On the zeros of Dirichlet 𝐿-functions. I

Akio Fujii

Type: Article

Publication Date: 1974-01-01

Citations: 26

DOI: https://doi.org/10.1090/s0002-9947-1974-0349603-2

Abstract

A mean value theorem for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="arg upper L left-parenthesis 1 slash 2 plus i left-parenthesis t plus h right-parenthesis comma chi right-parenthesis minus arg upper L left-parenthesis 1 slash 2 plus i t comma chi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>arg</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>+</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>χ</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−</mml:mo> <mml:mi>arg</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>χ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\arg L(1/2 + i(t + h), \chi ) - \arg L(1/2 + it, \chi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is established. This yields mean estimates for the number of zeros of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis s comma chi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>χ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L(s, \chi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in small boxes.

Locations

Transactions of the American Mathematical Society - View - PDF

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