# What is an intuitive explanation for the congruence class

## Intuitive reasoning and reference points for assuming the correctness of the Landau problems

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1 Intuitive reasoning and pointers for assuming the correctness of the Landau problems Stadler Alexander 8CN Wirtschaftskundliches Realgymnasium Salzburg Josef-Preis-Allee 5, 5020 Salzburg Supervisor: Mag. Wolfgang Schnessl February 17, 2016

2 Abstract This thesis provides an intuitive access to the prime number theorem and the four Landau problems. For this purpose, a method is explained on the basis of the prime number theorem, which firstly makes the proposition plausible through rough, intuitive arguments, secondly through derivations with a few intuitive arguments and thirdly through detailed examination of these intuitive approaches. This method is then transferred to Goldbach's Hypothesis and outlined for the other Landau problems. The estimation formulas obtained in this way are then compared with empirical data for small number ranges. In addition, additional intuitive arguments are occasionally introduced. The method is a form of Selberg's sieve and is based on a combination of the inclusion-exclusion principle and Möbius inversion, which provides an estimate for certain relevant functions. These derivations are neither consistently strict mathematical inferences nor approaches for solid evidence. They are intended to provide insight into the structure of the problems. 2

3 Foreword A mathematical conjecture is usually created by playing around with or trying things out on the part of a mathematician who discovers a regularity and asks himself whether it can be generalized. For example, the mathematician Christian Goldbach was happy to invent new conjectures, for the solution of which he then consulted his friend Euler. It was the same with Goldbach's Hypothesis, which Goldbach formulated in a letter to Euler in 1742 in a slightly different form than that by which we understand Goldbach's Hypothesis today. To this day, this assumption has not been proven. 1 However, in all attempts to solve this problem, the question arises as to why mathematicians are so convinced that this conjecture must be correct. I would like to answer this question in this thesis, with the most varied of means and through several approaches, which are intended to reveal themselves in particular to the intuitive mathematical understanding, and thus to make the correctness of the assumptions plausible, because I am convinced that the potential of intuitive mathematics should by no means be underestimated. So this work should not only serve as an explanation of the plausibility of the assumptions, but also give an insight into the structure of the mathematics that is behind it. This internal structure is astonishingly similar in the case of the four problems, since they are all problems of additive number theory. This means that they examine objects of multiplicative number theory, namely the prime numbers, for their additive properties. For example in relation to their sum (Goldbach's conjecture), difference (prime number twin conjecture) or how and where they occur (prime number theorem, Legendre's conjecture, n² + 1). 2 I would like to thank Professor Schnessl very much for the numerous support in the writing process and the help with math problems. 1 Cf. Stewart, Ian: Professor Stewart's mathematical hodgepodge. Hamburg: Rowohlt Taschenbuch Verlag p. 192f. 2 Cf. Halupczok, Karin: Introduction to Additive Number Theory. Albert Ludwig University of Freiburg: 2007/08. S. 3. As download: (Accessed) 3

4 Table of contents 1 Introduction Prime number theorem Two prime number theorems Inclusion-Exclusion-Principle Möbius-Inversion Goldbach's Hypothesis The Assumption Calculation with Probabilities Intuitive Reasoning Comparison with the Empirical Data The other Landau-Problems The Prime Twin-Assumption The Legendresche-Assumption The n² + 1-Assumption Problems & Discussion Prime number theorem Goldbach's assumption Conclusion List of figures List of tables List of references

5 1 Introduction At the International Congress of Mathematicians in 1912, Edmund Landau presented the four, in his opinion, most important and at the time not open to attack problems of number theory, namely the Goldbach conjecture, the prime number twin conjecture, the Legendre conjecture and the n² + 1- conjecture together, which is why these are known today as Landau problems. 3 All four were unproven at the time and still are today. However, there are already various results that come very close to them and promise an early solution to one of the problems. However, the question arises as to why it should be assumed in the first place that these assumptions are even true, which is what this work is about. This work is mainly based on the author's personal derivations, which are a variant of the interpretation of Selberg's sieving method. 4 However, basic theorems and terms of number theory are used, most of which come from the lecture notes for the lecture Introduction to Number Theory by Peter Hellekalek. For certain more detailed sentences, a textbook summarized and cited in this script and written by Tom Apostol on analytical number theory was used. In particular, the first chapters on elementary number theory are used here, for example for the Möbius inversion. The aim of this work is to create an intuitive access to the Landau problems and it is an attempt to answer the research question how the Landau problems can be explained intuitively and whether these considerations agree with the empirically recorded data. First of all, it should be noted that only few references are made to actual research in this area, as this is usually not very understandable and very technical. However, it must not be overlooked that the mere progress made in the direction of the problems suggests their correctness. 3 Cf. Pintz, Jànos: Landau s Problems on Primes. Budapest. S. 1. Download: (Accessed) 4 Cf. Halupczok, Karin: Introduction to Additive Number Theory. Albert Ludwig University of Freiburg: 2007/08. P. 60ff. As a download: (Accessed) 5

