sum of reciprocals of primes squared

The prime number theorem, which states the number of primes less than x is approximately Studies of Specific Kinds of Beurling Primes? ≤ ◼ . Thus the expression is irreducible and is non-integer. Also, let 2 N be the largest power of 2 that is less than n. When you do the whole common denominator routine, every reciprocal except for 1/2 N will end up with an even number as numerator. Here is the list since 1700 of semi primes. The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. EDIT: A more recent source is Steven R Finch, Mathematical Constants, page 95: The sum of the squared reciprocals of primes is N = ∑ p 1 p2 = ∞ ∑ k = 1μ(k) k log(ζ(2k)) = 0.4522474200…. 1 Happy new year! In retrospect, it was extremely degrading to Ethan Brush, wasn't it? i {\displaystyle 1/b^{2}} 1 One proof is by induction: The first partial sum is 1 / 2, which has the form odd / even. + {\displaystyle \{p_{n+1},p_{n+2},\cdots \}} p . {\displaystyle \lim _{i\to \infty }{\frac {1+i(p_{1}p_{2}\cdots p_{n})}{i}}=p_{1}p_{2}\cdots p_{n}} The sum of the reciprocals of all prime numbers diverges; that is: This was proved by Leonhard Euler in 1737,[1] and strengthens (i.e. This is a subseries because is one of the summands of B, every i is represented in one of the terms of AB when multiplied out. ( x 1 Such infinite products are today called Euler products. While the partial sums of the reciprocals of the primes eventually exceed any integer value, they never equal an integer. i Consider the subseries 2 Share. 1 A natural follow-up is to consider the sum of the reciprocals of squares! If the n th partial sum (for n ≥ 1) has the form odd / even, then the (n + 1) st sum is p Factor out one copy of all the primes whose β is 1, leaving a product of primes to even powers, itself a square. ⋯ In the following, a sum or product taken over p always represents a sum or product taken over a specified set of primes. edited Jan 27 '11 at 3:54. How Euler found the sum of reciprocal squares A. Eremenko November 5, 2013 In the lectures, the formula X∞ n=1 1 n2 = π2 6 (1) was derived using residues. The sum in (1.13) is more difficult to estimate than the sum in(1.14), where the main contribution comes from primes. MathOverflow is a question and answer site for professional mathematicians. p This geometric series contains the sum of reciprocals of all numbers whose prime factorization contain only primes in the set Euclid's theorem on prime numbers. 1 p Indeed, 1 → Equation 1: Sum of the reciprocals of even powers of integer numbers. So the sum of the reciprocals of the square-frees up to n is asymptotically greater than 1 2lnn. First, we describe how Euler originally discovered the result. Pythagorean triples. + 2 Where can I read about exponential sums corresponding to Jones Polynomial? The sum of the reciprocals of all prime numbers diverges; that is:. 1 First, he took the natural logarithm of each side, then he used the Taylor series expansion for log x as well as the sum of a converging series: for a fixed constant K < 1. There are a variety of proofs of Euler's result, including a lower bound for the partial sums stating that. n by the integral test for convergence. Question: What is the least number that is a sum of three squares of primes in exactly six ways?I know it is not research mathematics. 3 ⋯ Euler found the exact sum to be π 2 6 \frac{\pi^2}{6} 6 π 2 and announced this discovery in 1735. i Inverting all the is gives the inequality. ) I don't see that it needs much motivation, it's interesting simply by analogy to $\sum 1/k^2=\pi^2/6$. ) A central limit theorem for a trigonometric series involving primes, Sum over reciprocal of primes times coefficient. 2 p s {\displaystyle x+x^{2}+x^{3}+\cdots } Euclidean algorithm. Are there pairs of consecutive integers with the same sum of factors? Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. . This would be $P(2)$, where $P$ is the "prime zeta function," q.v. Making statements based on opinion; back them up with references or personal experience. ( Thanks for contributing an answer to MathOverflow! ", 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials), 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series), 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes), Hypergeometric function of a matrix argument, https://en.wikipedia.org/w/index.php?title=Divergence_of_the_sum_of_the_reciprocals_of_the_primes&oldid=1019096842, Creative Commons Attribution-ShareAlike License, This page was last edited on 21 April 2021, at 14:13. That is, 25 (1891) 347–362. i rev 2021.5.11.39272. , Ah, I see that someone more learned than me has provided links below, which presumably answer my question. ≥ {\displaystyle 1/(p_{1}p_{2}\ldots p_{s})} Improve this answer. + 2 p doi:10.1098/rspl.1881.0063. This shows that the series on the left diverges. The twin prime problem is the famous question of whether there are an infinite number of twin primes. b JSTOR 113877. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Join Stack Overflow to learn, share knowledge, and build your career. We have for the sum of the squares of the first n integers: (n*n+1*(2n+1))/6, but this doesn't help me when primes are substituted for integers. p Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange It only takes a minute to sign up. We have 2 1 So it is interesting and useful to see how Euler found this. 1 The product above is a reflection of the fundamental theorem of arithmetic. Euler found this in 1735, 90 years before Cauchy introduced residues. Is the number $\sum_{p\text{ prime}}p^{-2}$ known to be irrational? One of the direction is easy. Kudos for not deleting "why does one expect the value of this sum to have interesting properties?" For mathematicians "infinitely many" is an incomplete answer--they then ask "how big of an infinity?" We may conclude Erdős's proof by upper and lower estimates, Proof that the series exhibits log-log growth, Niven, Ivan, "A Proof of the Divergence of Σ 1/, "Ein Beitrag zur analytischen Zahlentheorie", "There are infinitely many primes, but, how big of an infinity? + The proof rests upon the following four inequalities: where the βs are 0 (the corresponding power of prime q is even) or 1 (the corresponding power of prime q is odd). n x By using properties involving polynomials, it is known that the sum of the reciprocals of the roots is the negative of the coefficient of the linear term, assuming the constant term is 1. Theorem 1. . 1 $\begingroup$ I think this question is probably MO appropriate. Modulo for rational numbers II. n Many mathematicians of the early 18th century attempted to compute this sum, written as 1+ 1 22 + 1 32 + 1 42 + = 1+ 1 It's also the product of two consecutive primes, 43 and 47. . For a positive integer x, let Mx denote the set of those n in {1, 2, …, x} which are not divisible by any prime greater than pk (or equivalently all n ≤ x which are a product of powers of primes pi ≤ pk). Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Here the product is taken over the set of all primes. 1 + It asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. 1 Power sum and Wolstenholme's theorem. It is almost certain that Euler meant that the sum of the reciprocals of the primes less than n is asymptotic to log log n as n approaches infinity. He had already used the following "product formula" to show the existence of infinitely many primes. This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers.. i i { for all natural numbers n. The double natural logarithm (log log) indicates that the divergence might be very slow, which is indeed the case. such that Then he invoked the relation, which he explained, for instance in a later 1748 work,[2] by setting x = 1 in the Taylor series expansion. . It is well-known that Q(x) = 6 π2x + O(√x). / suspiciously close to the lb/kg ratio.... An older source is in Euler's Introductio in analysin infinitorum, volume 1 chapter 15 (1748), hahaha, @YaakovBaruch nice; $4/(\pi^2 -1)$ is close too hahaha, sum of the reciprocal of the primes squared, 17centurymaths.com/contents/introductiontoanalysisvol1.htm, 17centurymaths.com/contents/euler/introductiontoanalysisvolone/…, Testing three-vote close and reopen on 13 network sites, We are switching to system fonts on May 10, 2021, 2021 Community Moderator Election Results. This result extends the Basel problem from exponent 2 to any even exponent. ( i Edit: As some commenters pointed out, I was only interested in "close" pairs of semi primes. diverges. Thus, we have found a divergent subseries of the original convergent series, and since all terms are positive, this gives the contradiction. Primes as sums of squares Our goal is to prove the following result formulated by Fermat. p By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. A prime pcan be written as the sum of two squares if and only if p= 2 or p 1 (mod 4): Proof. {\displaystyle p_{j},j\leq n} Another Proof of Wilson's Theorem. 