Mertens conjecture
The Mertens conjecture is a statement in analytic number theory about the bounded growth of the Mertens function M(x) = \sum_{n=1}^{\lfloor x \rfloor} \mu(n), where \mu(n) denotes the Möbius function, which assigns values of 0, 1, or -1 to positive integers based on their prime factorization.[1] The conjecture asserts that |M(x)| < \sqrt{x} holds for all real numbers x > [1](/page/1).[1] First suggested in an 1885 letter from Thomas Stieltjes to Charles Hermite claiming a related bound implying the Riemann hypothesis, it was independently and formally proposed by Franz Mertens in 1897 based on extensive numerical evidence up to x = 10^4.[2] Despite initial computational support extending far beyond Mertens's calculations—verifying the inequality up to x = 10^{13} in 1994—the conjecture's analytic foundations were questioned as better estimates for the distribution of primes emerged.[2] In 1985, Andrew Odlyzko and Herman te Riele disproved it through an indirect proof combining classical analytic number theory with lattice basis reduction algorithms to compute discrepancies in the zeta function's zeros, establishing that \limsup_{x \to \infty} |M(x)| / \sqrt{x} > 1.06.[1] Their work showed the first counterexample occurs at an enormously large x, with subsequent refinements placing it below \exp(1.96 \times 10^{19}) as of 2025, though no explicit value has been computed due to its scale.[3] The Mertens conjecture's falsity highlights the subtleties in bounding summatory functions like M(x), which is central to the prime number theorem and sieve theory; under the Riemann hypothesis, M(x) = O(\sqrt{x} (\log x)^{3/2}), a weaker but still unproven growth rate.[1] Its disproof did not impact the Riemann hypothesis but spurred advances in computational number theory, including explicit counterexample searches and improved bounds on M(x)'s oscillations.[2]Foundations
Möbius function
The Möbius function, denoted \mu(n), is a fundamental arithmetic function in number theory defined on the positive integers n. Specifically, \mu(n) = 1 if n is a square-free positive integer with an even number of distinct prime factors (including n=1), \mu(n) = -1 if n is square-free with an odd number of distinct prime factors, and \mu(n) = 0 if n has a squared prime factor (i.e., n is not square-free).[4] This definition captures the parity of the number of prime factors for square-free integers while vanishing on numbers with higher powers of primes.[5] Introduced by the German mathematician August Ferdinand Möbius in 1832, the function bears his name and marked an early systematic study of such arithmetic functions, though implicit appearances trace back to Euler's work on series expansions.[4] Möbius's contribution arose in the context of exploring transcendental functions and their connections to number-theoretic sums.[6] The Möbius function is multiplicative, satisfying \mu(ab) = \mu(a) \mu(b) whenever a and b are coprime positive integers, a property that facilitates its use in products and convolutions over the integers.[4] A central feature is its role in the Möbius inversion formula, which provides a way to recover functions from their divisor sums: for arithmetic functions f and g related by g(n) = \sum_{d \mid n} f(d), the inverse relation is f(n) = \sum_{d \mid n} \mu(d) g(n/d). This stems from the key identity \sum_{d \mid n} \mu(d) = \begin{cases} 1 & \text{if } n = 1, \\ 0 & \text{otherwise}. \end{cases} [5][4] In the context of prime distribution, the Möbius function appears prominently in the Dirichlet series for the reciprocal of the Riemann zeta function. For \operatorname{Re}(s) > 1, the Euler product formula \zeta(s) = \sum_{n=1}^\infty n^{-s} = \prod_p (1 - p^{-s})^{-1} (over primes p) implies that its reciprocal is \frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s}, linking the function directly to the distribution of primes via the zeros and poles of \zeta(s).[4] The Mertens function, defined as the partial sum M(x) = \sum_{n \leq x} \mu(n), builds upon this by accumulating these values.Mertens function
The Mertens function, denoted M(x), is defined for real x \geq 0 as the summatory function of the Möbius function \mu(n), given by M(x) = \sum_{n \leq x} \mu(n), where the sum is over positive integers n up to the floor of x. This function arises naturally in analytic number theory as the partial sum of the coefficients in the Dirichlet series expansion \sum_{n=1}^\infty \mu(n) n^{-s} = 1/[\zeta(s)](/page/Riemann_zeta_function) for \Re(s) > 1, with \zeta(s) the Riemann zeta function; thus, M(x) serves as the summatory analogue capturing the cumulative behavior of these coefficients.[7] From the prime number theorem, which states that the number of primes up to x satisfies \pi(x) \sim x / \log x as x \to \infty, it follows unconditionally that M(x) = o(x), or more precisely, M(x)/x \to 0 as x \to \infty. This asymptotic reflects the cancellation in the sum due to the oscillatory nature of \mu(n), ensuring that the Mertens function grows slower than any positive multiple of x.[7] Assuming the Riemann hypothesis, stronger bounds hold: M(x) = O(x^{1/2 + \varepsilon}) for every \varepsilon > 0, where the implied constant depends on \varepsilon. This conditional result ties the growth of M(x) directly to the location of the non-trivial zeros of \zeta(s) on the critical line \Re(s) = 1/2.[7] Early computations of M(x) revealed its oscillatory behavior, with frequent sign changes indicating the irregular alternation between positive and negative values. In 1897, Franz Mertens calculated values up to x = 10{,}000, observing multiple sign changes that suggested bounded growth relative to \sqrt{x}, though the function exhibited chaotic fluctuations. For instance, the following table illustrates early values in M(n) for small n:| n | M(n) | Sign |
|---|---|---|
| 1 | 1 | + |
| 2 | 0 | 0 |
| 3 | -1 | - |
| 4 | -1 | - |
| 5 | -2 | - |
| 6 | -1 | - |
| 7 | -2 | - |
| 10 | -1 | - |
| 11 | -2 | - |
| 13 | -3 | - |