On the real line with x > 1, the Riemann zeta function can be defined by the integral or by the sum ζ(x) = Σ_{n=1}^∞ 1/n^x. This series diverges for x ≤ 1. However, the zeta function can be analytically continued to a meromorphic function on the whole complex plane except for a simple pole at x = 1. In this extended sense, the value of ζ(−1) can be computed and is equal to −1/12.

The series 1 + 2 + 3 + 4 + ⋯ is a divergent series, meaning that its sequence of partial sums, 1, 3, 6, 10, 15, ..., tends to infinity, so that the series does not have a sum in the usual sense of the word. However, in the early 20th century, the series was studied by Srinivasa Ramanujan, who used a technique now known as Ramanujan summation to assign to the series a finite value of −1/12. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of −1/12, which is expressed by the famous formula 1 + 2 + 3 + 4 + ⋯ = −1/12, where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned methods and not as the sum of an infinite series in its usual meaning.

Zeta function regularization is a powerful tool to assign finite values to divergent sums arising in quantum field theory. However, in some cases formal manipulations with “zeta function” regularization (assuming linearity of sums) lead to correct results, while in other cases they may fail. The method relies on replacing divergent sums by values of zeta functions defined by convergent series and then analytically continued. For instance, in one dimensional problems the energy levels may give rise to sums like ∑_{k=0}^{∞} k, whose zeta-regularized value is expressed in terms of ζ(−1). In the paper the authors stress that although zeta function regularization assigns specific finite values such as ζ(−1) = −1/12 to divergent series like 1 + 2 + 3 + ⋯, these are not ordinary sums in the sense of limits of partial sums, and care is required in manipulating them.

For Re(s) > 1, the Riemann zeta function is defined as the convergent series ζ(s) = Σ_{n=1}^∞ 1/n^s. For nonpositive integers, the series does not converge, but via analytic continuation one can show that ζ(−n) = −B_{n+1}/(n+1) for n ≥ 0, where B_k are Bernoulli numbers. Another particular value is ζ(−1) = −1/12. This gives a pretext for assigning a finite value to the divergent series 1 + 2 + 3 + 4 + ⋯ in certain summation methods such as Ramanujan summation.

In general, for negative integers one has ζ(−n) = −B_{n+1}/(n+1), where B_k denotes the k-th Bernoulli number. The first few values for negative odd integers are ζ(−1) = −1/12, ζ(−3) = 1/120, ζ(−5) = −1/252, …. Thus the analytic continuation of the Riemann zeta function assigns the value −1/12 to ζ(−1).

No, of course the natural numbers can't be summed. 1 + 2 + 3 + … has no sum; or we might just as well say that it sums to infinity. By convention we then say that the sum tends to infinity; although you can't find infinity on a number line, mathematicians do supplement the number system with a "number" called infinity, and so it can be said that 1 + 2 + 3 + … equals infinity. Just remember the bottom line: neither mathematics nor physics says that 1 + 2 + 3 + … equals anything other than infinity, in the usual sense of summing an infinite series.

First of all, the infinite sum of all the natural number is not equal to -1/12. For example, for n=1000 you get S_n = 500,500, and for n = 100,000 you get S_n = 5,000,050,000. This is why mathematicians say that the sum 1+2+3+4+… diverges to infinity. Or, to put it more loosely, that the sum is equal to infinity. So where does the -1/12 come from? … He had been working on what is called the Euler zeta function… This analytic continuation gives S(-1) = 1+2+3+4+… = ζ(-1) = -1/12. This is one way of making sense of Ramanujan's mysterious expression, but it should not be interpreted as the usual sum of the series.

Ramanujan summation is a technique invented by the Indian mathematician Srinivasa Ramanujan to assign finite values to divergent infinite series. For a function f, the classical Ramanujan sum of the series ∑_{k=1}^{∞} f(k) is defined as the constant term that arises in the Euler–Maclaurin formula for the partial sums of the series. The Ramanujan sum of 1 + 2 + 3 + 4 + ⋯ is also −1/12. However, this value is not a sum in the classical sense, but a regularized value associated to the series using Ramanujan's method.

Zeta function regularization is a technique by which divergent sums and products are assigned finite values through analytic continuation of zeta functions. Given a sequence of positive numbers {λ_n}, one considers the spectral zeta function ζ(s) = ∑_{n} λ_n^{−s}, which converges for Re(s) large enough and can often be analytically continued to other values. In the simple case of the positive integers 1, 2, 3, ⋯, the associated zeta function is the Riemann zeta function ζ(s) = ∑_{n=1}^{∞} n^{−s}. Its analytic continuation satisfies ζ(−1) = −1/12. In the context of zeta function regularization, this is interpreted as assigning the divergent series 1 + 2 + 3 + ⋯ the regularized value −1/12. The document emphasizes that zeta-regularized sums are not convergent sums in the usual sense but rather values of analytically continued functions, and this distinction is important when using such regularized quantities in physics and mathematics.