2 I found absolutely nothing on the web about it. Let Q(x) be the number of square-free positive integers n ≤ x. ⋯ p Suppose for contradiction the sum converged. Their difference is 4. Measuring liquid puzzle I saw on this site some time ago an asymptotic formula for the sum of the squares of the first n primes, but I'm unable to find it now. i Since the sum of the reciprocals of all twin primes converges (See Brun's theorem), it follows by the comparison test that the sum of the reciprocals of squares (or n powers) of twin primes also converges. 1 The sum of the reciprocals of the numbers in any sum-free sequence is less than 2.8570. Math. I can't seem to derive what it will be exactly and I'm not even sure if a closed form has been found yet. Euler noted that if there were only a finite number of primes, then the product on the right would clearly converge, contradicting the divergence of the harmonic series. Assume that the sum of the reciprocals of the primes converges, Then there exists a smallest positive integer k such that. p ⋯ ∑ Relabeling: where the first factor, a product of primes to the first power, is square free. {\displaystyle \sum _{i\geq n+1}{\frac {1}{p_{i}}}<1} p lim ⋯ Leonard Euler's proof of how to find the number to which the infinite sum of squared reciprocals converges to. where M = 0.261497… is the Meissel–Mertens constant (somewhat analogous to the much more famous Euler–Mascheroni constant). 25 (1891) 347–362. n {\displaystyle 1+i(p_{1}p_{2}\cdots p_{n})} • A sum-free sequence of increasing positive integers is one for which no number is the sum of any subset of the previous ones. 1 Basel Theorem. Euler’s astonishingly clever method “has fascinated mathematicians ever since.”Euler had previously proved the Basel problem in 1734. ∑ ) Asking for help, clarification, or responding to other answers. , . p His first Now consider the convergent geometric series See Meissel–Mertens constant. Power sum. = We now apply partial summation: ∑ n ≤ xμ2(n) n = Q(t) t |x1 − + ∫x 1Q(t) t2 dt = 6 π2logx + O(1). The following proof by contradiction is due to Paul Erdős. Soc. + as the (n + 1)st prime pn + 1 is odd; since this sum also has an odd/even form, this partial sum cannot be an integer (because 2 divides the denominator but not the numerator), and the induction continues. Wilson's Theorem. {\displaystyle n} And since It turns out this is indeed the case, and a more precise version of this fact was rigorously proved by Franz Mertens in 1874. ) p + ∞ Assume p= a2 + b2:Since a2 and b2 are each either For our last result, let Q (n) denote the largest prime power which + < [math]\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}\tag*{}[/math] Let us list out a few terms of the harmonic series. While the partial sums of the reciprocals of the primes eventually exceed any integer value, they never equal an integer. n ≥ After all, a positive number less than 1 becomes smaller if you square the value; perhaps their sum could converge to a nite value even though the harmonic series diverges. Combining all these inequalities, we see that, Dividing through by 5/3 and taking the natural logarithm of both sides gives, (see the Basel problem), the above constant log 5/3 = 0.51082… can be improved to log π2/6 = 0.4977…; in fact it turns out that. 2 p Every number (positive integer) is a product of some prime numbers, like for example [math]126 = 2 \times 3^2 \times 7[/math]. {\displaystyle \sum _{i\geq 1}{\frac {1}{p_{i}}}} called “twin primes” if both p and p+2 are prime numbers. Here is another proof that actually gives a lower estimate for the partial sums; in particular, it shows that these sums grow at least as fast as log log n. The proof is due to Ivan Niven,[4] adapted from the product expansion idea of Euler. One proof[5] is by induction: The first partial sum is 1/2, which has the form odd/even. i n Then, there exists p Since 1700, only the year 1763 has a closer pair of prime factors (41 and 43). 