DOCSLIB.ORG

Distinct Zeros of Functions in the Selberg Class 1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Distinct Zeros of Functions in the Selberg Class 1 Distinct zeros of functions in the Selberg Class 1 K. Srinivas 1. Introduction. Selberg [S] defined a class of Dirichlet series that admit analytic continuation, functional equation and an Euler product. The proto- typical example of Selberg's class is the classical Riemann zeta function. In the same paper, Selberg formulated several fundamental conjectures. These conjectures have spectacular consequences ( see for example the paper by Murty [MR] and Kaczorowski and Perelli [KP] ). In this paper we study a question suggested by the properties of the Selberg class. More precisely, the Selberg class S ( see [S] ) is defined by the following axioms: P1 −s (i) ( Dirichlet series ) Every F 2 S is a Dirichlet series F (s) = n=1 a(n)n ; ( with s = σ + it) absolutely convergent for σ > 1: (ii) (Analytic continuation) There exists an integer m ≥ 0 such that (s − 1)mF (s) is an entire function of finite order . 1AMS classification 11M41 (11M26) (iii)( Functional equation ) F 2 S satisfies a functional equation of the type '(s) = w'(1 − s) where d s Y '(s) = Q Γ(λis + µi)F (s); i=1 here w is a complex number with absolute value 1, Q(> 0); λi(> 0); <µi(≥ 0) are certain contants. (iv) (Ramanujan Hypothesis ) For every > 0; a(n) = O(n): P1 −s (v) (Euler product) F 2 S satisfies log F (s) = n=1 b(n)n ; where m θ 1 b(n) = 0; unless n = p with m ≥ 1; and b(n) = O(n ) for some θ < 2 . Pd For F 2 S define deg F = 2 i=1 λi to be the degree of F . Let F; G 2 S . Define the counting function D(T ; F; G) of the distinct non-trivial zeros, counted with multiplicity, of the functions F and G by X D(T ; F; G) = max(mF (ρ) − mG(ρ); 0): 0≤<ρ≤1 0≤=ρ≤T where ρ runs over the zeros of F (s)G(s) and is counted without multiplicity and mF (ρ); mG(ρ) denotes the multiplicities of zeros of F and G respectively. Also define X D(T ) = D(T ; F; G) + D(T ; G; F ) = jmF (ρ) − mG(ρ)j; 0≤<ρ≤1 0≤=ρ≤T 2 with the same convention about ρ. In [MM], Ram Murty and Kumar Murty proved that if F 6= G, then D(T ) = Ω(T ): Assuming that F and G are orthogonal in the sense of Selberg (see [S] ) and a certain density hypothesis, Bombieri and Perelli ( see [BP]) proved that both D(T ; F; G) and D(T ; G; F ) are T log T; provided deg F = deg G. The reader may refer to the excellent survey articles by Kaczorowski and Perelli [KP] and Ram Murty [MR] for general information on Selberg class and also for a discussion on the distinct zeros and independence results for zeros of functions in the Selberg class. In this note we propose to study the gaps between the zeros counted by the function D(T ; F; G): More precisely we prove the following Theorem. Let F and G be two distinct functions in S: Assume that deg F ≥ deg G: Then for every sufficiently large T, there exists a zero ρ = β + iγ of FG with mF (ρ) > mG(ρ) such that jT − γj ≤ C1 log log T; where C1 is a large positive constant . 3 We obtain the following Corollary 1. Under the hypotheses of the theorem, let γ1 ≤ γ2 ≤ · · · denote the ordinates of the zeros of FG with mF (ρi) > mG(ρi); i = 1; 2; ··· : Then γn+1 − γn = O (log log(jγnj + 100)) : Corollary 2. Under the hypotheses of the theorem, T D(T ; F; G) ≥ C ; 2 log log T where C2 > 0 is a constant. Remarks. (1) The global number of distinct zeros given completely uncon- ditionally by Murty and Murty ( [MM] ) is larger than ours. However, their method only deals with the symmetric difference of zeros reflected by the counting function D(T ), whereas the present paper deals with the asymmet- ric difference between the zeros reflected by D(T;F,G), which is a difficult problem and therefore seems interesting. (2) The main idea involved in this paper has its origin in the papers [RS] and [BRSS]1, where it has been shown that under suitable conditions the quotient of finite products of translates of Riemann zeta function has in- 4 finitely many poles. Moreover, the gaps between the poles have also been obtained (see also [BRSS]2, [BRSS]3; for various results in this direction). Acknowledgements. The author thanks Professor Alberto Perelli for proposing the problem, suggesting the relevant literature and encourage- ment. He also thanks Professors R. Balasubramanian, M. Ram Murty and K. Ramachandra for their support. 