For Re(s) > 1, the Riemann zeta function is defined by the convergent series ζ(s) = ∑_{n=1}^{∞} 1/n^{s}. In this fashion, we define the Riemann zeta function over the entire complex plane. It is a meromorphic function, with a single, simple pole at z = 1. The analytic continuation satisfies ζ(−1) = −1/12, even though the original defining series 1 + 2 + 3 + 4 + ⋯ diverges and does not converge to −1/12.

The series 1 + 2 + 3 + 4 + ⋯ is divergent in the usual sense: its sequence of partial sums tends to infinity. Nevertheless, in physics and in some areas of mathematics it is common to assign a finite value to certain divergent series using regularization schemes. In zeta-function regularization, one defines the value of 1 + 2 + 3 + 4 + ⋯ to be ζ(−1), which equals −1/12, but this is a regularized value rather than a sum in the sense of classical analysis.

Recently the identity 1+2+3+… = −1/12 has appeared in a number of popular expositions. Mathematically, the series ∑_{n=1}^{∞} n is divergent; its partial sums grow without bound. The assignment of −1/12 to this series is based on analytic continuation of the Riemann zeta function and on Ramanujan’s summation techniques. We emphasize that these generalized summation methods do not alter the fact that, in the usual sense of summing an infinite series, 1+2+3+… does not have the sum −1/12.

The series ∑_{n=1}^{∞} n is divergent, but in quantum field theory and string theory it is often assigned the value −1/12 via zeta-function regularization. In this approach, one defines the regularized sum of ∑_{n=1}^{∞} n to be ζ(−1), where ζ(s) is the analytically continued Riemann zeta function. We stress that this regularized value is not a sum in the classical sense but a useful convention that allows divergent quantities to be handled consistently in physical calculations.

We discuss the identity 1 + 2 + 3 + · · · = −1/12 in light of classical summation theory. The series ∑_{n=1}^{∞} n is divergent; its sequence of partial sums tends to infinity. Nevertheless, several generalized summation methods assign the value −1/12 to this series. These methods rely on analytic continuation and do not imply that the ordinary sum of the natural numbers is −1/12. Confusing the regularized value with the classical sum leads to conceptual errors in analysis.

In mathematics, the infinite series 1 + 2 + 3 + 4 + ⋯ is divergent, meaning that its sequence of partial sums does not tend to a limit.[1] Although the series does not have a sum in the usual sense, it is possible to assign it a finite value, which is −1/12, using methods such as Ramanujan summation or zeta function regularization. In the context of the analytic continuation of the Riemann zeta function, one sometimes writes 1 + 2 + 3 + 4 + ⋯ = −1/12 as a shorthand for the identity ζ(−1) = −1/12.

An infinite series ∑_{n=1}^{∞} a_n is said to converge if the sequence of its partial sums S_N = ∑_{n=1}^{N} a_n converges to a finite limit. If S_N → ∞, the series diverges. For example, the series 1 + 2 + 3 + 4 + … has partial sums S_N = N(N + 1)/2, which tend to infinity as N → ∞, so the series diverges. In particular, it does not converge to −1/12 or any other finite number under the standard definition of convergence.

The series 1 + 2 + 3 + ⋯ has no sum in the usual sense: the sequence of partial sums tends to infinity, so the series diverges. Some physicists informally write 1 + 2 + 3 + ⋯ = −1/12 because −1/12 appears in certain regularization schemes applied to this series, for example in zeta-function regularization connected with the Riemann zeta function at −1. Baez emphasizes that neither standard mathematics nor physics literally says that 1 + 2 + 3 + ⋯ equals −1/12 as an ordinary sum of positive integers; rather, −1/12 arises as a value in a particular regularization procedure that is useful in physics.

