A Rigorous Proof of the Riemann Hypothesis from

A quick reading guide for my proof of the Riemann Hypothesis Ref: A Rigorous Proof of the Riemann Hypothesis from First Principles by Kumar Eswaran [ResearchGate Preprint January 2018] This note is for those who are looking for a reading guide that links the key steps in the proof together in a logical manner. It is strongly suggested that the Abstract and the Extended Abstract (which reveals the plan) of the above paper, should be first read before perusal of this Guide. For any serious reader, it is critical to understand the concept of “Towers” in the paper, which leads up to Theorem 3 that essentially proves that an equal number of integers have odd and even numbers of prime factors (multiplicity included) over the entire number system. This can be interpreted as implying that any randomly chosen integer will have an equal probability of having an even or odd number of prime factors. A simpler and more intuitive proof of Theorem 3, but perhaps not as rich in latent potentialities as the one in the formal paper, is given in the write-up in page 5 (below) which is adapted from lecture notes for my undergraduate students. You can start by reading it (instead of sections 2,3,4 in the formal paper). The outline of the proof is as follows: 1. My paper proves the equivalence statement given on pg 6 of Borwein et al (2006): The Riemann Hypothesis is equivalent to the statement that for every fixed  > 0 limN →∞

λ(1) + λ(2) + .... + λ(N ) 1

N 2 +

=0

(1)

where the λ’s are the the Liouville integer function defined for all positive integers, n = 1, 2, 3, 4, ...., as λ(n) ≡ (−1)Ω(n) , where Ω(n) is the number of prime factors, multiplicity included, in the integer n. Integers with an odd number of prime factors will have λ = −1, and those with an even number of prime PN factors will have λ=1. We also use the summatory Liouville function L(N ) ≡ n=1 λ(n). 2. That the statement (1) is actually equivalent to the Riemann Hypothesis has been proved in Section 5 of my paper, following Littlewood (1912). The proof of (1), and so the RH itself, is Theorem 4 in my paper. 3. Equation (1) is equivalent to 1

limN →∞ |L(N )| = CN 2 +dN

(2)

because the symmetry of the Riemann zeros around s = 1/2 does not allow the exponent on the right to go below 1/2. The deviation from the 1/2 power-law, dN , should be vanishingly small as N → ∞, for the equivalence of (1) and (2).

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4. Now consider a standard random walk, whose distance travelled in N steps can be represented as: N X X(N ) = xi (3) i=1

where the xi ’s are independent random numbers with an equal probability of being either +1 or -1, i.e., essentially “coin-tosses”, with H = 1 and T = −1. It is a well-known result that the expected value of the absolute value of X(N ), for large N , is limN →∞ E(|X(N )|) = C0 N 1/2 (4) The further line of advance of my proof is to show now that Equation (4) applies to L(N ) as well, and so proves Equation (1), and thereby the RH. To do this I have to prove that the L(N ) series is essentially a “random walk”, i.e., the λ(n)’s are essentially “coin-tosses”. 5. To show that the λ’s behave as coin tosses, we have to show that (i) their probabilities of being either +1 of -1 are equal, as was proved by Theorem 3. Further (ii) we have to show that the λ’s appearing in the natural sequence, n = 1, 2, 3, ..., are independent of each other — i.e., that the value of λ(n) has no influence on the value of λ(n + 1), say. This seems counter-intuitive, as the λ’s are obviously deterministically linked1 . Nevertheless, their independence in the natural sequence is shown by two different approaches: (a) In Appendix III, it is proved that the sequence of λ(n), n = 1, 2, 3, ... is noncyclic. This would preclude any dependence of the type λn = f (λn−1 , λn−2 , λn−3 , ..., λn−M ) because any finite series of +1’s and -1’s of length M would have a finite number of permutations P , so the series λn−1 , λn−2 , λn−3 , ..., λn−M must repeat itself after at most P numbers, and thereafter become cyclic if such a dependence relationship exists between the λ’s2 . (b) In Appendix IV, another approach is taken. It is shown that from merely these two rules: λ(p) = −1 for a prime p, and λ(pq) = λ(p) × λ(q), where q is any integer, the entire sequence of λ’s for n = 1, 2, 3, ..., can be obtained without determining the number of prime factors of n. It is then argued that for any two integers n > m, λ(n) is dependent on λ(m), if the latter is required to find the former, and independent if not. It is then shown that, as n → ∞, any finite sequential strip of λ’s will be independent of each other, thus essentially making them equivalent to coin-tosses. 1