6 A similar method should be applied to all four problems, which also forms the main part of the thesis. The tools required for this, namely the inclusion-exclusion principle and Möbius inversion, are first explained in the second chapter using the example of the prime number theorem. Then, as an example for all Landau problems, the procedure is applied to Goldbach's conjecture in the third chapter. Then, in the fourth chapter, an analogous approach is offered for the other problems, but the derivation is not carried out explicitly. Instead, for some problems, another intuitive argument is made that this method does not use. The fifth chapter deals with some of the intuitive arguments in more detail and also presents their problems. The derivations that are presented in this work are not to be understood as conclusive formulas, but are characterized by distinctive intuitive arguments, most of which are not scientifically proven which appear plausible or are explained by heuristic arguments and empirical data. Arguments of this kind are always clearly emphasized in order to highlight the weak points and the difference to an actual proof, but also to point out that every derivation is mathematically logical and meaningful except for these gaps. In order to clearly emphasize this distinction, a notation is used in this work that is of great importance in this context. The symbol x y is intended to represent an approximation of the two sides of the equation made plausible by an intuitive argument and is only used if such an argument was used in the development of the formula (the term estimate is used in this context). Equations that are actually valid are not marked as such, but the sign "=" is used consistently here. Finally, the relation f ~ g between functions f and g is used, which clarifies neither equality nor approximation, but the mathematically precise statement that two functions behave asymptotically the same. This means that the following applies: f (x) lim x g (x) = 1 However, this relation between functions is used in the work to justify the intuitive formula f g. 6th

7 2 Prime number theorem In this chapter two prime number theorems are presented, whereby the intuitive derivation method is developed for the first prime number theorem, which is used in the later chapters. For this purpose, the inclusion-exclusion principle and the Möbius inversion, as well as some necessary terms and results, are listed. 2.1 Two sets of prime numbers In order to be able to continue to use intuitively justified expressions in a meaningful way, two important results of number theory are listed here. The prime number theorem, empirically discovered and assumed by Gauss and Legendre on the basis of tables of prime numbers and logarithms, was independently proven in 1896 by Hadamard and de la Vallèe-Poussin using analytical number theory. 5 Its formulation reads: Theorem: (Prime number theorem) 6 Let π (x) # {p x p prim} denote the number of prime numbers less than or equal to x, called the prime number counting function. Then: π (x) lim x x logx This gives an asymptotic description of the growth of the number of prime numbers under a given bound, which is suitable as a rough approximation. One often writes π (x) ~ x logx grow for the above statement. In particular, the approximation π (x) should serve formulas. = 1 and means that the functions are at the same speed x logx for evaluating those obtained later. In 1949, Erdös and Littlewood proved the prime number theorem with elementary means, which, however, turns out to be much more difficult than an analytical proof. 7 In fact it was for 5 Cf. Apostol. Tom M .: Introduction to Analytic Number Theory. Pasadena: Springer p. 8f. 6 Cf. Apostol: Introduction to Analytic Number Theory. P. 8f. 7th

8 the analytical proof of the prime number theorem is sufficient to show that the analytical continuation of the Riemann zeta function ζ, defined by ζ (s) 1 ns for all s C, with Re (s)> 1, which exists on all of C except in 1, does not have any Has zeros on the line s = 1 + ib with b R. This is a consequence of the results of Riemann himself on this subject. 8 An equivalent formulation of the prime number theorem is also given here. The following definition of the logarithm integral is necessary: Definition: 9 n = 1 x Li (x) dt logt Because of the relationship Li (x) ~ x logx, the prime number theorem is equivalent to the statement: Li (x) ~ π (x) 2 Another prime number theorem will be necessary for an intuitive justification of Goldbach's conjecture. This makes a statement about the number of prime numbers in an arithmetic progression, i.e. a sequence of the form (a + bn) n = 1. A necessary condition for prime numbers to occur in such a sequence is that ggt (a, b) = 1, because otherwise every term in the sequence is divisible by the number ggt (a, b)> 1 and is therefore not a prime number. 7 Cf. Apostol: Introduction to Analytic Number Theory. P. 8f. 8 Cf. Edwards, H.M .: Riemann s Zeta Function. Mineola, New York: Dover Publications, Inc pp. 68ff. 9 Cf. Hellekalek, Peter: Introduction to number theory. Paris Lodron University Salzburg, p. 27f. (Accessed) 8

9 However, this requires the definition of a function that will be used later. At the same time, two results are noted for this function: Definition: (Euler's φ-function) 10 The Euler's φ-function φ: NN is defined by φ (m) # {km: ggt (k, m)} and give the number of m are coprime numbers m. The first values of the function are: n φ (n) Table 1 theorem: (Euler-Fermat formula and theorem) 11 Let n = k α pii = 1 i be the prime factorization of a natural number n and a prime to n. Then we have : k 1) φ (n) = n (1 1 i = 1) pi 2) a φ (n) 1 mod n 3) φ is multiplicative, i.e. for coprime a and b the following applies: φ (ab) = φ ( a) φ (b) In particular, this theorem for prime numbers p and a, a does not say a multiple of p, from (1) and 2) for n = p): ap 1 1 mod p With this function, the theorem proved by Dirichlet be formulated: 10 Cf. Hellekalek: Introduction to number theory. P. 40ff. 11 Cf. Hellekalek: Introduction to number theory. P. 40ff. 9