2. Notation. Throughout the paper the capital letters C1;C2;C3; ··· will denote positive constants. We write f(x) g(x) or f(x) = O(g(x)) to mean jf(x)j < C3g(x). All the implied constants arising from and O are effective. We fix H = D log log T (except in lemma 1) , where D is a large positive constant. 3. Basic Lemmas. We need the following lemmas for the proof of the theorem. Lemma 1. Let A > 0 be an integer constant, B = A + 2; a1; a2; ··· be r 1 A 8 complex numbers with a1 = 1; janj ≤ (n) H ; where 0 < ≤ 2 ; r ≥ −1 1 [(200A + 200) ] , let λ1; λ2; ··· be real numbers with λ1 = 1 and C ≤ P1 an λ −λ ≤ C where C ≥ 1 is a constant. Then f(s) = s is analytic n+1 n n=1 λn 1− 12A in σ ≥ A + 2: Let K ≥ 30;U1 = H 2 : Assume that K1 = (HC) K; H ≥ 5 100 2 2A+4 2 r 20 120(A + 2) C (4rC ) + (100r(A + 2)) log log K1; and that there exist T1;T2 with 0 ≤ T1 ≤ U1;H − U1 ≤ T2 ≤ H such that uniformly in σ ≥ 0 we have j f(σ + iT1) j + j f(σ + iT2 |≤ K; where f(s) is assumed to be analytically continualble in (σ ≥ 0; 0 ≤ t ≤ H): Then Z H 1 2 2 − −1 X 2 j f(it) j dt ≥ 1 − 10rC H 4 − 100BH log log K1 j an j : 0 H n≤H1− Proof. This result is due to R. Balasubramanian and K. Ramachandra. See page 45 of [R]. P1 an Lemma 2. Suppose the Dirichlet series f(s) = n=1 ns ; with an = O(n ) admits an analytic continuation in an infinite system of rectangles R(T − H; T + H) defined by fσ ≥ 0;T − H ≤ t ≤ T + Hg and that j f(s) j= 2 O (exp(C4(log T ) )) there. Then for all H ≥ C5 log log T; we have T +H 1 Z 2 X 2 j f(it) j dt ≥ C6 j an j : H T −H 1 n≤H 2 6 Proof.The proof follows from lemma 1 on taking λn = n; C = 1 and fixing 1 = 2 in lemma 1. Lemma 3. Let F 2 S. If ρ = β + iγ runs through the zeros of F (s), F 0(s) 1 = X + O(log jtj); F (s) jt−γ|≤1 s − ρ uniformly for −2 ≤ σ ≤ 2; jtj ≥ 30: Proof. This result is well known for the Riemann zeta-function. The proof for the functions in S is exactly the same since Riemann-von Mangoldt formula holds for Selberg class. See page 217 of [T]. Lemma 4. Let F 2 S: In any subinterval of length 1 in [T − H; T + H] there are lines t = t0 such that −1 2 jF (σ + it0)j = O(exp(C7(log T ) )); uniformly in σ ≥ −2. Proof. Let I0 denote a typical rectangle of unit height in −2 ≤ σ ≤ 2;T −H ≤ t ≤ T +H. Let ρ = β +iγ denote a zero of F (s) in I0 . Then the number of such zeros , by the Riemann-von Mangoldt formula, is less than C8 log T . Therefore , if we break the rectangle I0 into equal sub-rectangles of height 1 , then by the pigeon-hole principle , there is atleast one 20C8 log T 7 sub-rectangle I1 of I0 where F (s) is zero-free. Let t = t0 denote the middle horizontal line in I1. 1 Clearly jt0 − γj > . 40C8 log T P 1 2 Therefore , jt0−γ|≤1 js−ρj < C9(log T ) : Now from lemma 3, it follows that F 0(σ + it ) 0 2 = O((log T ) ) F (σ + it0) uniformly for σ ≥ −2: Now to complete the proof we observe that for σ ≥ −2, we have Z 2 0 F (r + it0) 2 log F (σ + it0) − log F (2 + it0) = dr = O((log T ) ) σ F (r + it0) and j log jF (σ + it0)j j = j< log F (σ + it0)j ≤ j log F (σ + it0)j: Therefore 1 2 log = O((log T ) ): F (σ + it0) Hence the lemma follows. Lemma 5. Let z = x + iy be a complex variable with jxj ≤ 1=4: Then we have j exp((sin z)2)j ≤ 2 for all y 8 and if jyj ≥ 2, then j exp((sin z)2)j ≤ 2(exp exp jyj)−1: Proof. This is a double order decaying kernel developed and extensively used by Ramachandra ( see lemma 2.1 of [R] for details ). Lemma 6. Let F and G be two distinct functions in S with deg F ≥ G(s−1) P1 ncn deg G: Let f(s) = F (s−1) = n=1 ns , where cn's are the coefficients of the G Dirichlet series for F .
