LIMITING DISTRIBUTIONS OF THE CLASSICAL ERROR TERMS OF PRIME NUMBER THEORY

By Amir Akbary, Nathan Ng, M. Shahabi. 2013

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Let � : (0,∞) → R and let y0 be a non-negative constant. Let (�n)n∈N be a non- decreasing sequence of positive numbers which tends to infin ity, let (rn)n∈N be a complex sequence, andc a real number. Assume thatis square-integrable on (0,y0) and fory ≥ y0, � can be expressed as independent over Q, we explicitly calculate the Fourier transform of the limit ing distribution mea- sure. Moreover, we prove general versions of the above results for vector-valued functions. Finally, we illustrate some applications of our general theorems by applying them to several classical error terms which occur in prime number theory. Examples include the error term in the prime number theorem for an automorphic L-function, weighted sums of the Mobius function, weighted sums of the Liouville function, the sum of the Mobius function in an arithmetic progression, and the error term in Chebotarev's density theorem.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/051b43bc835b1ffb4c6940d1a25a2bf818c8b355