Explicit large sieve inequalities for square moduli

Abstract:

The large sieve inequality is ubiquitous in analytic number theory and is a crucial ingredient in big results such as the Bombieri-Vinogradov Theorem and the Grand Density Theorem. Roughly speaking, the inequality states that the average size of a particular trigonometric polynomial over a well spaced subset of (0,1](0,1] is small relative to the size of its coefficients. It has its origins in 1941 with J. V. Linnik, whose method was studied extensively by A. R\'enyi in 1948. Much progress was made in the 1960s and 1970s by K. F. Roth, E. Bombieri and others, to the point that A. Selberg and P. Cohen (independently) proved the sharpest possible bound of general form. That being said, the large sieve inequality for certain sets is still an active area of research. In particular, for the set SQSQ of fractions in (0,1](0,1] of square denominator ≤Q2≤Q2. It is in this case that L. Zhao (2003) was the first to improve upon the standard bounds. This was followed by a bound of S. Baier (2005), prompting a collaborative effort (2008) to combine their methods so as to yield an even sharper bound. Here, we give an explicit version of said collaborative bound and novel numerical results on the spacing of SQ