The mean square of the product of the Riemann zeta-function with Dirichlet polynomials
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Bibliographic record
Abstract
Abstract Improving earlier work of Balasubramanian, Conrey and Heath-Brown [1], we obtain an asymptotic formula for the mean-square of the Riemann zeta-function times an arbitrary Dirichlet polynomial of length <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>T</m:mi> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>+</m:mo> <m:mi>δ</m:mi> </m:mrow> </m:msup> </m:math> {T^{\frac{1}{2}+\delta}} , with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>δ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>0.01515</m:mn> <m:mo></m:mo> <m:mi>…</m:mi> </m:mrow> </m:mrow> </m:math> \delta=0.01515\dots . As an application we obtain an upper bound of the correct order of magnitude for the third moment of the Riemann zeta-function. We also refine previous work of Deshouillers and Iwaniec [8], obtaining asymptotic estimates in place of bounds. Using the work of Watt [19], we compute the mean-square of the Riemann zeta-function times a Dirichlet polynomial of length going up to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>T</m:mi> <m:mfrac> <m:mn>3</m:mn> <m:mn>4</m:mn> </m:mfrac> </m:msup> </m:math> {T^{\frac{3}{4}}} provided that the Dirichlet polynomial assumes a special shape. Finally, we exhibit a conjectural estimate for trilinear sums of Kloosterman fractions which implies the Lindelöf Hypothesis.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.008 | 0.003 |
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| Bibliometrics | 0.000 | 0.001 |
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