limits 
Value approximated 
Sand/Mertens 
1/p 
2,x 
lnln(x) 
x is the upper limit of the running variable p (p<=x). x
is real and may be prime or not.
***
Later Werner Sand added two contributions:
A. Another
complement to the table:
Assumption: Σ (ln
p)/p (p=2…x) ~ ln п(x)
Proof:
Σ (ln p)/p = (ln
2)/2 + (ln 3)/3 + (ln 5)/5 + …+
(ln x)/x =
1/(2/(ln 2)) +
1/(3/(ln 3)) + 1/(5/(ln 5)) + ... + 1/(x/(ln x)) = (PNT)
1/1 + 1/2 + 1/3 +
1/4 + ... + 1/(x/(ln x)) ~ (PNT)
1/1 + 1/2 + 1/3 +
1/4 + ... 1/п(x)
~ (Euler)
ln п(x)), q.e.d.
B.
The approximate relations are not
necessarily onetoone. E.g.
Σ (ln p)^{r}/p
(p=2…x) ~ (ln x)^{r}/r (*)
Then if r=1
Σ (ln p)/p
(p=2…x) ~ ln x or
Σ (ln p)/p
(p=2…x) ~ ln п(x) (as shown
in A) (**)
(*)
Just by trial. No proof yet. It's
a conjecture.
Let R = Σ {[(ln p)^r]/p}/{[(ln
x)^r]/r}. I have tested R with MuPAD for r from 0.5 to 1000 incl.
ratios and irratios as sqrt(2) and for p up to 32.452.843 resp. 2*10^6
steps. R is sometimes oscillating, so I cannot state with absolute
certainty that R is going to 1. But I am nearly sure. I would be glad if
someone else could calculate.
(**) x is indeed unequal pi(x),
but nevertheless ln x ~ ln pi(x). Proof:
[ln pi(x)]/ln x ~ [ln(x/ln x)]/ln x = [(ln x  ln ln x)/ln x = 1(ln lnx)/ln
x].
(ln x)/(ln ln x) is approximately the number of primes up to ln x, which
means goes to infinity with growing x. Hence the reciprocal (ln ln x)/ln
x
goes to zero, and the above ratio goes to 1, q.e.d.
You can check that by pocket computer: lim [(ln x)/(ln pi(x))] ~ [lim(ln
x)/(ln(x/ln x))]. Just insert a few values of x, e.g. x=10, x= 100,
x=1000 and so on.
***
Anton Vrba wrote (May 09)
Regarding Werner's new conjecture Σ
(ln p)^{r}/p (p=2…x) ~ (ln x)^{r}/r  the following:
I have done some plots. The filename
indicates the r value.
Most interesting is the lower bound value of the ratio Σ (ln p)^{r}/p / (ln
x)^{r}/r as can be seen for r>>
I think for x>infinity the ratio 1 will be attained for all r>0, his so
called oscillations are getting asymptotically smaller as can be seen
from these plots.