Using analytic continuation of the Riemann zeta function beyond Re(s) > 1, one can assign finite values to certain divergent series. In particular, one obtains: 1 + 1 + 1 + ⋯ = ζ(0) = −1/2 and 1 + 2 + 3 + ⋯ = ζ(−1) = −1/12. Here the equalities are to be understood in the sense of the analytically continued zeta function (or Ramanujan summation), not as ordinary sums of series.

Zeta function regularization is a powerful tool to assign finite values to divergent sums and products appearing in quantum field theory. A classical example is the series 1+2+3+4+… which is divergent in the usual sense. Using the analytic continuation of the Riemann zeta function ζ(s), one finds ζ(−1) = −1/12, and in zeta function regularization this finite value is associated with the divergent series. Such regularized sums appear, for instance, in the computation of vacuum energies and Casimir forces.

The Riemann zeta function is initially defined for Re(s) > 1 by the series Σ_{n=1}^∞ 1/n^s but can be analytically continued to a meromorphic function on the entire complex plane except s = 1. For the negative integers this analytic continuation can be expressed in terms of Bernoulli numbers via ζ(−n) = −B_{n+1}/(n+1). Using B_2 = 1/6 we obtain ζ(−1) = −1/12 as the analytically continued value at s = −1, even though the defining series Σ_{n=1}^∞ n diverges.

Zeta function regularization is a technique that assigns finite values to otherwise divergent sums and products via analytic continuation of zeta functions. For example, one defines the zeta-regularized sum of the positive integers by considering the Dirichlet series ζ(s) = ∑_{n=1}^{∞} n^{−s}, which converges for Re(s) > 1 and defines the Riemann zeta function. The functional equation for the Riemann zeta function extends ζ(s) to a meromorphic function on the whole complex plane. Evaluating this analytic continuation at s = −1 gives ζ(−1) = −1/12, which in the context of zeta function regularization is interpreted as the regularized value of the divergent series 1 + 2 + 3 + 4 + ⋯. Thus, in zeta function regularization the expression 1 + 2 + 3 + 4 + ⋯ is not a convergent sum in the usual sense, but is assigned the finite value −1/12 via the analytic continuation of ζ(s) to s = −1.

The Riemann zeta function ζ(s) is defined for Re(s) > 1 by the Dirichlet series ζ(s) = Σ_{n=1}^∞ 1/n^s, which diverges for Re(s) ≤ 1. Using analytic continuation and the functional equation one can extend ζ(s) to a meromorphic function on ℂ with only a simple pole at s = 1. For negative integers s = −n this gives explicit values in terms of Bernoulli numbers, namely ζ(−n) = −B_{n+1}/(n+1). For example, ζ(−1) = −1/12 and ζ(−3) = 1/120.

There is a real sense in which adding up all of the natural numbers (numbers 1, 2, 3…) really does give you minus twelve, despite all the reasons this should be impossible. However, there is also a real sense in which it does not, and cannot, do any such thing. In string theory, one way to compute the required dimensions of space and time ends up giving you an infinite sum, a sum that goes 1+2+3+4+5+…. In context, this result is obviously wrong, so we regularize it. In particular, we say that what we’re really calculating is the Riemann Zeta Function, which we happen to be evaluating at −1. Then we replace 1+2+3+4+5+… with −1/12. So can you really add up all the natural numbers and get −1/12? No. But if a calculation tells you to add up all the natural numbers, and it’s obvious that the result can’t be infinite, then it may secretly be asking you to calculate the Riemann Zeta Function at −1. And that, as we know from complex analysis, is indeed −1/12.

In the video the presenter explains that the Riemann zeta function ζ(s) is first defined by a convergent series for Re(s) > 1, but then extended to other values of s by analytic continuation. By using the functional equation and Bernoulli numbers, he notes that 'we can assign a value to zeta of zero; this is minus one half' and then that the same method gives the values 'at all the negative integers'. For s = −1 this analytic continuation yields ζ(−1) = −1/12, even though the original series 1 + 2 + 3 + ⋯ diverges.

Here we will see 1+2+3+… "will be" -1/12 if you use the Ramanujan summation, which is a way to assign a value to a divergent series. … just do a little multiplications you end up with negative 1 over 12, aha just like that. So in other words … I will actually write down 1 plus 2 plus 3 plus 4 plus dot dot dot, put R on the top, and then you have negative 1 over 12. This right here once again it's just a way to assign values to divergent series.

