ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES
We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface $M$ . Actually, we study the zeros of $(Z_{M}H_{M})^{\prime }$ , where $Z_{M}$ is the Selberg zeta function and $H_{M}$ is the Dirichlet series compo...
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| Veröffentlicht in: | Nagoya mathematical journal 2017-12, Vol.228, p.21-71 |
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| creator | JORGENSON, JAY SMAJLOVIĆ, LEJLA |
| description | We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface
$M$
. Actually, we study the zeros of
$(Z_{M}H_{M})^{\prime }$
, where
$Z_{M}$
is the Selberg zeta function and
$H_{M}$
is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite volume hyperbolic Riemann surface
$M$
. Our main results address finiteness of number of zeros of
$(Z_{M}H_{M})^{\prime }$
in the half-plane
$\operatorname{Re}(s) |
| doi_str_mv | 10.1017/nmj.2016.52 |
| format | Article |
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$M$
. Actually, we study the zeros of
$(Z_{M}H_{M})^{\prime }$
, where
$Z_{M}$
is the Selberg zeta function and
$H_{M}$
is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite volume hyperbolic Riemann surface
$M$
. Our main results address finiteness of number of zeros of
$(Z_{M}H_{M})^{\prime }$
in the half-plane
$\operatorname{Re}(s)<1/2$
, an asymptotic count for the vertical distribution of zeros, and an asymptotic count for the horizontal distance of zeros. One realization of the spectral analysis of the Laplacian is the location of the zeros of
$Z_{M}$
, or, equivalently, the zeros of
$Z_{M}H_{M}$
. Our analysis yields an invariant
$A_{M}$
which appears in the vertical and weighted vertical distribution of zeros of
$(Z_{M}H_{M})^{\prime }$
, and we show that
$A_{M}$
has different values for surfaces associated to two topologically equivalent yet different arithmetically defined Fuchsian groups. We view this aspect of our main theorem as indicating the existence of further spectral phenomena which provides an additional refinement within the set of arithmetically defined Fuchsian groups.</description><identifier>ISSN: 0027-7630</identifier><identifier>EISSN: 2152-6842</identifier><identifier>DOI: 10.1017/nmj.2016.52</identifier><language>eng</language><publisher>Cambridge, UK: Cambridge University Press</publisher><subject>Dirichlet problem ; Equivalence ; Existence theorems ; Mathematics ; Riemann surfaces ; Vertical distribution</subject><ispartof>Nagoya mathematical journal, 2017-12, Vol.228, p.21-71</ispartof><rights>2016 by The Editorial Board of the Nagoya Mathematical Journal</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c336t-d3eb68c821e0664c16cd22c2c1fbebbdf0491a07e5155473d31034da1d40357a3</citedby><cites>FETCH-LOGICAL-c336t-d3eb68c821e0664c16cd22c2c1fbebbdf0491a07e5155473d31034da1d40357a3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.cambridge.org/core/product/identifier/S0027763016000520/type/journal_article$$EHTML$$P50$$Gcambridge$$H</linktohtml><link.rule.ids>164,314,780,784,27923,27924,55627</link.rule.ids></links><search><creatorcontrib>JORGENSON, JAY</creatorcontrib><creatorcontrib>SMAJLOVIĆ, LEJLA</creatorcontrib><title>ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES</title><title>Nagoya mathematical journal</title><addtitle>Nagoya Math. J</addtitle><description>We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface
$M$
. Actually, we study the zeros of
$(Z_{M}H_{M})^{\prime }$
, where
$Z_{M}$
is the Selberg zeta function and
$H_{M}$
is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite volume hyperbolic Riemann surface
$M$
. Our main results address finiteness of number of zeros of
$(Z_{M}H_{M})^{\prime }$
in the half-plane
$\operatorname{Re}(s)<1/2$
, an asymptotic count for the vertical distribution of zeros, and an asymptotic count for the horizontal distance of zeros. One realization of the spectral analysis of the Laplacian is the location of the zeros of
$Z_{M}$
, or, equivalently, the zeros of
$Z_{M}H_{M}$
. Our analysis yields an invariant
$A_{M}$
which appears in the vertical and weighted vertical distribution of zeros of
$(Z_{M}H_{M})^{\prime }$
, and we show that
$A_{M}$
