{'fraction_non_alphanumeric': 0.07532467532467532, 'fraction_numerical': 0.011688311688311689, 'mean_word_length': 4.244897959183674, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
AN ALTERNATE ARGUMENT FOR THE ARITHMETIC LARGE SIEVE INEQUALITY

the notation ∑[ and ∑∗ denoting, respectively, a sum over squarefree integers, and one over integers coprime with the (implicit) modulus, which is q here. By work of Montgomery-Vaughan and Selberg, it is known that one can take ∆ = Q − 1 +N (see, e.g., [IK, Th. 7.7]). There are a number of derivations of (1) from (2); for one of the earliest, see [M1, Ch. 3]. The most commonly used is probably the argument of Gallagher involving a “submultiplicative” property of some arithmetic function (see, e.g., [K, §II.2] for a very general version). We will show in this note how to prove (1) quite straightforwardly from the dual version of the harmonic large sieve inequality: ∆ is also any constant for which