2 research outputs found
Distribution in coprime residue classes of polynomially-defined multiplicative functions
An integer-valued multiplicative function is said to be
polynomially-defined if there is a nonconstant separable polynomial with for all primes . We study the distribution
in coprime residue classes of polynomially-defined multiplicative functions,
establishing equidistribution results allowing a wide range of uniformity in
the modulus . For example, we show that the values , sampled over
integers with coprime to , are asymptotically
equidistributed among the coprime classes modulo , uniformly for moduli
coprime to that are bounded by a fixed power of .Comment: edited paragraph following Theorem 1.3, correcting a claim in the
discussion of condition (i
The distribution of intermediate prime factors
Let denote the middle prime factor of
(taking into account multiplicity). More generally, one can consider, for any
, the -positioned prime factor of ,
. It has previously been shown that has normal order , and its values follow a
Gaussian distribution around this value. We extend this work by obtaining an
asymptotic formula for the count of for which ,
for primes in a wide range up to . We give several applications of these
results, including an exploration of the geometric mean of the middle prime
factors, for which we find that , where is the golden
ratio, and is an explicit constant. Along the way, we obtain an extension
of Lichtman's recent work on the ``dissected'' Mertens' theorem sums
for large values of