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

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.