Nonnegative Multiplicative Functions on Sifted Sets, and the Square Roots of −1 Modulo Shifted Primes

Pollack, Paul

doi:10.1017/s0017089519000041

Cited by 5 publications

(3 citation statements)

References 9 publications

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“…The following variant of Shiu's theorem, which is a consequence of a result due to Pollack [22], will allow us to work conveniently with sums over shifted primes in Section 6.…”

Section: Auxiliary Results About Multiplicative Functionsmentioning

confidence: 99%

Divisor-bounded multiplicative functions in short intervals

Mangerel

2023

Res Math Sci

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We extend the Matomäki–Radziwiłł theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a function f in typical intervals of length $$h(\log X)^c$$ h ( log X ) c , with $$h = h(X) \rightarrow \infty $$ h = h ( X ) → ∞ and where $$c = c_f \ge 0$$ c = c f ≥ 0 is determined by the distribution of $$\{|f(p) |\}_p$$ { | f ( p ) | } p in an explicit way. We give three applications. First, we show that the classical Rankin–Selberg-type asymptotic formula for partial sums of $$|\lambda _f(n) |^2$$ | λ f ( n ) | 2 , where $$\{\lambda _f(n)\}_n$$ { λ f ( n ) } n is the sequence of normalized Fourier coefficients of a primitive non-CM holomorphic cusp form, persists in typical short intervals of length $$h\log X$$ h log X , if $$h = h(X) \rightarrow \infty $$ h = h ( X ) → ∞ . We also generalize this result to sequences $$\{|\lambda _{\pi }(n) |^2\}_n$$ { | λ π ( n ) | 2 } n , where $$\lambda _{\pi }(n)$$ λ π ( n ) is the nth coefficient of the standard L-function of an automorphic representation $$\pi $$ π with unitary central character for $$GL_m$$ G L m , $$m \ge 2$$ m ≥ 2 , provided $$\pi $$ π satisfies the generalized Ramanujan conjecture. Second, using recent developments in the theory of automorphic forms we estimate the variance of averages of all positive real moments $$\{|\lambda _f(n) |^{\alpha }\}_n$$ { | λ f ( n ) | α } n over intervals of length $$h(\log X)^{c_{\alpha }}$$ h ( log X ) c α , with $$c_{\alpha } > 0$$ c α > 0 explicit, for any $$\alpha > 0$$ α > 0 , as $$h = h(X) \rightarrow \infty $$ h = h ( X ) → ∞ . Finally, we show that the (non-multiplicative) Hooley $$\Delta $$ Δ -function has average value $$\gg \log \log X$$ ≫ log log X in typical short intervals of length $$(\log X)^{1/2+\eta }$$ ( log X ) 1 / 2 + η , where $$\eta >0$$ η > 0 is fixed.

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“…The following variant of Shiu's theorem, which is a consequence of a result due to Pollack [22], will allow us to work conveniently with sums over shifted primes in Section 6.…”

Section: Auxiliary Results About Multiplicative Functionsmentioning

confidence: 99%

Divisor-bounded multiplicative functions in short intervals

Mangerel

2023

Res Math Sci

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“…In order to bound the sum of ρ(v) A , we use the following Brun-Titchmarsh-like theorem for multiplicative functions (the k = 1, y = x cases of [15], [12]).…”

Section: Outlinementioning

confidence: 99%

Irreducible quadratic polynomials and Euler's function

Lebowitz-Lockard

2018

Preprint

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“…It is a testimony to the flexibility of Hooley's approach that this restriction on k does not lead to significant complications of the analysis. As further evidence for the reach of Hooley's method, we mention that this approach was recently used in [18] to show that the square roots of −1 mod k are equidistributed as k ranges over the shifted primes p − 1.…”

Section: Introductionmentioning

confidence: 97%

The smallest root of a polynomial congruence

Crişan

Pollack

an³

2020

Mathematical Research Letters

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Fix f (t) ∈ Z[t] having degree at least 2 and no multiple roots. We prove that as k ranges over those integers for which the congruence f (t) ≡ 0 (mod k) is solvable, the least nonnegative solution is almost always smaller than k/(log k) c f . Here c f is a positive constant depending on f . The proof uses a method of Hooley originally devised to show that the roots of f are equidistributed modulo k as k varies.

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Nonnegative Multiplicative Functions on Sifted Sets, and the Square Roots of −1 Modulo Shifted Primes

Cited by 5 publications

References 9 publications

Divisor-bounded multiplicative functions in short intervals

Divisor-bounded multiplicative functions in short intervals

Irreducible quadratic polynomials and Euler's function

The smallest root of a polynomial congruence

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