For example, if n = p × m and p is a prime, then λ(n) = −λ(m) This is because any dependence of the type indicated, involving M entities occurring in the argument of the function f (λn−1 , λn−2 , λn−3 , ..., λn−M ) cannot be always different for all values of n, there will always be some integer n0 such that all its preceding M entities λn0 −1 , λn0 −2 , λn0 −3 , ..., λn0 −M are the same as the above, thus making λ(n0 ) = λ(n) and starting a cycle. 2

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Some Clarifications: (a) Because the statement Theorem 4 is written immediately after Theorem 3 in my paper, it may mislead you into thinking that I assume that the λ’s are coin-tosses, merely because they have an equal probability of being +1 or -1. That is not true. In fact, two appendices, Appendix III and IV are devoted to showing the independence (as in independent random numbers3 ) of the λ’s. (b) Also, while the re-ordering of the number system was used to prove the equal probability of +1 and -1 of the λ’s over the entire number system, my proof henceforth focuses on showing the behavior of L(N ) in the natural sequence, n = 1, 2, 3, ..., (i.e. in increasing order) as expressed in Equation (2). 6. Having shown, as above, that the λ’s are essentially coin-tosses, I then argue (see Section 5.2, p. 13 of the main paper), that the ‘expected’ value of |L(N )| in Equation (1) above should follow the same rule as E(X(N )) in (4), and so scale as N 1/2 . However L(N ) is not a stochastic quantity, but a purely deterministic one, and so we have to deal with the actual value of L(N ), not its ‘expected’ one. This requires an examination of the deviation dN in Equation (2). However, the random walk argument still applies (with the λ’s of the natural number system being one realization of a random walk). So I use (towards the end of Section 5) Khinchin and Kolmogorov’s law4 of the iterated logarithm [which gives the maximum possible deviation of |X(N )| from the expected value E(|X(N )|)] to show dN → 0 as N → ∞ so the equivalent statement of RH, Equation (1), and the Riemann Hypothesis is proved! 7. Wait, there is more, as the TV ads say: In Appendix V, starting from Littlewood’s ansatz, that L(N ) ∼ N a , for N → ∞, I argue that the statistics of the λ’s must become “self-similar”, i.e., independent of N for large N . I partition the natural number systems into contiguous subsets each of which contain consecutive integers whose lambda values, are independent because they belong to different towers and thus should behave like coin tosses. I then argue that L(N ) is nothing but a summation over all such subsets and therefore L(N ) will collectively behave like the cumulative sum of N coin tosses thus leading to the result a = 21 . While I believe this to be a ‘physicist’s proof’, as I call it, of the RH, I would be interested to know what mathematicians think of it. 8. Finally, in Appendix VI, I confirm by considering the λ’s from n = 1 to 176 trillion, that their sequence is statistically indistinguishable (using the χ2 statistical test) from coin-tosses in each of the subsets mentioned above and also over the entire set of numbers considered5 . While this is merely a ‘verification’, not a ‘proof’, this fact has not been reported in the literature, and by itself, requires an explanation (which I have provided) given its surprising nature. 3 It worthy of mention that it is never claimed that the λ’s are actually random numbers, but only that they behave as independent random numbers, just as the fully deterministic L(N ) series behaves as a random walk. 4 I am very thankful to Dr. S.V. Ramanan, formerly of the Physics Dept.of SUNY, Stonybrook, for drawing my attention to this law of the Iterated Logarithm. 5 God seems to have “played dice” at least once, when he created the natural number system!

