On Cornacchia's Algorithm
We give an endorsement for Cornacchia's famous algorithm. Thus we do not claim anything new but an approach which is supposed to be simpler than those of previous works written with the same aim.
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arXiv subjects
Yoichi Motohashi
Publications and source records attributed to Yoichi Motohashi.
The twin prime conjecture
This is an exposition of recent developments in the theory of bounded differences between primes. Readers are expected to be beginners of analytic number theory. The present text is a substantially improved and augmented version of the one that I had prepared for my talk which I delivered at the Annual Meeting of the Mathematical Society of Japan on 15 March 2014.
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Spectral Analysis of the Zeta and L-Functions
This is my talk delivered at the workshop 'Automorphic L-Functions and related prpblems' (March 10--13, 2012, Tokyo University). We showed an instance of applications of the theory of automorphic representations to a genuinely traditional problem in the theory of the zeta and allied functions. We restricted ourselves to very basic issues and results, because of the purpose of the workshop.
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On sums of Hecke-Maass eigenvalues squared over primes in short intervals
We prove a uniform estimate for sums of Hecke--Maass eigenvalues squared over primes in short intervals that can be regarded as an analogue of Hoheisel's classical prime number theorem for all real analytic cusp forms. Our argument is modelled after our treatment of Linnik's least prime number theorem for arithmetic progressions (Tata LN 72) and depends on recent works in the theory of automorphic representations. We stress that constants in the present work, including those implicit, are all universal and effectively computable, although we pay no particular attention to numerical precision. In this third version, we have made an improvement upon the main assertion and added greater details. In this new version some augmentation is made concerning the bounds for logarithmic derivatives of relevant symmetric L-functions. It is, however, only to make our argument more accessible.
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A Smoothed GPY Sieve
We show a smoothed version of Goldston-Pintz-Yildirim's sifting argument to detect small gaps between primes, which has a higly flexible error term. Our argument is applicable to high dimensional Selberg sieve situations as well, although the relevant details are not stated explicitly. In the present v.2, we have added Appendix in which we made a minor correction to our reasoning following formula (5.14). We are indebted to Professor Terrence Tao for pointing out the necessity of this amendment. The text of the original version has not been changed, except for a correction of the title in the item [2] of the references.
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An extension of the Linnik phenomenon. II
The (Deuring-Heilbronn-) Linnik phenomenon is extended to L-functions associated with real analytic automorphic forms. The repelling effect of exceptional zeros of Dirichlet L-functions are felt not only by those L-functions themselves but also by automorphic L-functions. This version 2 contains substantial changes of notation as well as a few corrections.
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An Extension of the Linnik Phenomenon
The Linnik phenomenon concerning the repelling effect of Siegel's zeros is extended to L-functions in a family wider than hitherto considered.
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On Some Improvements of the Brun-Titchmarsh Theorem. IV
We show a new large sieve version of the Brun-Titchmarsh theorem.
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Elements of Automorphic Representations
This is an attempt at a practical and essentially self-contained theory of automorphic representations in the framework $$\hbox{$L^2(\varGamma\backslashrG)$ with $rG=\r{PSL}(2,\B{R})$ and $\varGamma=\r{PSL}(2,\B{Z})$.}$$
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Spectral Theory of the Riemann Zeta-Function: Chapter 6: Appendix
The main aim of this article is to develop, in a fully detailed fashion, a {\bf unified} theory of the spectral theory of mean values of individual automorphic L-functions which is a natural extension of the fourth moment of the Riemann zeta-function but does not admit any analogous argument and requires a genuinely new method. Thus we first develop a relatively self-contained account of the theory of automorphic representations, especially highlighting the Kirillov model, with which we resolve the problem on the mean value of those L-functions. As another reward, we gain a geometrical understanding of sum formulas involving Kloosterman sums, which is in fact a considerably simplified account of Cogdell-Piatetski-Shapiro's method. Our reasoning is quite explicit in contrast to theirs.
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The Riemann Zeta-Function and Hecke Congruence Subgroups. II
This is a rework of our old file, which has been left unpublished since September 1994, on an explicit spectral decomposition of the fourth power moment of the Riemann zeta-function against a weight which is the square of a Dirichlet polynomial. At this occasion we add an explicit treatment of generalized Kloosterman sums associated with arbitrary Hecke congruence subgroups (Section 15), which might have an independent interest. At the end (Section 36) of our discussion, we set out a few problems on the distribution of eigenvalues of the hyperbolic Laplacian, which appear to us to be related to the nature of the sixth power moment of the Riemann zeta-function. The contents of this work were presented in a worshop at RIMS Kyoto University on October 18, 2007. In this second version, some corrections are made in the part on generalized Kloosterman sums, and in Addendum a mention is made concerning a recent work by C.P. Hughes and M.P. Young (0709.2345).