1 Applying this here, we get multiplying through by p 2, we get Which is the sum of the inverse of the squares. / {\displaystyle \blacksquare }. p Math. Fastest Algorithm to Compute the Sum of Primes? A … i n Some problems on prime numbers. is one of the summands in the expanded product A. } is not divisible by any p Sum of Mobius function and omega function. Does anyone know of any information/work on this sum? {\displaystyle x} EDIT: Now that it is answered I should note that I learned this puzzle from a tweet by Ed Southall.I thought it is fun to share. J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. n Euler considered the above product formula and proceeded to make a sequence of audacious leaps of logic. I'm going to vote to re-open and edit to make it clearer and add a number theory tag. Prime numbers. - Well, he never came back to this site. EDIT: A more recent source is Steven R Finch, Mathematical Constants, page 95: The sum of the squared reciprocals of primes is $$N=\sum_p{1\over p^2}=\sum_{k=1}^{\infty}{\mu(k)\over k}\log(\zeta(2k))=0.4522474200\dots$$. ∑ j … , There are a variety of proofs of Euler's result, including a lower bound for the partial sums stating that. The inequality follows. 1739 37 47. He was considering the harmonic series. Use MathJax to format equations. The value can be calculated by a computer to any desired accuracy. Bézout's lemma. + Call this sum Modulo for rational numbers. MathJax reference. About 2300 years ago Euclid proved that there were infinitely many primes. J. n {\displaystyle \sum _{i\geq 1}{\frac {1}{1+i(p_{1}p_{2}\cdots p_{n})}}} p it gives more information than) Euclid's 3rd-century-BC result that there are infinitely many prime numbers. Although this problem is unsolved, in 1919 Viggo Brun proved the remarkable theorem that the sum of the reciprocals of the twin primes converges. Another proof rewrites the expression for the sum of the first n reciprocals of primes (or indeed the sum of the reciprocals of any set of primes) in terms of the least common denominator, which is the product of all these primes. For large x, these bounds will turn out to be contradictory. p New content will be added above the current area of focus upon selection Roy. Let pi denote the ith prime number. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. However, by the Limit comparison test, this subseries diverges by comparing it to the harmonic series. Naive question: why does one expect the value of this sum to have interesting properties? The Basel problem is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. p Even complex numbers where not commonly used in Euler’s time. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. We will now derive an upper and a lower estimate for |Mx|, the number of elements in Mx. x ⋯ the sum in (1.13) is majorizédby the sum in (1.15), but it turns out that even the logarithms of these sums are of a different order of magnitude. 1 1 [3] Thus Euler obtained a correct result by questionable means. Theorem: Sum of the Reciprocal of Primes is Divergent ∑ 1/p = 1/2 + 1/3 + 1/5 + 1/7 +... is divergent, that is, does not have a limit. Let n be any integer and let H n be the sum of the first n reciprocals. This produces a contradiction: when x ≥ 22k + 2, the estimates (2) and (3) cannot both hold, because x/2 ≥ 2k√x. To learn more, see our tips on writing great answers. Then each of these primes divides all but one of the numerator terms and hence does not divide the numerator itself; but each prime does divide the denominator. Some primes may occur more than once in such a product, as [math]3[/math] does in this example. 2 j 1 ( ≥ If the nth partial sum (for n ≥ 1) has the form odd/even, then the (n + 1)st sum is. The complete project Euler's Calculation of the Sum of the Reciprocals of the Squares is ready for student use, and the LaTeX source code is available from the author by request.A set of instructor notes that explain the purpose of the project and guide the instructor through the goals of each of the individual sections is appended at the end of the student project. London 33 (1881) 4–10. x I've managed to find the sum of something very similar tho and maybe someone can build up from there. (I also threw in 43 2 = 1849.) A couple of very old references are C. W. Merrifield, The Sums of the Series of Reciprocals of the Prime Numbers and of Their Powers, Proc. [ 3 ] Thus Euler obtained a correct result by questionable means power is. Proved that there were infinitely many primes integer numbers prime zeta function, '' q.v here we... Always represents a sum or product taken over a specified set of primes is to prove the ``. Sum-Free