CC BY
51
Section 4. Mathematics
Section 4. Mathematics
Drushinin Victor Vladimirovich, Lazarev Alexey Alexandrovich, Lapin Maksim Sergeevich, National Research Nuclear University "MEPHI”, Sarov physical-technical Institute E-mail: vvdr@newmail.ru
The formula for the function n(N) for the number of primes
Abstract: We analytically obtained the formula for the number of primes in the given range in the concept of two buffer zones. It goes into the empirical formula of Legendre in the limit, i. e., for the first time we gave its rationale. We’ve done the calculations and comparison with experiment.
Keywords: formula of distribution of Prime numbers, functions of Gauss and Legendre.
The question of the distribution of Prime numbers on the number line has a long history and to the present time is as follows. There are two empirical (adjustable) formulas for number of primes on the interval [2;N]:Legendre n L(N) and Gauss П G (N) which is given in all textbooks and problem books on number theory [1-3]:
nL (N) =----N-----; nG (N) = f—. (1)
LV / l^vr 1 r\Q GV / J lnx
lnN -1,08
nR (N ) = 1+ У-rV ' ^nq(n +1)
In addition, there are exact analytical formula of the Riemann:
(2)
n=1 !Ц'(П + 1) n!
where the summation is over all zeros of the Riemann function g(s). Formula (2) is almost impossible to use, as the nontrivial zeros of the Riemann function are complex numbers, hypothetically having the real part of s equal to Res = 0.5 . They are considered by numerical methods. Thus in (2) is an infinite sum.
In the article of the first author Druzhinin [4], exact formula number of primes on an arbitrary interval, which is more simpler than (2), was obtained. However, it also requires lengthy calculations.
In this article we proposed a simple approximate formula for n DL (N), obtained analytically from the theory of probability, and for large N this formula becomes n L (N). Thus for the first time we analytically substantiated Legendre formula. For very large N correctness nL (N) was proved Chebyshev [3].
Our conclusion nDL (N) based on the concept of two buffer zones on the sequence of Prime numbers. Let us number the primes in ascending order:
p = P =3- =5;- =7- = Th---;— =71;-}. The
first buffer area A(n) = [pn+1;p2n+l - 2] is segment numeric axis. If we remove all numbers are multiples of previous primes 2 < p < Pn , then only prime numbers will on
the interval A(n). For example, n = 3, A(3) = [7;23].Onthis segment prime numbers remain:
p = P =1 ь - =13 - =17 -„ =19 - =23}.
The second buffer zone B (n) =
1; Пр
is interval, in-
cluding (for n > 2) the first buffer area A(n). A characteristic feature of the second buffer zone is that if we remove all the numbers are multiples of 2 < P < Pn and these primes, the second buffer zone B (n) will remain the exact number of
numbers, as simple, compound, namely, Cn = U(Pk -1)).The
k=1
number Cn is the famous Euler function on the number of coprime integers to the set pn]. For example,
B(3) = [l;30] and the number of numbers relatively prime to {;3;5} on this segment equal to C3 = 1 • 2 • 4 = 8. This number {1;7;11;13;17;19;23;29}.
After that, we can find the probability of not remote on the interval B (n) which is the number as the ratio of the number of favourable events to all possible events:
m = -
C
П
pk-1
П
1 -1-
(3)
nipk “ pk
The calculation of such works was engaged by Legendre and Chebyshev. They proposed the following formula:
a.
a = -
lnPn
(4)
In (4) an slowly increases with n values from 0.49 to 0.561 in the limit n^ro. Our analysis of numerical calculations for a>n showed that a more adequate ratio is:
a =
в
lnpn
, ßn ^ " V Г
1 + -Ln- +
lnpn
в
lnpn
в
lnpn
ßn
lnpn - ßn
(5)
Parameter ßn changes slower than an, and in the limit of large primes becomes slightly smaller than an.
The next step is to define a new function nDL (N). Let N lies in a first buffer zone i. e. pn+1 < N < p2n+l - 2. Probabilistic
k=1
14
The formula for the function n(N) for the number of primes
value on the interval [pn+1; N] are A Nn numbers that are not removed by the sieve of Eratosthenes, and lying in a first buffer zone, i. e. they are the next primes:
ANn = 1 + —в— (N - pn+l). (6)
lnpn - pn
Since in considering there are already the first n «initial» primes then:
”+1(7)
Analyze (7). When N = pn+1 we have the correct number of primes nDL(pn+1 ) = n +1 . For large N = pN -2, N = p2n+1 - 2, N n p2, pn xtJN . Next, in (7) neglect (n +1), lnpn = lm/N = InN /2. Then (7) given that ß ~ 0.54, is converted to the form:
/,л 2ßnN N ,0,
n°L( )_ ln(N)-2ßn ~ln(N)-1.08.
In this case our formula is completely converted into Legendre formula (1), which very well describes the distribution of primes. So far we have not seen in the literature, the derivation of the Legendre formula, so it seems to us that we are first who logically substantiated it. The obtained function (7) is a recurrent formula. It allows you to consistently calculate the previous value of n with a predetermined value.
In table 1 there are comparative calculations for the three formulas. The data of (7) are not shown, since they almost coincide with the exact value of n (N). It is seen that the approximate formulas (1) and (8) give very close values.
Table 1. - N - interval of numeric axis [2; N]; n(N) is the exact value of the number of primes on the interval; nL (N)- calculate Legendre formula (1); nG (N) - calculate by the formula of Gauss (1); nDL (N) calculation
formula the authors (8)
N n(N) n, (N) П (N ) Th (N)
1000 168 171.593 176.565 151.838
2000 303 306.706 313.765 282.055
3000 430 433.127 441.715 406.617
4000 550 554.474 564.32 525.791
5000 669 672.297 683.236 646.381
6000 783 787.452 799.37 755.117
7000 900 900.476 913.286 858.92
8000 1007 1011.74 1025.37 971.36
9000 1117 1121.5 1135.9 1091.66
10000 1229 1229.96 1245.09 1200.23
11000 1335 1337.28 1353.1 1295.03
12000 1438 1443.58 1460.05 1387.89
13000 1547 1548.96 1566.06 1491.96
14000 1652 1653.51 1671.21 1605.76
15000 1754 1757.3 1775.58 1720
16000 1862 1860.39 1879.22 1834.68
17000 1960 1962.83 1982.2 1934.42
18000 2064 2064.66 2084.56 2033.38
19000 2158 2165.93 2186.33 2128.33
20000 2262 2266.68 2287.57 2225.82
21000 2360 2366.92 2388.29 2335.37
22000 2464 2466.7 2488.54 2448.85
23000 2564 2566.03 2588.33 2524.94
24000 2668 2664.95 2687.68 2633.79
25000 2762 2763.46 2786.63 2729.21
26000 2860 2861.59 2885.19 2837.88
27000 2961 2959.36 2983.37 2928.69
28000 3055 3056.78 3081.2 3018.94
29000 3153 3153.87 3178.69 3126.94
References:
1. Druzhinin V. V The Determinant Criteria for Divisibility. - Sarov, Russia, 2012. - Р. 24.
2. Sizii S. V. Lectures on number theory. - Moscow: Fizmatlit, 2007. - Р. 81.
3. Syshkevich A. K. Number theory. - Kharkov: IHGU,1956. - Р. 181.
4. Druzhinin V V., Holushkin V S. Scientific and technical Bulletin of the Volga region. -№ 2 - 2014. - Р. 14.
15