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A note on the mean value of the zeta and L-functions. XV
The aim of the present article is to render the spectral theory of mean values of automorphic $L$-functions -- in a unified fashion. This is an outcome of the investigation commenced with the parts XII and XIV, where a framework was laid on the basis of the theory of automorphic representations and a general approach to the mean values was envisaged. We restrict ourselves to the situation offered by the full modular group, solely for the sake of simplicity. Details and extensions are to be published elsewhere. In this new version we added details to all aspects of our argument especially for the treatment of the discrete series and the continuous spectrum.
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Mean Values of Zeta-Functions via Representation Theory
The aim of the present article is to reveal a structure shared by two basic zeta-functions in their fourth power moments through the view point of representation theory of Lie groups, relying specifically upon the Kirillov model. It might induce one to ponder over the possibility to go beyond.
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Prime numbers -- your gems
Prime numbers or primes are man's eternal treasures that have been cherished for several millennia, until today. As their academic ancestors in ancient Mesopotamia, many mathematicians are still trying hard to see primes better. I shall relate here a part of my impressions that I have gathered in a corner of my mind through my own research and excursions outside my profession. This is in essence a translation of my Japanese article that was prepared for my public talk at the general assembly of the Mathematical Society of Japan, Spring 2005, and is now available on the home page of MSJ. At this opportunity I have made substantial revisions, and changed the title into a more appropriate one. This is also an enlarged version of my public talk to be delivered at TECHNION, Haifa, on Dec 20, 2005.
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Uniform Bound for Hecke L-Functions
Our principal aim in the present article is to establish a uniform hybrid bound for individual values on the critical line of Hecke $L$-functions associated with cusp forms over the full modular group. This is rendered in the statement that for $t\ge0$ $$ \eqalignno{H_j(\txt{1\over2}+it)&\ll(κ_j+t)^{1/3+ε},& (1.1)\cr H_{j,k}(\txt{1\over2}+it)&\ll (k+t)^{1/3+ε},&(1.2)\cr} $$ with the common notation to be made precise in the course of discussion. Talks on this and relevant results were delivered by the authors at MF Oberwolfach on September 20, 2004, and at General Assembry of Math. Soc. Japan on March 30, 2005.
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The moments of the Riemann zeta-function. Part I: The fourth moment off the critical line
In this paper, the first part of a larger work, we prove the spectral decomposition of $$ \int_{-\infty}^\infty|ζ(\s+it)|^4g(t){\rm d}t\qquad(\hf < σ< 1 {\rm {fixed}}), $$ where $g(t)$ is a suitable weight function of fast decay. This is used to obtain estimates and omega results for the function $$\eqalign{E_2(T,σ) &: =\int_0^T|ζ(σ+it)|^4{rm d}t - {ζ^4(2σ)\overζ(4σ)}T -{T\over3-4σ}{({T\over2π} )}^{2-4σ}{ζ^4(2-2σ)\overζ(4-4σ)}\cr& - T^{2-2σ}(a_0(σ) + a_1(σ)\log T + a_2(σ)\log^2T),\cr} $$ the error term in the asymptotic formula for the fourth moment of $|ζ(σ+it)|$.
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A New Approach to the Spectral Theory of the Fourth Moment of the Riemann Zeta-Function
The aim of the present work is to exhibit a new proof of the explicit spectral expansion for the fourth moment of the Riemann zeta-function that was established by the second named author a decade ago. Our proof is new, particularly in the sense that it dispenses completely with the Kloostermania, the spectral theory of sums of Kloosterman sums that was used in the former proof. The argument is now constructed precisely upon the spectral structure of the Lie group PSL(2,R). Main ingredients in our argument are the theory of automorphic representations as well as the harmonic analysis on the big Bruhat cell. In essence, this work of ours indicates a new way to view the Riemann zeta-function.
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A note on the mean value of the zeta and L-functions. XIV
The aim of the present note is to develop a study on the feasibility of a unified theory of mean values of automorphic L-functions, a desideratum in the field. This is an outcome of the investigation commenced with Part XII of this series, where a framework was laid on the basis of the theory of automorphic representations, and a general approach to the mean values was envisaged. Specifically, it is shown here that the inner-product method, which was initiated by A. Good and greatly enhanced by M. Jutila, ought to be brought to perfection so that the mean square of the L-function attached to any cusp form on the upper half-plane could be reached within the notion of automorphy. The Kirillov map is our key implement. Because of its geometric nature, our argument appears to extend to bigger linear Lie groups. This paper is essentially self-contained.
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