7,975 research outputs found
Biases in prime factorizations and Liouville functions for arithmetic progressions
We introduce a refinement of the classical Liouville function to primes in
arithmetic progressions. Using this, we discover new biases in the appearances
of primes in a given arithmetic progression in the prime factorizations of
integers. For example, we observe that the primes of the form tend to
appear an even number of times in the prime factorization of a given integer,
more so than for primes of the form . We are led to consider variants of
P\'olya's conjecture, supported by extensive numerical evidence, and its
relation to other conjectures.Comment: 25 pages, 6 figure
Explicit Formulae and Trace Formulae
In this thesis, motivated by an observation of D. Hejhal, we show that the explicit formulae of A. Weil for sums over zeroes of Hecke L-functions, via the Maass-Selberg relation, occur in the continuous spectral terms in the Selberg trace formula over various number fields. In Part I, we discuss the relevant parts of the trace formulae classically and adelically, developing the necessary representation theoretic background. In Part II, we show how show the explicit formulae intervene, using the classical formulation of Weil; then we recast this in terms of Weil distributions and the adelic formulation of Weil. As an application, we prove a lower bound for these explicit formulae using properties of the trace formula, in the spirit of Weil\u27s criterion for the Riemann hypothesis
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