Load more

Recommended publications
  • Selberg's Orthogonality Conjecture and Symmetric
    SELBERG'S ORTHOGONALITY CONJECTURE AND SYMMETRIC POWER L-FUNCTIONS PENG-JIE WONG Abstract. Let π be a cuspidal representation of GL2pAQq defined by a non-CM holomorphic newform of weight k ¥ 2. Let K{Q be a totally real Galois extension with Galois group G, and let χ be an irreducible character of G. In this article, we show that Selberg's orthogonality conjecture implies that the symmetric power L-functions Lps; Symm πq are primitive functions in the Selberg class. Moreover, assuming the automorphy for Symm π, we show that Selberg's orthogonality conjecture implies that each Lps; Symm π ˆ χq is automorphic. The key new idea is to apply the work of Barnet-Lamb, Geraghty, Harris, and Taylor on the potential automorphy of Symm π. 1. Introduction Thirty years ago, Selberg [21] introduced a class S of L-functions, called the Selberg class nowadays. The class S consists of functions F psq of a complex variable s enjoying the following properties: (Dirichlet series and Euler product). For Repsq ¡ 1, 8 a pnq 8 F psq “ F “ exp b ppkq{pks ns F n“1 p k“1 ¸ ¹ ´ ¸ ¯ k kθ 1 with aF p1q “ 1 and bF pp q “ Opp q for some θ ă 2 , where the product is over rational primes p. Moreover, both Dirichlet series and Euler product of F converge absolutely on Repsq ¡ 1. (Analytic continuation and functional equation). There is a non-negative integer mF such that ps ´ 1qmF F psq extends to an entire function of finite order. Moreover, there are numbers rF , QF ¡ 0, αF pjq ¡ 0, RepγF pjqq ¥ 0, so that rF s ΛF psq “ QF ΓpαF pjqs ` γF pjqqF psq j“1 ¹ satisfies ΛF psq “ wF ΛF p1 ´ sq for some complex number wF of absolute value 1.
    [Show full text]
  • Analytic Theory of L-Functions: Explicit Formulae, Gaps Between Zeros and Generative Computational Methods
    Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2016 Analytic theory of L-functions: explicit formulae, gaps between zeros and generative computational methods Kühn, Patrick Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-126347 Dissertation Published Version Originally published at: Kühn, Patrick. Analytic theory of L-functions: explicit formulae, gaps between zeros and generative computational methods. 2016, University of Zurich, Faculty of Science. UNIVERSITÄT ZÜRICH Analytic Theory of L-Functions: Explicit Formulae, Gaps Between Zeros and Generative Computational Methods Dissertation zur Erlangung der naturwissenschaftlichen Doktorwürde (Dr. sc. nat.) vorgelegt der Mathematisch-naturwissenschaftlichen Fakultät der Universität Zürich von Patrick KÜHN von Mendrisio (TI) Promotionskomitee: Prof. Dr. Paul-Olivier DEHAYE (Vorsitz und Leitung) Prof. Dr. Ashkan NIKEGHBALI Prof. Dr. Andrew KRESCH Zürich, 2016 Declaration of Authorship I, Patrick KÜHN, declare that this thesis titled, ’Analytic Theory of L-Functions: Explicit Formulae, Gaps Between Zeros and Generative Computational Methods’ and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree • at this University. Where any part of this thesis has previously been submitted for a degree or any • other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly at- • tributed. Where I have quoted from the work of others, the source is always given. With • the exception of such quotations, this thesis is entirely my own work.