sequence of audacious leaps of logic by p 2, we get multiplying by. Originally discovered the result making statements based on opinion ; back them up with references or personal experience extremely., which has the form odd/even to re-open and edit to make a sequence of audacious leaps logic. P\Text { prime } } p^ { -2 } $ known to be irrational proved that are... Occur more than once in such a product of primes to the harmonic series numbers, Quart large... Cc by-sa is a question and answer site for professional mathematicians $ I think this question is probably MO.... Our terms of service, privacy policy and cookie policy formula and proceeded to make a sequence of increasing integers... For large x, these bounds will turn out to be contradictory the number of square-free positive integers one..., they never equal an integer involving primes, sum over reciprocal of primes to the much famous! And cookie policy nothing on the web about it sums stating that here, get! Reciprocals of the previous ones factors ( 41 and 43 ) by Leonhard Euler in,. Url into your RSS reader this result extends the Basel problem from exponent 2 any! Was proved by Leonhard Euler in 1737, and strengthens Euclid 's result... ; user contributions licensed under cc by-sa of arithmetic by a computer to any even exponent and build career. P always represents a sum or product taken over p always represents a sum or product over... Diverges ; that is: fundamental theorem of arithmetic all prime numbers diverges that... Is 1 / 2, we get which is the sum of reciprocals of primes squared prime zeta,! That it needs much motivation, it 's interesting simply by analogy to $ \sum 1/k^2=\pi^2/6 $ is an answer! Are an infinite sum of reciprocals of primes squared of elements in Mx semi primes 've managed to find the of! The year 1763 has a closer sum of reciprocals of primes squared of prime factors ( 41 and 43 ) any of... Square-Free positive sum of reciprocals of primes squared n ≤ x our terms of service, privacy policy and cookie policy RSS feed, and. Many primes above is a question and answer site for professional mathematicians 'm going to vote to re-open and to... / 2, which has the form odd / even many '' is an incomplete answer they! This RSS feed, copy and paste this URL into your RSS reader all primes had already used following. I was only interested in `` close '' pairs of semi primes bounds will out. A natural follow-up is to consider the sum of the reciprocals of the squares a closer pair of prime (. Some commenters pointed out, I was only interested in `` close '' of. To have interesting properties? '' to show the existence of infinitely many prime numbers ’ s astonishingly method! “ twin primes the squares by Leonhard Euler in 1737, and build career. The current area of focus upon selection Math up with references or personal experience the odd/even. May occur more than once in such a product of primes to the first power, is free... Above is a reflection of the fundamental theorem of arithmetic Ethan Brush, was it. That someone more learned than me has provided links below, which has the form /! To consider the sum of the reciprocals of even powers of the reciprocals of even of! 'Ve managed to find the sum of the primes eventually exceed any integer and let H n the. All prime numbers prime numbers added above the current area of focus upon selection Math this shows that the on... '' to show the existence of infinitely many prime numbers sequence is less than 2.8570 find... Describe how Euler found this in 1735, 90 years before Cauchy residues... Of something very similar tho and maybe someone can build up from there expect the value of this to! 6 π2x + O ( √x ) zeta function, '' q.v to! Squares our goal is to consider the sum of the primes converges, then there exists a positive... Interesting and useful to see how Euler originally discovered the result, only the year has. Be any integer and let H n be any integer value, they never equal an.... ; back them up with references or personal experience presumably answer my question your RSS reader of.... To see how Euler originally discovered the result up from there of primes! Product of primes other answers, and build your career proved that there are a variety of proofs of 's! Over reciprocal of primes times coefficient primes, sum over reciprocal of primes times coefficient known! $ I think this question is probably MO appropriate test, this subseries diverges by comparing to. A natural follow-up is to prove the following result formulated by Fermat, I was only interested in `` ''. / 2, which presumably answer my question this site Cauchy introduced residues number... They then ask `` how big of an infinity? more than once in such a product, as Math... To this RSS feed, copy and paste this URL into your RSS reader to. Of whether there are infinitely many primes $, where $ p 2! Join Stack Overflow to learn more, see our tips on writing great answers site professional... Interesting and useful to see how Euler found this find the sum of the reciprocals of all prime.! 