It is not a convergent series at all: the sum does not become close to -1/12 no matter how many terms you add together. "Ramanujan summation" is a way of assigning a finite value to certain divergent series; for the particular series 1+2+3+4+…, that value is −1/12. This does not mean that the usual sum of the series is −1/12; in the ordinary sense, the series diverges to +∞.

The zeta regularization method can be used to obtain a finite definition for otherwise divergent integrals and series, apparently violating the usual summation properties. The method exploits properties of the Riemann and Hurwitz zeta functions to evaluate sums of powers of integers. For integer m one uses relations involving Bernoulli polynomials and zeta values to define sums over k^m, and the usual definition of zeta regularization gives finite results such as ζ(−1) = −1/12 for the sum over k, i.e. for 1 + 2 + 3 + ⋯. These finite values arise from analytic continuation and are applied in the computation of physical quantities like the Casimir energy, even though the original series 1 + 2 + 3 + ⋯ diverges in the standard sense.

In mathematical physics and number theory, the statement '1 + 2 + 3 + 4 + ⋯ = −1/12' is typically shorthand for the fact that the analytic continuation of the Riemann zeta function satisfies ζ(−1) = −1/12 and that, in zeta-function regularization, this value is associated to the divergent series Σ_{n=1}^∞ n. Physicists use this assignment in contexts such as string theory and Casimir energy, but in standard real analysis the series Σ_{n=1}^∞ n diverges and does not have a finite sum.

As we know that Sir Ramanujan gave the solution of sum of all natural numbers up to infinity and said that the sum of all natural numbers till infinity is -1/12. In this paper, we study Ramanujan summation method and prove that the sum of natural numbers 1+2+3+… to infinity is -1/12 under this method. It is emphasized that this is according to Ramanujan’s summation technique, not in the usual sense of the limit of partial sums.

In this lecture the presenter introduces the Riemann zeta function ζ(s) = ∑_{n=1}^{∞} 1/n^s and explains that it converges for Re(s) > 1 but can be analytically continued to other values of s. They then use this continuation to define regularized sums and products of positive integers. Around the discussion of divergent series, the speaker notes that zeta function methods can be used to "add up" all the integers in a regularized sense: by evaluating ζ(s) at s = −1 one obtains ζ(−1) = −1/12, which is used in physics as the regularized value of the sum 1 + 2 + 3 + ⋯. The video emphasizes that this is a regularized value derived from analytic continuation, not an ordinary convergent sum of the series 1 + 2 + 3 + ⋯.

A commenter explains that the series 1 + 1/2^s + 1/3^s + ⋯ defines the Riemann zeta function only when the real part of s exceeds 1. The Riemann zeta function ζ(s) extends this series to the rest of the complex plane (except s = 1) by analytic continuation. This analytic continuation yields the value ζ(−1) = −1/12. The post stresses that this does not mean the ordinary sum 1 + 2 + 3 + 4 + ⋯ equals −1/12 in the usual sense, but that −1/12 is the value of the analytically continued zeta function at s = −1.

The sum of all natural numbers is equal to -1/12. This blog post is not just about mathematical trickery either. The equation in the title is actually a very important result used in theoretical physics, particularly in string theory. … So, there are ways to define the sums of non-converging infinite series so that they do not lead to contradictions. The one that leads legitimately to the conclusion that 1 + 2 + 3 + 4 … = -1/12 is called Ramanujan summation.

A user explains: "The sum is formally divergent: 1+2+3+4+... is obviously infinity. The point is that while the definition of the zeta function, zeta(s) = ∑_{n=1}^{∞} 1/n^s, only converges for Re(s) > 1, you can analytically continue it to other s values. In that extended definition, ζ(−1) = −1/12." They continue: "So in zeta function regularization, we replace a divergent sum like ∑ n with the value of ζ(s) at a point where the analytic continuation is defined. That’s why people say ‘1 + 2 + 3 + ... = −1/12’ — it’s shorthand for ‘the zeta-regularized value of this divergent series is −1/12’."

Because -1/12 being the Ramanujan summation of all natural numbers doesn't mean 1+2+3+...=-1/12. The Ramanujan Summation acts like a function f, if you want, such as f(1+2+3+...)=-1/12. 1+2+3=-1/12 is obviously false, but the value -1/12 can be associated to the sum 1+2+3+... (thanks to the Riemann Zeta function, in fact). The way I like to clarify it is that 1 + 2 + 3 + ... + N = kN^2 – 1/12, and in Ramanujan summation or zeta regularization one discards the divergent term and keeps the constant −1/12.