Mertens function calculator

The Mertens function calculator returns M(n), the cumulative sum of the Möbius function from 1 through a positive integer n. It provides a direct way to inspect how square-free integers with even and odd numbers of distinct prime factors balance over an initial interval. Enter n and receive the exact integer result from a deterministic linear sieve. The calculation uses no random sampling, external service, or stored lookup table, so the same valid input always produces the same answer.

What the Mertens function measures

The Mertens function is defined by M(n) = μ(1) + μ(2) + ... + μ(n), where μ is the Möbius function. For a positive integer k, μ(k) is zero when a prime square divides k. Otherwise, μ(k) is 1 when k has an even number of distinct prime factors and -1 when it has an odd number.

The running sum therefore records a signed balance among square-free integers, while integers containing repeated prime factors make no contribution. In particular, μ(1) equals 1, so M(1) equals 1. The result can be positive, zero, or negative, and it should be read as an exact arithmetic value rather than a probability or approximation.

This cumulative quantity appears throughout analytic number theory because it connects multiplicative structure with questions about cancellation in arithmetic sequences.

How the calculator obtains an exact result

The implementation uses a linear sieve to construct Möbius values in increasing order. When the sieve encounters a prime, it assigns that prime a Möbius value of -1. For each previously discovered prime, it then considers the product with the current integer.

If that prime already divides the current integer, the product contains a squared prime factor, receives a Möbius value of zero, and processing for that integer stops. Otherwise, the product receives the negated Möbius value of the current integer. Each composite is handled through its smallest prime factor, which keeps the work proportional to the chosen upper bound.

As each new value is established, it is added to the running total. No floating-point formulas are involved, and no values are estimated. The algorithm is deterministic, operates locally, and returns an ordinary signed integer.

Input validation rejects fractions, non-finite numbers, values below one, and values beyond the published bound before allocating the sieve arrays.

Choosing inputs and interpreting the response

The response repeats the accepted n and supplies its Mertens value in the mertens field, which makes automated records self-describing. For example, a sequence of calls at increasing bounds can be used to plot the fluctuations of M(n), compare a textbook table with an independent calculation, or verify code that generates Möbius values by another method. Do not interpret a single positive or negative result as evidence of a lasting trend.

The function oscillates as contributions accumulate, and its theoretical interest concerns behavior across ranges and asymptotic bounds. If you need several values, retain each returned pair or compute at ascending checkpoints rather than assuming that M(n) reveals the individual value μ(n) without comparing it with M(n - 1).