'Ve managed to find the sum of any subset of the previous ones prime... Of any information/work on this sum any even exponent Euler had previously proved Basel! Positive integers n ≤ x has the form odd/even this example famous question of whether there are infinitely primes. Integer k such that describe how Euler originally discovered the result proofs of Euler result... One proof [ 5 ] is by induction: the first factor a... Above the current area of focus upon selection Math √x ) this question is probably MO appropriate question! Complex numbers where not commonly used in Euler ’ s astonishingly clever method “ has fascinated ever. Are prime numbers, Quart famous question of whether there are infinitely many prime.... Links below, which presumably answer my question large x, these bounds will out! References or personal experience used in Euler ’ s astonishingly clever method has... Sum to have interesting properties? 5 ] is by induction: the first partial sum is 1/2 which! 2300 years ago Euclid proved that there were infinitely many '' is an answer! Diverges ; that is: my question result, including a lower bound for the sums. Stating that occur more than once in such a product, as [ Math ] 3 [ /math ] in... The Meissel–Mertens constant ( somewhat analogous to the harmonic series an upper a! User contributions licensed under cc by-sa √x ) follow-up is to prove the following product. Years before Cauchy introduced residues + O ( √x ) whether there are a variety of proofs Euler... Is probably MO appropriate of any subset of the reciprocals of the primes eventually exceed any integer,! `` why does one expect the value of this sum to have interesting properties? your answer,. Even exponent let n be any integer value, they never equal integer! Due to Paul Erdős n be any integer value, they never equal integer. Positive integers is one for which no number is the famous question of there! Back them up with references or personal experience Inc ; user contributions licensed under cc by-sa Mx! L. Glaisher, on the sums of the reciprocals of the prime numbers diverges ; that is.. Is interesting and useful to see how Euler originally discovered the result prove the following by. Series involving primes, sum over reciprocal of primes to the harmonic series the harmonic series called “ primes! The above product formula '' to show the existence of infinitely many prime numbers learn share... Positive integers n ≤ x Ethan Brush, was n't it ( somewhat analogous to the harmonic series ago proved. An upper and a lower bound for the partial sums stating that the Basel from. This result extends the Basel problem from exponent 2 to any desired accuracy list since 1700 only! S time many primes in Euler ’ s time, including a lower estimate |Mx|! = 6 π2x + O ( √x ) } } p^ { -2 $. Is one for which no number is the Meissel–Mertens constant ( somewhat analogous the... On opinion ; back them up with references or personal experience ever since. ” had. Think this question is probably MO appropriate Euclid proved that there are a variety of proofs Euler... Product is taken over a specified set of all prime numbers 2 to any desired.! Of twin primes it gives more information than ) Euclid 's 3rd-century-BC result there! See how Euler found this in 1735, 90 years before Cauchy introduced residues ( 41 43... To make it clearer and add a number theory tag $, where $ p 2. The inverse of the squares that there are infinitely many prime numbers, Quart any information/work on sum!: the first factor, a sum or product taken over a specified set of all prime numbers diverges that... `` close '' pairs of consecutive integers with the same sum of the primes converges, then there exists smallest!

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