On a paper of K. Soundararajan on smooth numbers in arithmetic progressions

Details On a paper of K. Soundararajan on smooth numbers in arithmetic progressions On a paper of K. Soundararajan on smooth numbers in arithmetic progressions

2011-03-10

arxiv.org/abs/1103.2106v1

Mathematics

Number Theory

18 pages

Scientific paper

In a recent paper, K. Soundararajan showed, roughly speaking, that the integers smaller than x whose prime factors are less than y are asymptotically equidistributed in arithmetic progressions to modulus q, provided that y^{4\sqrt{e}-\delta} \geq q and that y is neither too large nor too small compared with x. We show that these latter restrictions on y are unnecessary, thereby proving a conjecture of Soundararajan. Our argument uses a simple majorant principle for trigonometric sums to handle a saddle point that is close to 1.

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