    [Show full text]
  • On the Selberg Class of L-Functions
    ON THE SELBERG CLASS OF L-FUNCTIONS ANUP B. DIXIT Abstract. The Selberg class of L-functions, S, introduced by A. Selberg in 1989, has been extensively studied in the past few decades. In this article, we give an overview of the structure of this class followed by a survey on Selberg's conjectures and the value distribution theory of elements in S. We also discuss a larger class of L-functions containing S, namely the Lindel¨ofclass, introduced by V. K. Murty. The Lindel¨ofclass forms a ring and its value distribution theory surprisingly resembles that of the Selberg class. 1. Introduction The most basic example of an L-function is the Riemann zeta-function, which was intro- duced by B. Riemann in 1859 as a function of one complex variable. It is defined on R s 1 as ∞ 1 ( ) > ζ s : s n=1 n It can be meromorphically continued to( the) ∶= wholeQ complex plane C with a pole at s 1 with residue 1. The unique factorization of natural numbers into primes leads to another representation of ζ s on R s 1, namely the Euler product = −1 1 ( ) ( ) > ζ s 1 : s p prime p The study of zeta-function is vital to( ) understanding= M − the distribution of prime numbers. For instance, the prime number theorem is a consequence of ζ s having a simple pole at s 1 and being non-zero on the vertical line R s 1. ( ) = In pursuing the analogous study of distribution( ) = of primes in an arithmetic progression, we consider the Dirichlet L-function, ∞ χ n L s; χ ; s n=1 n ( ) for R s 1, where χ is a Dirichlet character( ) ∶= moduloQ q, defined as a group homomorphism ∗ ∗ χ Z qZ C extended to χ Z C by periodicity and setting χ n 0 if n; q 1.
    [Show full text]
  • Arxiv:1711.06671V1 [Math.NT] 17 Nov 2017 Function Enetnieysuid[1 Nrcn Ie.Tesuyo Au Dist Value of Study the of Times
    VALUE DISTRIBUTION OF L-FUNCTIONS ANUP B. DIXIT Abstract. In 2002, V. Kumar Murty [8] introduced a class of L-functions, namely the Lindel¨of class, which has a ring structure attached to it. In this paper, we establish some results on the value distribution of L-functions in this class. As a corollary, we also prove a uniqueness theorem in the Selberg class. 1. Introduction In 1992, Selberg [10] formulated a class of L-functions, which can be regarded as a model for L-functions originating from arithmetic objects. The value distribution of such L-functions has been extensively studied [11] in recent times. The study of value distribution is concerned with the zeroes of L-functions and more generally, with the set of pre-images L−1(c) := {s ∈ C : L(s)= c} where c is any complex number, which Selberg called the c-values of L. For any two meromorphic functions f and g, we say that they share a value c ignoring multiplicity(IM) if f −1(c) is the same as g−1(c) as sets. We further say that f and g share a value c counting multiplicity(CM) if the zeroes of f(x) − c and g(x) − c are the same with multiplicity. The famous Nevanlinna theory [9] establishes that any two meromorphic functions of finite order sharing five values IM must be the same. Moreover, if they share four values CM, then one must be a M¨obius transform of the other. The numbers four and five are the best possible for meromorphic functions.
    [Show full text]
  • Uncoveringanew L-Function Andrew R
    UncoveringaNew L-function Andrew R. Booker n March of this year, my student, Ce Bian, plane, with the exception of a simple pole at s = 1. announced the computation of some “degree Moreover, it satisfies a “functional equation”: If 1 3 transcendental L-functions” at a workshop −s=2 γ(s) = ÐR(s) := π Ð (s=2) and Λ(s) = γ(s)ζ(s) at the American Institute of Mathematics (AIM). This article aims to explain some of the then I (1) (s) = (1 − s). motivation behind the workshop and why Bian’s Λ Λ computations are striking. I begin with a brief By manipulating the ζ-function using the tools of background on L-functions and their applications; complex analysis (some of which were discovered for a more thorough introduction, see the survey in the process), one can deduce the famous Prime article by Iwaniec and Sarnak [2]. Number Theorem, that there are asymptotically x ≤ ! 1 about log x primes p x as x . L-functions and the Selberg Class Other L-functions reveal more subtle properties. There are many objects that go by the name of For instance, inserting a multiplicative character L-function, and it is difficult to pin down exactly χ : (Z=qZ)× ! C in the definition of the ζ-function, what one is. In one of his last published papers we get the so-called Dirichlet L-functions, [5], the late Fields medalist A. Selberg tried an Y 1 L(s; χ) = : axiomatic approach, basically by writing down the 1 − χ(p)p−s common properties of the known examples.