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I believe that I have described and clarified every crucial step leading to my proof of RH. I hope that a careful reading of this note along with the paper will convince you, my reader, of the veracity of the statements made and of the validity of the proof. And as for me, if in this task I have helped in providing a place from where you could view the eternal beauty inherent and ever present in mathematics, I will consider myself very fortunate; I cannot ask for more! P.S.: If I may, let me point out that there are many things that are truly beautiful in my paper: 1. In 1761 Johann Lambert proved that the sequence of digits that constitute the number π never repeats, because π is irrational. (Later Lindemann proved it is a transcendental number.) In Appendix III, I prove a theorem that the factorisation sequence of all natural numbers similarly, never repeats. That is, when the sequence of natural numbers: 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, ... are each factorized into primes giving the sequence 1, 2, 3, 2 × 2, 5, 2 × 3, 7, 2 × 2 × 2, 3 × 3, 2 × 5, 11, 2 × 2 × 3, 13, 2 × 7, 3 × 5, ... and when each of the numbers in the sequence is replaced by +1 if the number has even number of prime factors and is replaced by -1 if it has an odd number of prime factors giving rise to the factorization sequence: 1, −1, −1, 1, −1, 1, −1, −1, 1, 1, −1, −1, −1, 1, 1, 1, −1, −1, −1, −1, ... This sequence never repeats. The above factorization sequence of all natural numbers is represented by using the Liouville functions: λ(1), λ(2), λ(3), λ(4), λ(5), λ(6), λ(7), ... in the paper. 2. I further prove that the number of +1s is exactly equal to the number of -1s in the above factorization sequence. 3. I also show that the +1 s and -1 s behave like tosses of coins. It is as if, God chose to toss coins when he decided to create the integers!

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Not every child can understand Lambert. However, I believe that a day will come, when the above results will be known and understood by most 5-year old children, who have just begun to be enthralled by the enigma that is Mathematics, and will ever remain in their memory to remind them of the beauty of numbers. Kumar Eswaran ACKNOWLEDGMENTS: I thank my brothers Professors Mukesh Eswaran (Univ. of British Columbia, Vancouver) and Vinayak Eswaran (IIT, Hyderabad) for their many discussions and criticisms and for their careful reading of this manuscript. April 4th, 2018 6

It was Leopold Kronecker (1823-91), who first suspected the ‘hand’ of God in the creation of the integers, when he (famously) said: “God, made the integers, everything else is the work of Man!”

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Borwein’s equivalence statement (This is a gist of an informal talk that I gave to some of my undergraduate students.) Let us go back to the assertion of Borwein and his co-authors: “...the Riemann Hypothesis is equivalent to the statement that an integer has an equal probability of having an odd number or an even number of distinct prime factors”. From its context, it is clear that multiplicities7 are included, by implication, among the factors. There is still the question of probability, as mentioned in that statement. Here it means that if we randomly choose a number, any number, it will have an equal chance of having an odd or an even number of prime factors (multiplicities included). This would be true if, in the infinite set of positive integers, the number of integers with an odd number of prime factors is equal to the number of integers with an even number of prime factors. This is what I proved that night. However, it so turns out that the above verbal statement of Borwein et al, stating that the RH is “equivalent” to an equal number of integers having odd and even numbers of prime factors (proved by my Theorem 3), is only a necessary condition for the RH to be true and not a sufficient condition. There are some more conditions which need to be satisfied for RH to be true - these are dealt with in my paper. To explain the basis of the proof, let us consider this analogous situation: You visit a planet which is populated by a species whose members have either pink hair (P) or blue hair (B). You suspect that the number of Ps are equal to the number of Bs, but they are too numerous to individually count in the time you have on hand. So how do you prove your conjecture? You could do it by establishing what is known as a one-to-one correspondence between the Ps and Bs. That is, we link them in pairs, so that for each P there is precisely one B, and for each B there is precisely one P, its partner in the pair. If that paring can be done8 so that each and every P, and similarly every B, is in a single unique9 pairing, no more or less, then you have shown that the numbers of Ps and Bs are equal. The Twin Towers10 To do such a pairing, I devised a scheme for placing each positive integer in its own particular ‘tower’, as I called them, based on its prime factorization. Composite numbers with multiplicity: Let us first consider the factorization of a composite number that has some multiplicity: n = pe11 pe22 pe33 ...pekk ...peqq peLL pi pj 7