10 Theorem: (Prime numbers in arithmetic progressions) 12 Let π (x; a, b) # {pxp prime and n with p = a + bn} denote the number of prime numbers less than or equal to x in the arithmetic progression (a + bn) n = 1. Let ggt (a, b) = 1. Then: π (x; a, b) lim = 1 x π (x) φ (b) Again the alternative formulation reads π (x; a, b) ~ π ( x). Intuitively formulated, this theorem means that the prime numbers are distributed as equally as possible to all coprime residual classes of b. In particular, this implies that there are infinitely many prime numbers in each of these remainder classes. For later intuitive arguments the rough approximation π (x; a, b) π (x) is noted. For the prime number theorem in particular, an intuitive justification is given in the next chapter, while the theorem about prime numbers in arithmetic progressions suggests a natural order of the prime numbers. A very unequal distribution among the various remainder classes would mean a gross irregularity in the distribution of the prime numbers. φ (b) φ (b) 2.2 Inclusion-Exclusion Principle For the following intuitive analysis of Goldbach's Hypothesis, this procedure will be of essential importance. A general formulation is given here as a basis for using the inclusion-exclusion principle. Theorem: (Inclusion-Exclusion-Principle) 13 Let A 1, A 2 ,, A n be finite sets. Then n n A k = (1) i + 1 α i k = 1 i = 1 12 Cf. Hellekalek: Introduction to number theory. S Cf. Wolf, Reinhard: Diskrete Mathematik (lecture script by Stadler Alexander from the VO of the same name). Paris Lodron University of Salzburg: 2013/14, Chapter I, Sentence I.7. 10

11 where: α i A j MP i ({1, .., n}) j M The operator P i (.) Is defined as follows: P i (A) {MAM = i} P i assigns every set A the set of i-element subsets of A. The principle of inclusion and exclusion is illustrated by the following two examples. 1) Venn diagram: 14 The pictorial representation of quantities is known as the Venn diagram. Given are three sets D, E, F with: D = 13, E = 8, F = 7 DE = 2, EF = 3, DF = 6 DEF = 2 With the help of the inclusion-exclusion formula, the number of elements in the amount DEF are calculated DEF = α 1 α 2 + α 3 where the following applies: α 1 = D + E + F = = 28 α 2 = DE + EF + DF = = 11 α 3 = DEF = 2 From this it follows: DEF = α 1 α 2 + α 3 = = See Wolf: Discrete Mathematics. Chapter D. 11

12 The formula can be viewed as a step-by-step addition of the individual numbers. First the elements of the three sets are added. However, since exactly those elements that are in two of the three quantities were counted twice, they must be subtracted again. In the end, this again subtracts the elements that are contained in all three quantities, so they have to be added up again. 2) Prime number theorem: Now, with the inclusion-exclusion principle, an intuitive explanation for the above-mentioned prime number theorem is to be found. If you want to get the prime numbers from 1 to 100, you start with a list of all integers from 1 to Figure 1. Now you cross out all the numbers in the list that are divisible by 2 (in red). The first number that remains is 3. So we continue the process for 3 by deleting all multiples of 3 (in blue). The numbers that are divisible by 2 as well as by 3, i.e. by 6, are crossed out twice (in purple). So you always get the next largest prime number than the first number that was not deleted. Once you have reached the multiples of 7, any deletion from then on becomes superfluous, since every composite number 100 has at least 12

13 has a prime divisor 7. So one finally gets all prime numbers between 1 and 100. This method is called Sieve of Eratosthenes. 15 This allows the number of prime numbers from x to x to be determined, in this example for x = 100. Namely with the help of the inclusion-exclusion principle. You start with the 90 possible prime numbers from 100 = 10 to 100. Then the multiples of 2 and 3 are subtracted. But because the multiples of 6 were deducted twice, these are added again once. Do the same with the multiples of 5. However, the multiples of 15 are now deducted twice, so they are added once again. But now the common multiples of 6 and 15, namely the multiples of 30, have been added twice, so they have to be subtracted again. This procedure continues for all further prime numbers up to the root of the desired number x, in this example up to 10. Now the prime numbers that are smaller than x have also been subtracted, which is why only the value π (x) π ( x) can be determined.That means: π (100) π (10) + 1 = 100 {n divides n} {n divides n} {n divides n} {n divides n} + {n divides n} + {n divides n} + {n divides n} + {n divides n} + {n divides n} + {n divides n} {n divides n} {n divides n} {n divides n} {n divides n} + {n divides n} On the left Page 1 must be added because 1 cannot be divided by any prime number and is therefore not filtered out, but it is not a prime number. The number of elements of these sets {n 100 d divides n}, which contain nothing other than the multiples of d less than or equal to n, is given by: And this results in the formula: {n 100 d divides n} = 100 d π ( 100) π (10) + 1 = (1) i 4 i = 0 r 1, r 2,, ri prime 10 with rirj for ij {n 100 r 1 r 2 ri divides n} = 15 Cf. Hellekalek: Introduction to number theory. S.

