which has been used in certain contexts (Ramanujan summation) such as string theory. Analogously, the particular value can be viewed as assigning a finite result to the divergent series 1 + 1 + 1 + 1 + ⋯. The value is employed in calculating kinetic boundary layer problems of linear kinetic equations. Although diverges, its Cauchy principal value exists and is equal to the Euler–Mascheroni constant . The demonstration of the particular value is known as the Basel problem. The reciprocal of this sum answers the question: What is the probability that two numbers selected at random are relatively prime? The value is Apéry's constant. Taking the limit through the real numbers, one obtains . But at complex infinity on the Riemann sphere the zeta function has an essential singularity. Various properties For sums involving the zeta function at integer and half-integer values, see rational zeta series. Reciprocal The reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius function : for every complex number with real part greater than 1. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series. The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of is greater than . Universality The critical strip of the Riemann zeta function has the remarkable property of universality. This zeta function universality states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable. The first proof of universality was provided by Sergei Mikhailovitch Voronin in 1975. More recent work has included effective versions of Voronin's theorem and extending it to Dirichlet L-functions. Estimates of the maximum of the modulus of the zeta function Let the functions and be defined by the equalities Here is a sufficiently large positive number, , , , . Estimating the values and from below shows, how large (in modulus) values can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip . The case was studied by Kanakanahalli Ramachandra; the case , where is a sufficiently large constant, is trivial. Anatolii Karatsuba proved, in particular, that if the values and exceed certain sufficiently small constants, then the estimates hold, where and are