(5)

That is, 4 (=2 × 2) will be counted as having two prime factors, not one. How this could be done is not crucial; for example, it could be due to the biological process of reproduction, say. 9 Here and henceforth, the word unique will be used in a mathematical sense, to mean one and only one. 10 with apologies to JRR Tolkien! 8

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where p1 < p2 < p3 ... < pq < pL < pi < pj are prime numbers in the order of size, and the ek are the integer exponents indicating the multiplicity of the respective primes pk . That is, ek = 1 if pk appears only once in the factorization, ek = 2 if it appears twice, and so on. In the factorization (5), pL is the largest prime with multiplicity, i.e., with an exponent exceeding 1. So the primes appearing after pL will all have an exponent of 1; though only two are shown above, there may be any finite number of them, or none. For example, the integer 15400 = 2 × 2 × 2 × 5 × 5 × 7 × 11 = 23 × 52 × 7 × 11, has p1 = 2, e1 = 3, pL = 5, eL = 2, pi = 7, pj = 11. Not all the primes below pL need necessarily have multiplicity (ek > 1). In fact, there may be no primes at all below pL appearing in the factorization. The number n, as factorized in (5) above, is placed in a tower Pm;p;u with the label m; p; u that is determined as: m = pe11 pe22 pe33 ...peqq ;

p = pL ;

u = pi p j .

(6)

That is, p is pL which is the largest prime with multiplicity in the factorization (5) of n, while m is the product of all prime factors of n that are smaller than pL , and u is the product of all factors greater than pL . If there are no primes below pL in the factorization, we assign m = 1; and if there are no primes greater than pL , we assign u = 1. The towers are identified by their labels (m; p; u), which take numerical values. For example, the integer 15400 (= 2 × 2 × 2 × 5 × 5 × 7 × 11) will be in the tower labelled by (8; 5; 77). Different towers will have different labels – differing in at least one of the triplet of numbers (m; p; u) – with, say, (8; 5; 77), (4; 5; 77), (8; 3; 77) and (8; 5; 7) all representing different towers. We will henceforth use the collective name of tower structure to refer to the entire (infinite) set of towers. However, each tower Pm;p;u will itself contain an infinite number of integers Pm;p;u = {mp2 u, mp3 u, mp4 u, ...}

(7)

with its members, shown within the curly brackets { }, being defined by n = mpeL u, with eL = 2, 3, 4, 5, 6, ... on to infinity11 . The tower label (m; p; u) and ‘membership number’ eL thus define each of these members uniquely. We can also reconstruct the original factorization (5) for each member, because the tower labels m and u will also eL−1 have unique factorizations as m = pe11 pe22 pe33 ...pL−1 and u = pi pj . Further, because of the uniqueness of the factorization (5), every composite number with multiplicity will be placed in a unique tower, with unique labels. That is, no such number can be a member of two different towers, and moreover will appear only once in each tower. Conversely, each member of each tower will represent one and only one such integer. However, we have as yet considered only those integers that are composite numbers with some multiplicity in their prime factorization. What about the the other types of 11

where mp2 u is the smallest member, because the prime p (=pL ) necessarily has multiplicity, as first assumed.