14 4 = (1) ii = 0 r 1, r 2,, ri prime 10 with rirj for ij 100 r 1 r 2 ri In order to further simplify the term obtained, a further procedure is necessary, which is described in more detail in the following chapter is discussed. 2.3 Möbius inversion The Möbius inversion is used in the intuitive justification of Landau problems to further process the terms resulting from the inclusion-exclusion formula, which is suggested by their natural structure. Again the method will be described in general terms. The following definition: Definition: 1) A function f: NC is called a number-theoretic function 16 2) The multiplicative Möbius μ-function is defined by: 17 k μ: N {1,0,1} Let n = α pii = 1 i the prime factorization of n, then: μ (n) 0 if i: α i 2 μ (n) (1) k if i: α i <2 (n square-free) 3) For two number theoretic functions f and g let the binary one Operation explained by: (fg) (n) f (d) g (nd) d divides n The expression is called the Dirichlet convolution of f and g. 18 4) The sum function S f of a number-theoretic function f is defined by: Cf. Apostol: Introduction to Analytic Number Theory. S Cf. Apostol: Introduction to Analytic Number Theory. S Cf. Apostol: Introduction to Analytic Number Theory. S Cf. Hellekalek: Introduction to number theory. S.

15 S f (n) f (d) d divides n The first values of the Möbius μ-function are: n μ (n) Table 2 As an example of a sum function, the Euler φ function defined above is given: 20 S φ (n ) = φ (d) = nd divides n Under Möbius inversion the following relationship is understood: Theorem: (Möbius inversion) 21 Let f be a number-theoretic function and S f the associated sum function. Then the following applies: f = S f μ The converse also applies, i.e. if the following applies to a number-theoretic function g: f = g μ Thus, g is the sum function S f. This formula is a consequence of the identity S μ (n ) = [1 n] This sentence should now be used to further transform the expression from 1.2. The following applies: π (100) π (10) = (1) i 4 i = 0 r 1, r 2,, r i prim 10 with r i r j for i j This expression should now be written more compactly. Define: 100 r 1 r 2 r i 20 Cf. Apostol: Introduction to Analytic Number Theory. S Cf. Hellekalek: Introduction to number theory. P. 38f. 15th

16 p x p p prim x Thus, in the above summation range, every selection of prime numbers r 1, r 2 ,, r i corresponds to a unique divisor d of p 100, which is square-free (because none of the prime numbers may appear twice). In addition, for square-free divisors d of p 100, the preceding factor (1) k agrees with μ (d). This results in: π (100) π (10) + 1 = μ (d) d divides p d How this formula can be obtained should now be clear from the example x = 100. Analogously, one obtains the general formula: π (x) π (x) + 1 = μ (d) x d d divides p x What now follows is an intuitive argument that is intended to greatly simplify this formula. The above expressions, which are used for totaling, are getting smaller and smaller in terms of amount. In addition, the signs change constantly, which is why one can assume that roughly the following applies: π (x) π (x) + 1 = μ (d) xd μ (d) xdd divides px = xpx μ (d) divides pxd pxdd divides px = x μ (d) dd divides px In fact, the error in this estimate may be rather large, which is discussed in Chapter 5. In this summation, d and p x each represent the complementary divisor of the number p x, so this is the Dirichlet convolution μ id, where the function id: N R is the identity with id (n) = n. The identity is the sum function of Euler's φ-function, which was already noted earlier. With the help of Möbius inversion we get φ = μ id. Accordingly, the above sum is simplified as follows: π (x) π (x) + 1 x p μ (d) p x = x x d p φ (p x) x d divides p x d 16

17 By evaluating φ (px) using the formula for Euler's φ function, taking into account the freedom from squares of px, we get: π (x) π (x) + 1 x (p 1) = xpxp divides p xp divides p x (p 1) p = x (1 1 p) p prim px With this formula, knowing the prime numbers up to x, the number of prime numbers under x can also be calculated. However, the above product can still be roughly simplified. Namely: (1 1 p) = p prim p x p prim p x 1 = 1 (1 1 p) p prim p x 1 1 n = 0 p n Where the formula for geometric series x n = 1 n = 0 was used for x <1. The multiplication in the denominator now results in: 1 1 n n 1 x A summation over the range {k N all prime divisors p of k x} should be indicated by 1 n. This is an infinite, but apparently convergent, sum, the value of which is now roughly estimated by x 1 n = 1 n. This is intuitively justified with the assumption that the summation values of n, which are not contained in both sums, become relatively small and are also roughly the same size. This sum can now be estimated using the corresponding integral: xx 1 n 1 x dx = logx log1 = logx n = 1 And thus the estimate for the product is obtained: 1 (1 1 p) p prim px = logx nn But how justified above, the relationship applies: 17

18 π (x) π (x) + 1 x (1 1 p) p prim p x x logx This is now the result of this intuitive calculation. In order to actually derive the prime number theorem, one can argue as follows: If the symbol, with which only an intuitive comparison is meant here, were replaced by a strict ~, this would result in: π (x) lim xx logx π (x) π ( x) + 1 π (x) 1 = lim xx + lim xx logx logx π (x) 1 = 1 + lim xx logx However, trivially, π (x) 1 x and thus: 0 lim x π (x) 1 x logx applies x lim xx logx logx = lim xx = 0 And that would result in: π (x) lim xx logx That is the statement of the prime number theorem. However, a more precise statement was required in the intuitive argumentation at the point mentioned above in order to be able to calculate with limit values = 1, but the above derivation suggests this. An even more intuitive version of this derivation reads: About half of all numbers from x to x are divisible by 2, so the number of numbers from x to x that cannot be divided by 2 is: x (1 1 2) In addition, about a third is that remaining numbers divisible by 3, so the number of numbers not divisible by 2 or 3 from x to x: x (1 1 2) (1 1 3) 18

