ON MOMENTS OF TWISTED L-FUNCTIONS
We study the average of the product of the central values of two L-functions of modular forms f and g twisted by Dirichlet characters to a large prime modulus q. As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over mul...
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| Veröffentlicht in: | American journal of mathematics 2017-06, Vol.139 (3), p.707-768 |
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| container_title | American journal of mathematics |
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| creator | Blomer, Valentin Fouvry, Étienne Kowalski, Emmanuel Michel, Philippe Milićević, Djordje |
| description | We study the average of the product of the central values of two L-functions of modular forms f and g twisted by Dirichlet characters to a large prime modulus q. As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of q, and we use the theory of Deligne and Katz to prove new bounds on bilinear forms in Kloosterman sums with power savings when both variables are near the square root of q. When at least one of the forms f and g is non-cuspidal, we obtain an asymptotic formula for the mixed second moment of twisted L-functions with a power saving error term. In particular, when both are non-cuspidal, this gives a significant improvement on M. Young's asymptotic evaluation of the fourth moment of Dirichlet L-functions. In the general case, the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in Kloosterman sums. |
| doi_str_mv | 10.1353/ajm.2017.0019 |
| format | Article |
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As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of q, and we use the theory of Deligne and Katz to prove new bounds on bilinear forms in Kloosterman sums with power savings when both variables are near the square root of q. When at least one of the forms f and g is non-cuspidal, we obtain an asymptotic formula for the mixed second moment of twisted L-functions with a power saving error term. In particular, when both are non-cuspidal, this gives a significant improvement on M. Young's asymptotic evaluation of the fourth moment of Dirichlet L-functions. 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In the general case, the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in Kloosterman sums.</description><subject>Algebra</subject><subject>Asymptotic properties</subject><subject>Convolution</subject><subject>Dirichlet problem</subject><subject>Energy conservation</subject><subject>Number theory</subject><subject>Spectral theory</subject><subject>Sums</subject><issn>0002-9327</issn><issn>1080-6377</issn><issn>1080-6377</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><sourceid>GUQSH</sourceid><sourceid>M2O</sourceid><recordid>eNpFkM1LwzAYxoMoOKdHj0LFc-ab7-Qoc9PB1h7a4TE0WQoWZ2fTHfzvTZno6eWB5-Plh9AtgRlhgj3W7X5GgagZADFnaEJAA5ZMqXM0AQCKDaPqEl3F2CYJCugE3Rd5tik2i7wqs2KZVW-rslo8Z2u83ObzalXk5TW6aOqPGG5-7xRtl4tq_orXxctq_rTGnjE6YCWcpEI6xj2oIDRQ4aXQfMdpEJQ3TNbOSOXq9JBxgTjwjeDe1I2Dehc0m6KHU--h776OIQ627Y79Z5q0RBvNjBYKkgufXL7vYuxDYw_9-77uvy0BO1KwiYIdKdiRQvLzv9Y2-GF_jOG_WAqlpLLlSGoERRRLYLhKsbtTrI1D1_9tcC5AG2PYD-CHZDs</recordid><startdate>20170601</startdate><enddate>20170601</enddate><creator>Blomer, Valentin</creator><creator>Fouvry, Étienne</creator><creator>Kowalski, Emmanuel</creator><creator>Michel, Philippe</creator><creator>Milićević, Djordje</creator><general>Johns Hopkins University Press</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7XB</scope><scope>8AF</scope><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>GNUQQ</scope><scope>GUQSH</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K7-</scope><scope>L6V</scope><scope>M2O</scope><scope>M2P</scope><scope>M7S</scope><scope>MBDVC</scope><scope>P5Z</scope><scope>P62</scope><scope>PADUT</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>Q9U</scope><scope>S0X</scope></search><sort><creationdate>20170601</creationdate><title>ON MOMENTS OF TWISTED L-FUNCTIONS</title><author>Blomer, Valentin ; 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As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of q, and we use the theory of Deligne and Katz to prove new bounds on bilinear forms in Kloosterman sums with power savings when both variables are near the square root of q. When at least one of the forms f and g is non-cuspidal, we obtain an asymptotic formula for the mixed second moment of twisted L-functions with a power saving error term. In particular, when both are non-cuspidal, this gives a significant improvement on M. Young's asymptotic evaluation of the fourth moment of Dirichlet L-functions. In the general case, the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in Kloosterman sums.</abstract><cop>Baltimore</cop><pub>Johns Hopkins University Press</pub><doi>10.1353/ajm.2017.0019</doi><tpages>62</tpages></addata></record> |
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| ispartof | American journal of mathematics, 2017-06, Vol.139 (3), p.707-768 |
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| subjects | Algebra Asymptotic properties Convolution Dirichlet problem Energy conservation Number theory Spectral theory Sums |
| title | ON MOMENTS OF TWISTED L-FUNCTIONS |
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