Ramanujan summation is a technique to assign finite values to divergent series. For the series 1 + 2 + 3 + 4 + ⋯, Ramanujan’s method associates the value −1/12, even though the ordinary sum diverges. In Ramanujan’s notation, if c denotes the Ramanujan sum of 1 + 2 + 3 + 4 + ⋯, he finds −3c = 1 − 2 + 3 − 4 + ⋯ = 1/(1 + 1)^2 = 1/4, so c = −1/12. This value agrees with the value obtained by analytic continuation of the Riemann zeta function ζ(s) to s = −1.

Terence Tao writes about Ramanujan's famous (but extremely unintuitive) formula 1+2+3+4+… = −1/12. He explains that from a post-rigorous perspective, such an equation should more accurately be rendered as 1+2+3+4+… = −1/12 + …, where the “…” denotes terms which could be infinitely large (or divergent) when interpreted classically, but which one wishes to view as negligible for one’s intended application. He further notes that as a rough first approximation, one should actually write 1+2+3+4+… = −1/12 + 1/2·infinity², indicating that in the classical sense there is an infinite divergence in addition to the −1/12 term.

The Riemann zeta function is initially defined for complex numbers s with real part greater than 1 by the convergent series ζ(s) = ∑_{n=1}^∞ 1/n^s. In this region, ζ(s) is the sum of an infinite series of positive terms. By analytic continuation, ζ(s) extends to a meromorphic function on the whole complex plane except for a simple pole at s = 1. In particular, the analytically continued value at s = −1 is ζ(−1) = −1/12. This value is often informally written as 1 + 2 + 3 + 4 + ⋯ = −1/12, but that identity relies on interpreting the divergent series via analytic continuation rather than as an ordinary convergent sum.

A highly upvoted answer on this Q&A explains that the series 1+2+3+4+⋯ diverges in the usual sense, but that in analytic continuation one considers the Riemann zeta function ζ(s) = 1+2^{-s}+3^{-s}+⋯, which converges for Re(s) > 1. This function has a unique analytic continuation to the whole complex plane except for a pole at s = 1, and this continuation satisfies ζ(−1) = −1/12. The answer stresses that saying 1+2+3+4+⋯ = −1/12 is shorthand for the statement that the analytic continuation of ζ(s) at s = −1 equals −1/12, not that the ordinary series of positive integers literally sums to −1/12.

Padilla explains that the series 1 + 2 + 3 + 4 + ⋯ is divergent in the usual sense: the partial sums grow without limit and “it truly does equal infinity” as an ordinary sum. He then discusses why mathematicians and physicists sometimes write that this series equals −1/12: using methods such as the Euler–Maclaurin formula and zeta-function regularization, one can extract a finite ‘regularized’ value from the divergent sum, and that finite part is −1/12. He stresses that this does not mean that adding all positive integers in the traditional way gives −1/12; instead, −1/12 is a regularized value that appears consistently across correct regularization schemes and has physical relevance.

In this educational video, mathematician Burkard Polster (Mathologer) analyzes the identity 1+2+3+⋯ = −1/12. He explains that if the series is treated formally and certain manipulations are justified, one can be led to the value −1/12, but he repeatedly stresses that the ordinary sum of the natural numbers diverges. Around the end of the video he connects the calculation to the analytic continuation of the Riemann zeta function, stating that the analytic continuation at s = −1 is −1/12, which is the precise sense in which the expression 1+2+3+⋯ = −1/12 is used in mathematics and physics.

The video demonstrates how Ramanujan summation can be applied to the divergent series 1+2+3+4+⋯ to obtain the value −1/12. The presenter notes early on that in a standard calculus class 1+2+3+4+⋯ is said to have sum infinity, but that Ramanujan summation is a different way to assign a value to divergent series. After working through the Ramanujan summation formula for f(n) = n, he arrives at −1/12 and writes 1+2+3+4+⋯ with an R above the summation to indicate the Ramanujan-summed value −1/12.