19 If you do this for all prime numbers x, you get the estimate: π (x) π (x) + 1 x (1 1 p) p prim p x Again the prime numbers x have also been subtracted. This explanation uses very rough arguments, the credibility of which this chapter proves with the inclusion-exclusion principle and Möbius inversion. 19th

20 3 Goldbach's Hypothesis This chapter deals with the first of the Landau problems. On the one hand, an intuitive consideration with probabilities is made, on the other hand, the procedure from the previous chapter is carried over. Finally, the estimation formulas obtained in this way are compared with empirical data. 3.1 The conjecture First and foremost, an intuitive justification for the correctness of Goldbach's conjecture is to be found among the Landau problems. In a letter to Euler, Goldbach made the following two conjectures: 22 1) Every even number 4 can be represented as the sum of two prime numbers 2) Every odd number 7 can be represented as the sum of three prime numbers The first conjecture is also known as Goldbach's binary conjecture, the second as Goldbach's ternary conjecture. It should be noted that the first claim implies the second, since any odd number n 7 can be represented as 3 + 2m, where 2m is even and 4, so 2m could be represented as the sum of two prime numbers p and q. Thus n could be represented as 3 + p + q. Today, however, Goldbach's conjecture is mainly understood to mean the following statement: For all even numbers 2m 4 there is at least one pair of prime numbers (p, q) with: 2m = p + q So far the conjecture could not be proven, but Vinogradov proved the in 1937 ternary conjecture for all sufficiently large odd numbers 23. For small numbers the conjecture has been confirmed empirically. 22 Cf. Halupczok, Karin: Introduction to Additive Number Theory. Albert Ludwig University of Freiburg: 2007/08. P. 28f. As download: (Accessed) 23 Cf. Halupczok: Introduction to Additive Number Theory. S.

21 The proofs of these results employ powerful methods of analytic number theory. The first result in particular uses the circle method, which is discussed in more detail in the fifth chapter. To have a handy form for the number of pairs of prime numbers that confirm the conjecture, define: Definition: Let the function G: NN be defined by G (n) # {pnp prim and 2n p prim} and give the number of pairs of prime numbers whose sum is equal to 2n. Goldbach's conjecture then reads: G (n)> 0 for all n calculations with probabilities A first method for estimating the above function is provided by the prime number theorem, which can also be understood in such a way that for an n the prime numbers 2n with a relative frequency of 1 log (2n) occur. 2n can be written in n ways as the sum of two numbers m + k, m n and n k 2n. The probability that both of these numbers are prime is given by the prime number theorem as follows: 1 log (2n) 2 Summed up over all these n decompositions, this results in an estimate of: G (n) n log (2n) ² 21

22 This derivation is carried out in a script on additive number theory. 24 However, this can be improved if it is taken into account that the probability of being prime for an mn is about 1 logn, and for a k with nk 2n: 2n log (2n) Which gives the estimate: nn log (n) = 1 logn (log2 1) logn G (n) n log (n) 2 (log2 1) logn It should be noted that the term in brackets for n approaches 1, and the term outside of it is asymptotically the same as the first estimate behaves. Thus, in general, the expression: G (n) n =: S (n) log (n) 2 This is an estimate for large n, but the second formula should provide a more precise approximation in lower ranges. This argument uses the prime number theorem, so it does not give much insight into the problem. Nevertheless, the estimate obtained here will be examined more closely later. However, a more intuitive derivation leads to a similar formula. 24 Cf. Halupczok: Introduction to Additive Number Theory. S.

23 3.3 Intuitive justification For an intuitive justification of Goldbach's conjecture, an analogous procedure as in 2.2 and 2.3 for the prime number theorem leads to the goal. A slightly modified form of Selberg's sieve is used for this purpose. 25 Let 2m be the number to be tested. We are looking for prime numbers p and q with 0 q m and m p 2m with p + q = 2m. Such pairs can be found by choosing a prime number p with m p 2m and testing whether the number 2m p is prime. We have 0 2m p m, therefore 2m p can only be a prime number if it is not divisible by a prime number r m. So we consider the set P = {p prim m p 2m and r m prim does not divide r 2m p}. It remains to be noted that all prime divisors p of the number 2m cannot be considered for such a pair, since the number 2m p cannot be a prime number because it is divisible by p. This must be taken into account in the inclusion-exclusion formula. The case that m = p is an exception to this, because then 2m p = m = p and thus prime, but it can generally be neglected if a downward estimate of the number of pairs is desired. In addition, those cases are omitted for which 2m = p + q and 0 qm and mp 2m apply, because here 2m p is divisible by a prime number m, namely by 2m p = q itself, which means that this p is not contained in P. . These cases should be neglected for the following consideration, as this visibly simplifies the formula. Accordingly, the empirical data must also be collected later. Obviously, however, the following applies: G (m) P = 1 Now this sum should be calculated in a different way. Namely, the prime numbers r m are summed, the prime divisors of the 2m being omitted. For each of these prime numbers the prime numbers p are counted with m p 2m, for which the following applies: r does not divide 2m p, which again yields P. Formally this means: p P 25 Cf. Halupczok: Introduction to Additive Number Theory. P. 60ff. 23

