Inhomogeneous convolutions

Bykovskii, V. A.; Vinogradov, A. I.

doi:10.1007/bf02342917

Cited by 9 publications

(30 citation statements)

References 8 publications

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“…However there is no such process at work in (1.3), leaving one to wonder if power-saving can be achieved. In [10], Bykovskiȋ and Vinogradov announce results implying…”

Section: Correlation Of Divisor Functionsmentioning

confidence: 93%

“…The author is indebted to R. de la Bretèche, É. Fouvry, V. Blomer, D. Milićević, S. Bettin, G. Tenenbaum, B. Topacogullari and A. Granville for valuable discussions and comments, and to an anonymous referee for helpful remarks and careful reading of the manuscript. The author is particularly grateful to V. Blomer for making a preprint of available, and for making him aware of the reference ; and finally to B. Topacogullari for correcting a significant oversight in an earlier version.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…However there is no such process at work in , leaving one to wonder if power‐saving can be achieved. In , Bykovskiĭ and Vinogradov announce results implying

{scriptT}_{k} false(x false) = x P_{k} false(prefixlog x false) + O_{k} false(x^{1 - δ / k} false) false(k ⩾ 4, x ⩾ 2 false)

for some absolute

δ > 0

, and sketch ideas of a proof. The proposed argument, in a way, is dual to the method adopted in (which is related to earlier work of Motohashi ).…”

Section: Introductionmentioning

confidence: 99%

“…We have not sought optimal values for

δ

in Theorems and . In the case of , the method of seems to yield much better numerical results.…”

Section: Introductionmentioning

confidence: 99%

“…In , the authors study the distribution of

τ_{k} false(n false)

in progressions of moduli up to

x^{1 / 2}

, while in the authors address the distribution of

τ (n)

in progressions of moduli up to

x^{1 - 1 / k}

.…”

mentioning

confidence: 99%

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Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

Drappeau

2017

Proc. London Math. Soc.

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International audienceWe prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the "smooth" summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. As a consequence, assuming the Riemann hypothesis for Dirichlet $L$-functions, we prove a power-saving error term in the Titchmarsh divisor problem of estimating $\sum_{p\leq x}\tau(p-1)$. Unconditionally, we isolate the possible contribution of Siegel zeroes, showing it is always negative. Extending work of Fouvry and Tenenbaum, we obtain power-saving in the asymptotic formula for $\sum_{n\leq x}\tau_k(n)\tau(n+1)$, reproving a result announced by Bykovski\u{i} and Vinogradov by a different method. The gain in the exponent is shown to be independent of $k$ if a generalized Lindel\"of hypothesis is assumed

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“…However there is no such process at work in (1.3), leaving one to wonder if power-saving can be achieved. In [10], Bykovskiȋ and Vinogradov announce results implying…”

Section: Correlation Of Divisor Functionsmentioning

confidence: 93%

Section: Acknowledgementsmentioning

confidence: 99%

“…However there is no such process at work in , leaving one to wonder if power‐saving can be achieved. In , Bykovskiĭ and Vinogradov announce results implying