has different values for surfaces associated to two topologically equivalent yet different arithmetically defined Fuchsian groups. We view this aspect of our main theorem as indicating the existence of further spectral phenomena which provides an additional refinement within the set of arithmetically defined Fuchsian groups.</description><subject>Dirichlet problem</subject><subject>Equivalence</subject><subject>Existence theorems</subject><subject>Mathematics</subject><subject>Riemann surfaces</subject><subject>Vertical distribution</subject><issn>0027-7630</issn><issn>2152-6842</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><sourceid>AFKRA</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><recordid>eNptkM1Kw0AUhQdRsFZXvsCAS0m9M5NM0mUaJ-1AmkAy6cLNkF9psT8mLejO1_D1fBKTtuDG1b2c-91z4CB0T2BEgNhPm_VqRIHwkUUv0IASixrcMeklGgBQ27A5g2t007YrAHDYmA3QRxRiNRP4WSYqlpNUyU6IfPwi4ijpl-NRxHLhKrkQvZKIYCLi6c_Xd9JhysV-GnrHPzdJIk-6SjxjFWFfhlIJvIiCdC5wLMXcDUOcpLHveiK5RVd19tZWd-c5RKkvlDczgmgqPTcwCsb43ihZlXOncCipgHOzILwoKS1oQeq8yvOyBnNMMrAri1iWabOSEWBmmZHSBGbZGRuih5Pvrtm-H6p2r1fbQ7PpIjUFsM0xtTl01OOJKppt2zZVrXfNcp01n5qA7qvVXbW6r1ZbtKONM52t82ZZvlZ_pv_xv8K6cY8</recordid><startdate>201712</startdate><enddate>201712</enddate><creator>JORGENSON, JAY</creator><creator>SMAJLOVIĆ, LEJLA</creator><general>Cambridge University Press</general><scope>AAYXX</scope><scope>CITATION</scope><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>AFKRA</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope></search><sort><creationdate>201712</creationdate><title>ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES</title><author>JORGENSON, JAY ; SMAJLOVIĆ, LEJLA</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c336t-d3eb68c821e0664c16cd22c2c1fbebbdf0491a07e5155473d31034da1d40357a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Dirichlet problem</topic><topic>Equivalence</topic><topic>Existence theorems</topic><topic>Mathematics</topic><topic>Riemann surfaces</topic><topic>Vertical distribution</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>JORGENSON, JAY</creatorcontrib><creatorcontrib>SMAJLOVIĆ, LEJLA</creatorcontrib><collection>CrossRef</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central UK/Ireland</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><jtitle>Nagoya mathematical journal</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>JORGENSON, JAY</au><au>SMAJLOVIĆ, LEJLA</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES</atitle><jtitle>Nagoya mathematical journal</jtitle><addtitle>Nagoya Math. J</addtitle><date>2017-12</date><risdate>2017</risdate><volume>228</volume><spage>21</spage><epage>71</epage><pages>21-71</pages><issn>0027-7630</issn><eissn>2152-6842</eissn><abstract>We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface
$M$
. Actually, we study the zeros of
$(Z_{M}H_{M})^{\prime }$
, where
$Z_{M}$
is the Selberg zeta function and
$H_{M}$
is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite volume hyperbolic Riemann surface
$M$
. Our main results address finiteness of number of zeros of
$(Z_{M}H_{M})^{\prime }$
in the half-plane
$\operatorname{Re}(s)<1/2$
, an asymptotic count for the vertical distribution of zeros, and an asymptotic count for the horizontal distance of zeros. One realization of the spectral analysis of the Laplacian is the location of the zeros of
$Z_{M}$
, or, equivalently, the zeros of
$Z_{M}H_{M}$
. Our analysis yields an invariant
$A_{M}$
which appears in the vertical and weighted vertical distribution of zeros of
$(Z_{M}H_{M})^{\prime }$
, and we show that
$A_{M}$
has different values for surfaces associated to two topologically equivalent yet different arithmetically defined Fuchsian groups. We view this aspect of our main theorem as indicating the existence of further spectral phenomena which provides an additional refinement within the set of arithmetically defined Fuchsian groups.</abstract><cop>Cambridge, UK</cop><pub>Cambridge University Press</pub><doi>10.1017/nmj.2016.52</doi><tpages>51</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0027-7630 |
| ispartof | Nagoya mathematical journal, 2017-12, Vol.228, p.21-71 |
| issn | 0027-7630 2152-6842 |
| language | eng |
| recordid | cdi_proquest_journals_2007492760 |
| source | Cambridge Journals |
| subjects | Dirichlet problem Equivalence Existence theorems Mathematics Riemann surfaces Vertical distribution |
| title | ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES |
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