24 P = {p prim: mp 2m with r does not divide 2m p} r prim mr does not divide 2m This means that the number of prime numbers that do not meet Goldbach's condition is given by: {p prim: mp 2m} \ P = {p prim: mp 2m} \ {p prim: mp 2m with r does not divide 2m p} r prim mr does not divide 2m = With the calculation rules for the difference of quantities this can be transformed to: r prim mr does not divide 2m {p prim: mp 2m} \ {p prim: mp 2m with r does not divide 2m p} = r prim mr does not divide 2m {p prim: mp 2m with r divides 2m p} And that means with the inclusion-exclusion principle: { p prim: mp 2m} \ P = π (2m) π (m) P = π (m) ω (2m) k (1) k + 1 {p prim: mp 2m with ri divides 2m p} k = 1 r 1, r 2,, rk prime m with rirj for ijri does not divide 2m i = 1 Where ω (n) # {p divides np prim} gives the number of disjoint prime divisors of n. Here again nothing else happens than that all prime numbers p are counted with m p 2m, for which 2m p is divisible by a prime number r m, which is not a divisor of 2m. But if 2m p is now divisible by r i and r j, then that p would be counted twice, so that p has to be subtracted again. The same goes for three, four and five, up to 24

25 to π (m) ω (2m) prime numbers, which is the number of all prime numbers r m without the prime divisors of 2m. Furthermore, the mean k {p prime: mp 2m with ri divides 2m p} i = 1 means nothing else than the set of all prime numbers p with mp 2m, for which applies: r 1 divides 2m p 2m p mod r 1 divides r 2 2m p 2m p mod r 2 rk divides 2m p 2m p mod rk All these congruences together are equivalent to 2m p mod r 1 r 2 rk since r 1 to rk are pairwise different prime numbers. That means: k {p prim: mp 2m with ri divides 2m p} i = 1 = {p prim: mp 2m with 2m p mod r 1 rk} And therefore: k {p prim: mp 2m divides 2m p} with ri = i = 1 = {p prim: mp 2m with 2m p mod r 1 rk} = = π (2m; 2m, r 1 r 2 rk) π (m; 2m, r 1 r 2 rk) The product pn becomes like is defined as follows: 25

26 pnpp prim np does not divide 2n This means that every product r 1 r 2 rk of prime numbers represents a divisor d 1 of the number pn. This simplifies the entire above expression to: π (2m) π (m) P = μ (d) (π (2m; 2m, d) π (m; 2m, d)) What can be transformed to: d divides pmd 1 P = μ (d) (π (2m; 2m, d) π (m; 2m, d )) d divides pm In section 1.1 it was noted that for large values of x, π (x; a, b) is given by π (x) φ (b) if ggt (a, b) = 1. The Approximation is very good on average, because the signs alternate because of the factor μ (d) and in particular the small terms are shortened at the end, whereas the actually decisive terms for small values of d are well matched by the above approximation, which is the value of P hardly adulterated. So one writes: P μ (d) (π (2m) φ (d) π (m) μ (d)) = (π (2m) π (m)) φ (d) φ (d) d divides pmd divides pm Again using the formula for the φ function, the last sum can be written as the product of the prime numbers r 1 to r π (m), which denote the first π (m) prime numbers, without the divisors of 2m: d divides pm μ (d) φ (d) π (m) = (1 1 rk 1) = rk 2 rk 1 k = 2 rk does not divide m π (m) k = 2 rk does not divide m 26

27 At this point it should be noted that this product already gives a good estimate of the problem that can also be applied in practice, but a closed form in m would be more desirable. In order to gain such, however, strong estimates are made, which reject the sharpness of this estimate. Chapter 3.4 provides a comparison. As with the prime number theorem, this product should now be represented differently: First, an extremely rough estimate is used. Since each factor in the product is <1, adding more such factors will make the product smaller. So: π (m) rk 2 rk 2 H (m) =: rk 1 rk 1 π (2m) π (m) k = 2 rk does not divide m π (m) This product is also discussed in Chapter 3.4 with the empirical data compared. The following applies to each of the factors with k 3: Because: rk (1 multiply by (rkrk 1) results in: k = 2 rk 2 rk 1 rk 1 rk 1 rkrk 1) rk (1 1 2) rk (rkrk 1 1) rkrk 1 Adding rkrk 1 and subtracting rk on both sides yields: rk ² 2r krkrk 1 rk 1 And finally with division by rk (rk 1) you get: rk 2 rk 1 rk 1 rk If you now estimate every factor from k in the above product = 3 in this way, we get: 27

28 π (m) rk 2 1 rk 1 2 rk 1 = 1 rk 2 3 k = 2 π (m) k = 3 r π (m) 3 2 m Because very rough estimates were made here, such as rkrk 1 2 for two consecutive prime numbers, it can be assumed that the estimate is falsified so much downwards that it is considered a bound. This means that the following applies: 3 G (m) P (π (2m) π (m)) 2 m The truth of this assertion is checked in Section 3.4 for small values of m. Using the prime number theorem, the inequality for large m is: 2m P (log2 + logm And consequently: m logm) 3 2 m = G (m) m logm 3 2 m (2 (log2 1) logm) m 2logm 3 m 2logm In particular, the correctness of this inequality would not only imply Goldbach's conjecture, but also a stronger conjecture, namely that the function G is unbounded. In addition, it also gives a measure of the growth in the number of the conjecture-confirming prime number pairs, which is relatively large. The correctness of this prognosis for the growth of the G-function in small number ranges will also be dealt with in the following chapter. It should only be noted here that this statement can be brought into line with the estimate from Chapter 3.2, because: And the inequality is true because for large n the following applies: m log (m) ² G (m) 3 m 2logm m log ( m) ² 3 m 2 logm 28