A highly upvoted answer explains that in standard analysis, the series 1 + 2 + 3 + 4 + ⋯ diverges to infinity and so does not equal −1/12 as an ordinary sum. The answer goes on to say that the identity 1 + 2 + 3 + 4 + ⋯ = −1/12 is meaningful only when one interprets the left-hand side via analytic continuation of the Riemann zeta function or through Ramanujan summation. In these generalized summation methods, one defines a linear functional on a space of series extending the usual notion of sum, and for that functional the assigned value of this series is −1/12.

In this popular video, the presenter derives −1/12 from manipulations of several divergent series and concludes with the equation S = −1/12 for S = 1 + 2 + 3 + 4 + ⋯. The speaker refers to a statement in a physics text that “the sum of all the integers…is minus a twelfth,” and then shows an informal derivation. Mathematicians later clarified in follow-up material that these series manipulations are not valid as ordinary algebra on convergent series; instead, the result should be interpreted within the framework of zeta-function regularization used in theoretical physics.

MathWorld notes that the series 1 + 2 + 3 + 4 + ⋯ is a classic example of a divergent series whose partial sums grow without bound. The entry explains that several summation methods, including zeta-function regularization and Ramanujan summation, can assign finite values to divergent series, and for 1 + 2 + 3 + 4 + ⋯ these methods give −1/12. It emphasizes that such an assigned value is not the sum in the usual sense of convergence but a generalized notion of summation defined by specific analytic procedures.

The Riemann zeta function is defined for complex s with Re(s) > 1 by the series ζ(s) = ∑_{n=1}^{∞} 1/n^{s}. Under this definition, the series at s = −1 becomes ∑_{n=1}^{∞} n, which diverges. Riemann extended ζ(s) analytically to a meromorphic function on the whole complex plane (except a simple pole at s = 1), and this extension yields ζ(−1) = −1/12. Thus, in modern number theory, the equation 1 + 2 + 3 + 4 + … = −1/12 is interpreted as a symbolic way to refer to the analytic continuation value ζ(−1), not a literal sum.

The blog claims that “the sum of all natural numbers is equal to −1/12” and presents an algebraic manipulation using the alternating series 1 − 2 + 3 − 4 + ⋯ to derive −1/12. Later in the article, the author concedes that the algebra is being applied to a divergent infinite series in a way that is “completely invalid” under the usual rules of analysis. They then state that the correct framework is Ramanujan summation and zeta-function regularization, where extending the functional that assigns sums to convergent series can assign the value −1/12 to 1 + 2 + 3 + 4 + ⋯ without that value being the sum in the traditional sense.

A commenter explains that the series 1+2+3+4+⋯ is not convergent at all: the sum does not become close to −1/12 no matter how many terms you add together, and in the usual sense it diverges to positive infinity. They go on to say that 'Ramanujan summation' is a different operation on series, not the usual notion of sum, under which this series is assigned the value −1/12. Thus, 1+2+3+4+⋯ = −1/12 only holds when the symbol is interpreted in the Ramanujan summation sense.

One highly upvoted reply states: "1+2+3+… doesn’t equal −1/12. 1+2+3+… diverges to positive infinity." The commenter clarifies that −1/12 is the Ramanujan summation (or zeta-regularized value) associated with the series, and suggests thinking of a function f such that f(1+2+3+…) = −1/12. They explain that by analyzing smoothed partial sums one gets 1+2+3+⋯+N = kN² − 1/12, and if one discards the divergent kN² term in a regularization procedure, one is left with the finite remainder −1/12, which is what Ramanujan summation and zeta regularization capture.

A top reply explains that the idea that 1 + 2 + 3 + 4 + ⋯ equals −1/12 is connected to extending the definition of the Riemann zeta function. The usual series ∑_{n=1}^∞ 1/n^s defining ζ(s) does not converge at s = −1, but using analytic continuation one defines ζ(s) on a larger domain, and this extended function has the value ζ(−1) = −1/12. The commenter stresses that this does not mean the ordinary sum of the natural numbers is −1/12; rather, −1/12 is the analytically continued zeta value associated with that divergent series.