The Distribution of the Summatory Function of the Möbius Function

By Nathan Ng. 2003

88 citations recorded by the source corpus.

The summatory function of the Möbius function is denoted M(x). In this article we deduce conditional results concerning M(x) assuming the Riemann hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the Riemann zeta function. Assuming these conjectures, we show that M(x), when appropriately normalized, possesses a limiting distribution, and also that a strong form of the weak Mertens conjecture is true. Finally, we speculate on the lower order of M(x) by studying the constructed distribution function. 2000 Mathematics Subject Classification 11M26, 11N56.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9b870db0f3d3251332aafe7aa55681de40ae6112

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

LIMITING DISTRIBUTIONS OF THE CLASSICAL ERROR TERMS OF PRIME NUMBER THEORY (2013, methodology): “Limiting distributions have been a useful tool in problems concerning summatory functions [18, 35], prime number races [11, 26, 37] and the distribution of values of L-functions [16, 19, 25].”
MOMENTS OF RANDOM MULTIPLICATIVE FUNCTIONS, I: LOW MOMENTS, BETTER THAN SQUAREROOT CANCELLATION, AND CRITICAL MULTIPLICATIVE CHAOS (2017, background): “In fact, a conjecture of Gonek based on the distribution of zeros of the Riemann zeta function (see Ng’s paper [21]) asserts that this sum should be O( √ x(log log log x)), and infinitely often as large as that.”
Squarefree polynomials and Mobius values in short intervals and arithmetic progressions (2015, background): “It is important that the length H of the interval be significantly smaller than its location, that is H < X1−ǫ, since otherwise one expects non-Gaussian statistics, see [23].”
Partial sums of the Möbius function (2007, unknown): “Ng [8]) has conjectured that ∞ > lim sup x→∞ M(x) √ x(log log log x) 5 4 > 0 > lim inf x→∞ M(x) √ x(log log log x) 5 4 > −∞.”