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integers? These fall in two categories, namely, (i) Primes and (ii) Composite numbers without multiplicity, such as q = p1 p2 p3 ...pL−1 pL , (8) where, once again, the prime factors are written in increasing order. Note that all the primes here occur only once in the factorization, including pL , which is merely the largest prime12 in the factorization. Composite numbers without multiplicity: Let us consider the latter case (8) first. Here we note that – from our consideration of composite numbers with multiplicity – there already would exist a tower Pm;p;u , with m = p1 p2 p3 ...pL−1 , p = pL , and u = 1, which would have as its members: Pm;p;1 = {mp2 , mp3 , mp4 , ...} Into this set, we now insert q, from (8), as the first member, so that the set now becomes Pm;p;1 = {mp, mp2 , mp3 , mp4 , ...}

(9)

This accommodates even the composite numbers without multiplicity in the towers structure that was previously created. It is to be noted that the members of this set, as before, are identified as n = m × peL × u, except that here u = 1, and the first member has eL = 1, rather than eL =2, as before. Prime numbers: As primes cannot be factorized, their factorization is merely the tautology: pL = pL . However, the powers of pL , that is, p2L , p3L , p4L are composite numbers with multiplicity that already would have their own tower Pm;p;u , with m = 1, p = pL , and u = 1, that would have as its members: P1;p;1 = {p2 , p3 , p4 , ...} Into this set, we now insert p as the first member, so that the set now becomes P1;p;1 = {p, p2 , p3 , p4 , ...}

(10)

so that even prime numbers are accommodated in the towers structure. Here too, the first member has eL = 1, rather than eL =2. We have now placed all the positive integers greater13 than 1 in ‘towers’ that each have unique triplet-number labels and contain an infinite number of integers. Every such integer — whether prime or composite with, or without, multiplicity — will be placed in a unique tower, and will have a unique ‘membership number’, eL , within that tower. However, some towers will have membership numbers starting from eL = 2, while others from eL = 1. The latter can be identified from their labels, as they will either14 have m as 12

and not the largest prime with multiplicity, as before. The number 1, is excluded as its tower P1;1;1 contains only one member. 14 which will contain a composite number without multiplicity, n = m × p, as its first member.

13

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a composite number without multiplicity and u=1, or15 have both m = 1 and u = 1; the remaining towers will all start from eL = 2. With this proviso, we can state that every positive integer greater than 1 will have a unique membership in the tower structure, and every member of the tower structure will be a unique integer greater than 1. Thus a oneto-one correspondence between the integers and the members of the towers is established. Why is this important? Because in the tower structure, within each tower, the members alternate between having an odd number of prime factors and an even number. For example, consider a tower that has its first member at eL = 1, so its members are m × p × u, m × p2 × u, m × p3 × u, m × p4 × u, m × p5 × u, .., and so on. So if the first member has, say, an odd number of prime factors16 , then the second member will have one more prime-factor — as we obtain it by merely multiplying the first number by p — hence adding one more prime-factor (recall we include multiplicity), and so will have an even number of prime factors. This can be repeated for the third member, which will have a prime-factor more than the second member, and so again an odd number of prime factors. This can go on ad infinitum. It is easy to see that — irrespective of which tower we consider, or which is its starting membership, or whether its first member has an even or odd number of prime factors — we can always pair, or ‘twin’, the consecutive members in each tower in groups of two, and be guaranteed that one twin will have an odd number of prime factors, while the other has an even number. As all the positive integers greater than 1 are included in the towers, all the positive integers greater than 1 can be twinned in unique pairs with odd and even numbers of prime factors. Hence, the ‘twins’ can be defined as (2n − 1, 2n) for towers with first eL = 1 and (2n, 2n + 1) for towers with first eL = 2, where n = 1, 2, 3, .... It is easy to see that this pairing will be unique with each partner being the twin of the other.17 Thus, the number of such integers with odd and even numbers of prime-factors (including multiplicity) are equal. So, the verbal ‘equivalence statement’ of Borwein and his co-authors has been proved!

*****

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which will contain the prime number p as its first member. which can be determined by prime-factorizing n = m × p × u 17 The Paper uses as an alternative definition for the word ‘twin’ and a different mapping scheme which has the advantage of demonstrating that Class I and Class II integers (see paper for definitions) separately satisfy Borwein’s assertion. 16

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