29 m 3 log (m) Comparison with the empirical data A computer-assisted study by Oliveira e Silva in April 2012 showed that Goldbach's hypothesis is correct for all n. 26 Although this fact says nothing about the general validity of the conjecture, it reinforces the intuitive conviction of its correctness. In this context, a graphic with reference to the previous two chapters will now be considered. This graph compares empirical values with two estimators derived above. Arguments for this were tested up to 2000 H P S Figure 2 26 Cf. Oliveira e Silva, Thomàs: Goldbach Conjecture Verification (accessed) 29

30 The red graph shows the value of P, which was used in the above chapters as the lower limit of G (n). The green graph describes the function S (n), which was introduced in Chapter 3.2 as: n S (n) = log (n) 2 The blue graph shows the values of the product from Chapter 3.3, which is denoted by H (n) were: π (n) H (n) = (π (2n) π (n)) rk 2 rk 1 The strong oscillation of the values of P is clearly visible here. Nevertheless, these describe a curve that is exceeded by the other two functions only for small arguments. For larger arguments in this graph, the two estimates are lower bounds. In particular, they also run below the individual outliers that do not follow the upward trend. This is to be expected for the estimation function H insofar as the product is estimated downwards in Chapter 3.3 because the factors that contain the prime divisors of n are included. Nevertheless, an asymptotic equality to the red graph is required of this function and not that it represents a lower bound, as this graphic suggests. The function S was also derived in such a way that it offers an estimate for the empirical values and not a limit. However, it runs further up than the function H and touches the graph of the empirical values several times in the range between 1000 and 1500. At about 700 it also crosses these, so it is not a lower bound. For this function, too, a downward estimate was made in Section 3.2. However, both functions are to be understood as lower estimates of the empirical values in such a way that the above derivations make it clear why they correspond to the empirical values. However, due to some downward estimates, they are generally below the empirical values. k = 2 30

31 Another graphic should now specifically show the lower limit given above for the function G: Estimation G 50 0 Figure 3 This graphic compares the empirical values of the function G with the values of the estimation from the chapter above: G (m) 3 m 2logm Here it can be clearly seen that the estimate forms a lower limit in this number range. However, fuzzy estimates were also made for these. However, in contrast to the two above, this function is actually intended to represent a lower bound. Such an estimate makes it clear that the values of the function G are apparently not only greater than zero, which the Goldbach hypothesis requires, but also follow a certain order. 31

32 4 The other Landau problems 32

33 Now that the principle with which an intuitive explanation for the conjectures is given has now been explained using the prime number theorem and Goldbach's conjecture, it will be shown in a sketchy manner how this can be transferred to the other three Landau problems. The resulting formulas are also compared with the empirical data. In addition, other intuitive arguments are given. 4.1 The Prime Twin Conjecture This conjecture reads: 27 There are infinitely many prime numbers p for which p + 2 is also a prime number An equivalent formulation reads: There are infinitely many pairs of prime numbers of the form (p, p + 2) These pairs are called prime twins . Again a counting function is introduced and noted with π 2: 28 π 2 (x) # {(p, p 2): px, p and p 2 prime} With this the prime twin conjecture reads: The function π 2 is unbounded, that means that for all C> 0 there is an x with π 2 (x)> C. An upward estimate is known for this function 29, namely there is a constant K such that x π 2 (x) K ( logx) ² Another result in connection with this assumption is the fact that the sum converges over the reciprocal values of the twin prime numbers 30. This is the sum: 27 Cf. Hellekalek: Introduction to number theory. S Cf. Hellekalek: Introduction to number theory. S Cf. Hellekalek: Introduction to number theory. S Cf. Hellekalek: Introduction to number theory. S.

34 (1 p + 1 p + 2) (p, p + 2) P² The divergence of this sum would already prove the conjecture. Since this is not the case, both options remain open, but intuitively speaking this result says that the number of prime twins decreases relatively quickly in the density of their occurrence, or that there are only a finite number of them. This fact makes a computer-assisted search for prime twins computationally intensive, since they occur less and less frequently. Now the method of the last chapter is to be adapted to this assumption. Under a given bound x, all numbers n that cannot be the greater of a prime twin should be sorted out. This is the case if and only if n or n 2 is divisible by at least one prime number p x, i.e. two numbers are sifted out instead of one. As with the prime number theorem and Goldbach's conjecture, however, the cases for which n is x are omitted here. Again, these are neglected. In addition, even numbers must also be filtered out by the factor 1, since n is even 2 if and only if n is also 2. This means that two numbers are filtered out for prime numbers and four numbers for composite numbers with two prime factors, since every combination of the remainders must be taken into account. In general, for factors with k prime factors, 2 k numbers have to be filtered out. The individual steps of the inclusion-exclusion principle now proceed analogously to the above calculations, one finally obtains the formula for the prime twins between x and x: π 2 (x) π 2 (x) 1 2 d divides pxd odd μ (d) 2 ω (n) xd This last sum can again be represented by Möbius inversion as a product, namely: x 1 2 (1 2 p) =: Z (x) pxp 2 34

