Divisor problem in arithmetic progressions modulo a prime power

Liu, Kui; Shparlinski, Igor E.; Zhang, Tianping

doi:10.1016/j.aim.2017.12.006

Cited by 24 publications

(20 citation statements)

References 13 publications

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“…• bounds on the zero-free regions of L-functions, which leads to results on the distribution of primes in arithmetic progressions (see [1,2,6,15]), • asymptotic formulas in the Dirichlet problem on sums with the divisor function over arithmetic progressions modulo p n (see [17,20]), • asymptotic formulas for moments of L-functions (see [21,22]).…”

Section: Such Results Includementioning

confidence: 99%

On a conjecture of Soundararajan

Banks

Shparlinski

2021

Bulletin of London Math Soc

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Building on recent work of Harper, and using various results of Chang and Iwaniec on the zerofree regions of L-functions Lps, χq for characters χ with a smooth modulus q, we establish a conjecture of Soundararajan on the distribution of smooth numbers over reduced residue classes for such moduli q. A crucial ingredient in our argument is that, for such q, there is at most one 'problem character' for which Lps, χq has a smaller zero-free region. Similarly, using the 'Deuring-Heilbronn' phenomenon on the repelling nature of zeros of L-functions close to one, we also show that Soundararajan's conjecture holds for a family of moduli having Siegel zeros. Contents

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Section: Such Results Includementioning

confidence: 99%

On a conjecture of Soundararajan

Banks

Shparlinski

2021

Bulletin of London Math Soc

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“…with r m i ,n being a solution of the congruence r 2 ≡ m i na mod q for i = 1, 2. Now we consider the inner sum over n. Note that for i = 1, 2, we have gcd(am i , q) = 1, then we use the following argument, which is similar to that in [6] and [11].…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

“…One of our principal technical tools is the following estimate from [11] on the cancellations amongst Kloosterman sums, Lemma 2.3. Let q = p k , k ∈ Z be a power of an odd prime p. Then for any fixed constant 0 < λ < 1 and q λ < N < q , there exist a constant K 0 (λ) depending only on λ and an absolute constant c > 0 such that for any k > K 0 (λ) and τ (λ) = cλ 3 we have…”

Section: Some Bounds On Kloosterman Sumsmentioning

confidence: 99%

Cancellations Between Kloosterman Sums Modulo a Prime Power With Prime Arguments

2019

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We obtain a nontrivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short interval, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However the underlying technique is different and allows us to obtain nontrivial results starting from much shorter ranges.2010 Mathematics Subject Classification. 11L05, 11T23.

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“…where q grows together with x and they are nontrivial for q ≪ x 2 3 −ε . For the function τ (n) K. Liu, I. Shparlinskii and T. Zhang ( [2]) obtained the extended region of non-triviality.…”

Section: Introductionmentioning

confidence: 99%