On Certain L-Functions

Abstract

and its Applications, and the Number Theory Foundation. Over 100 mathematicians attended, and there were 23 one-hour lectures. The conference focused on several aspects of the Langlands program, including some exposition of Shahidi's work, recent progress, and future avenues of investigation. Far from being a retrospective, the conference emphasized the vast array of significant problems ahead. All lecturers were invited to contribute material for this volume. In addition, some important figures who were unable to attend or deferred on speaking at the conference were invited to submit articles as well. We hope this resulting volume will serve as a modest tribute to Shahidi's legacy to date, but should not be considered the final word on this subject.The editors wish to thank all of the authors for their willingness to contribute manuscripts of such high quality in honor of our colleague. We also wish to thank the anonymous referees for their conscientious reading of these manuscripts and their helpful comments to authors which have improved the contents. We wish to express our deep gratitude to

Oscillations of Fourier coefficients of modular forms

“…We therefore prove: Remark. This result is partially contained in Shahidi [15,Theorem 5.3]. Our proof is different.…”

“…Our proof is different. For our application, the fact that La(1)4:0 is crucial and this is not contained in [15].…”

Riemann’s zeta function and beyond

“…Through (8.11) and the known holomorphy of E(s, f, g) on the line Re s = 0 (which follows from the general spectral analysis), one also obtains a new proof of the famous result that ζ(s) never vanishes along the line Re s = 1 (see [75], [120], [153], [161]). Among other things, this result is the key to the standard proof of the Prime Number Theorem -as was originally outlined by Riemann himself in [145]!…”

“…There, analysis of the constant term and first Fourier coefficient already sufficed for the analytic continuation and functional equation of ζ(s) via Selberg's method. Langlands proposed studying the non-trivial Fourier coefficients in general, and Shahidi has now worked that theory out ( [161], [162], [163], [164], [165], [166], [167], [168]) along with Kim and others. In general it has been a difficult challenge to prove the L-functions arising in the constant terms and Fourier coefficients are entire.…”

Infinite Dimensional Groups and Automorphic L-Functions

“…We shall assume π is not monomial. It then follows that either Theorem 5.1 is proved by applying a version of converse theorems of Cogdell and Piatetski-Shapiro [10,11] to certain triple product L-functions L(s, (π 1 ⊠ π 2 ) × σ) whose analytic properties are obtained from the Langlands-Shahidi method [19,27,48,49,50,51,52].…”