35 Again, this opens up a much more intuitive approach, since every factor of the form (1 2 p) can be understood as the sifting out of those numbers n for which either n or n 2 is divisible by p. It should now be shown that this function is unlimited. For n 2 the following applies: p n + 1 pn + 2 And therefore the following applies: 1 2 p n + 1 = p n + 1 2 p n + 1 pnp n + 1 From this the estimate for the product follows: Z (x) = x 1 2 (1 2 p) x 6 pn = xpnpxp 2 π (x) 1 n = 2 p π (x) x 2 x = 1 2 x Why Z is unbounded, because 1 x for x. 2 So the estimate π 2 (x) π 2 (x) Z (x) implies that the function π 2 would also have to be unbounded, which is the statement of the conjecture. A table of empirical values should clarify the usefulness of this estimate in the lower number ranges. x π 2 (x) π 2 (x) Z (x) π 2 (x) π 2 (x) Z (x), 6 1,,,,,,,,,

36,, Table 3 The column on the far right gives the relationship between the empirical value and the estimate. The former fluctuates rather strongly and levels off between and apparently at 1 for values of x, but then tends to decrease again. However, this decline slows down to the point of. Nevertheless, this table suggests that the estimated value Z (x) for large x is too large, which can be explained by the fact that the specification of the values screened out produces an irregularity. However, due to the rapidly slowed down decrease up to and including the convergence of the ratio towards a value of 0.94, such small number ranges are extremely unsuitable for more general assumptions. A renewed increase in the values is also possible. If they nevertheless converge, this value is essential for determining the quality of the above estimate and in particular the question of whether, if the sequence of these values converges, the limit value is positive or zero. In the latter case, the estimate is completely useless. 4.2 The Legendre's Conjecture The Legendre's conjecture is: 31 For all n N the following applies: There is a prime number between n² and (n + 1) ² Expressed with the prime number counting function this means: For all n N the following applies: π ((n + 1) 2 ) π (n 2)> 0 A similar and already proven conjecture is Bertrand's postulate, which reads: 32 For all n 2 the following applies: 31 Cf. Weisstein, Eric W .: Legendre's Conjecture. MathWorld-A Wolfram Web Resource. (Accessed) 32 Cf. Aigner, Martin / Ziegler, Günter M .: The Book of Evidence. Berlin Heidelberg: Springer-Verlag S.

37 There is a prime number between n and 2n Or with the prime number counting function: For all n N the following applies: π (2n) π (n)> 0 In fact, Bertrand's postulate is intuitively even stronger, because it says that for every nn successive There are numbers that contain a prime number. On the other hand, the correctness of Legendre's conjecture would only say that for every n 2n + 1 there are consecutive numbers that contain a prime number. That seems like a weaker statement. However, both Bertrand's postulate and Legendre's conjecture also make a statement about the exact location of these consecutive numbers. And it is well known that prime numbers decrease in the density of their occurrence, because according to the prime number theorem, the proportion of prime numbers among all numbers up to a limit x is approximately: π (x) xx logx x = 1 logx That tends for x towards 0. That but n² is so much larger than n, it makes it less likely that among the 2n + 1 numbers n², n² + 1 ,, n² + 2n + 1 is a prime number, because these numbers are larger than the n numbers n, n + 1 ,, n + n and because the prime number density is lower there. A first intuitive argument, which does not justify the strict statement as credible, but suggests the correctness of the same on the whole, deals with the sums of the reciprocal of two consequences. The first sequence is that of the square numbers: 1 n² n = 1 This sum is convergent because of the following argument: 33 Consider the partial sums and estimate them as follows: 33 Cf. Revers, Michael: Analysis II (lecture script by Stadler Alexander von der VO WS 2014/15 of the same name). 37

38 m 1 n² = 1 + (1 n (n 1) n 1 1 n) n = 1 mn = 2 The sum on the left is a so-called telescope sum, which is simplified as follows: m (n 1 1 n) = n 1 1 n = nn = m = 2 1 mn = 2 mn = 2 The following applies to the original series: mn = 2 mn = 2 m 1 n = 1 0 = lim 0 lim 1 mm n² lim 2 1 mmmn = 1 what proves the convergence of the series and also shows that the value of the series is between 0 mn = 2 = 2 and 2. Euler determined this value to be exactly π² 6 in 1734.34 The second sequence is that of the prime numbers: p prim This series is divergent according to Euler. 35 In a book about the mathematician Jacques Hadamard this fact is interpreted as follows: 1 p The divergence of the sum of the reciprocals of the primes shows that the primes are distributed quite densely among the natural numbers, for instance more densely than the squares of natural numbers, for which the sum of reciprocals gives a convergent series. 36 The fact that the series of the reciprocal of the square numbers converges and that of the prime numbers diverges suggests that the prime numbers are closer than the square numbers. With an analogous argument it could also be asserted that a more general formulation of the Legendre's conjecture applies, namely: 34 Cf. Aigner / Ziegler: The Book of Proofs. S Cf. Aigner / Ziegler: The Book of Evidence. P. 5f. 36 Maz ja, Vladimir G. / Shaposhnikova T. O .: Jacques Hadamard: A Universial Mathematician. American Mathematical Soc S

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