    [Show full text]
  • Arxiv:1904.03123V2 [Math.NT] 30 Oct 2019 N Any and J Uhta Nteregion the in That Such Then Hoe 1.1
    ZEROS OF THE EXTENDED SELBERG CLASS ZETA-FUNCTIONS AND OF THEIR DERIVATIVES RAMUNAS¯ GARUNKSTISˇ Abstract. Levinson and Montgomery proved that the Riemann zeta-function ζ(s) and its derivative have approximately the same number of non-real zeros left of the critical line. R. Spira showed that ζ′(1/2+ it) = 0 implies ζ(1/2+ it) = 0. Here we obtain that in small areas located to the left of the critical line and near it the functions ζ(s) and ζ′(s) have the same number of zeros. We prove our result for more general zeta-functions from the extended Selberg class S. We also consider zero trajectories of a certain family of zeta-functions from S. 1. Introduction Let s = σ + it. In this paper, T always tends to plus infinity. Speiser [17] showed that the Riemann hypothesis (RH) is equivalent to the absence of non-real zeros of the derivative of the Riemann zeta-function ζ(s) left of the critical line σ = 1/2. Later on, Levinson and Montgomery [11] obtained the quantitative version of the Speiser’s result: Theorem (Levinson-Montgomery) Let N −(T ) be the number of zeros of ζ(s) − ′ in R :0 < t < T, 0 <σ< 1/2. Let N1 (T ) be the number of zeros of ζ (s) in R. − Then N1 (T )= N(T )+ O(log T ). − Unless N (T ) > T/2 for all large T there exists a sequence Tj , Tj as j such that N −(T )= N −(T ). { } →∞ →∞ 1 j j arXiv:1904.03123v2 [math.NT] 30 Oct 2019 Here we prove the following theorem.
    [Show full text]
  • Arxiv:1506.07630V2 [Math.NT] 31 Mar 2016 Itiuino H Eo N Ihidpnec Results
    SOME REMARKS ON THE CONVERGENCE OF THE DIRICHLET SERIES OF L-FUNCTIONS AND RELATED QUESTIONS J.KACZOROWSKI and A.PERELLI Abstract. First we show that the abscissae of uniform and absolute convergence of Dirichlet series coincide in the case of L-functions from the Selberg class S. We also study the latter abscissa inside the extended Selberg class, indicating a different behavior in the two classes. Next we address two questions about majorants of functions in S, showing links with the distribution of the zeros and with independence results. Mathematics Subject Classification (2000): 11M41, 30B50, 40A05, 42A75 Keywords: Dirichlet series, Selberg class, almost periodic functions 1. Introduction Let ∞ a(n) F (s)= ns n=1 be a Dirichlet series which converges somewhereX in the complex plane. It is well known that there are four classical abscissae associated with F (s): the abscissa of convergence σc(F ), of uniform convergence σu(F ), of absolute convergence σa(F ) and of boundedness σb(F ). It may well be that σc(F )= −∞, in which case the other three abscissae equal −∞ as well. From the theory of Dirichlet series we know that σc(F ) ≤ σb(F )= σu(F ) ≤ σa(F ), and in general this is best possible, i.e. inequalities cannot be replaced by equalities. We refer to Maurizi-Queff´elec [15] for a modern reference for this sort of problems. Our first result is that σb(F )= σa(F ) for an important class of Dirichlet series, namely those defining the L-functions of the Selberg class S. We recall that the axiomatic class S contains, at least conjecturally, most L-functions from number theory and automorphic forms theory, arXiv:1506.07630v2 [math.NT] 31 Mar 2016 and that σb(F )= σa(F ) is known in some special cases like the Riemann or the Dedekind zeta functions.
    [Show full text]
  • LARGE VALUES of L-FUNCTIONS from the SELBERG CLASS 3 Stronger Form
    LARGE VALUES OF L-FUNCTIONS FROM THE SELBERG CLASS CHRISTOPH AISTLEITNER ANDLUKASZ PANKOWSKI´ Abstract. In the present paper we prove lower bounds for L-functions from the Selberg class, by this means improving earlier results obtained by the sec- ond author together with J¨orn Steuding. We formulate two theorems which use slightly different technical assumptions, and give two totally different proofs. The first proof uses the “resonance method”, which was introduced by Soundararajan, while the second proof uses methods from Diophantine ap- proximation which resemble those used by Montgomery. Interestingly, both methods lead to roughly the same lower bounds, which fall short of those known for the Riemann zeta function and seem to be difficult to be improved. Additionally to these results, we also prove upper bounds for L-functions in the Selberg class and present a further application of a theorem of Chen which is used in the Diophantine approximation method mentioned above. 1. Introduction. It is well known that the absolute value of the Riemann zeta function ζ(σ + it) takes arbitrarily large and arbitrarily small values when t runs through the real numbers and σ [1/2, 1) is a fixed real number. However, the growth of the Riemann zeta function∈ as a function of t (for fixed σ) cannot be too fast, since its absolute value is bounded by a power of t. More precisely, if µζ (σ) denotes the infimum over all c 0 satisfying ζ(σ + it) tc for sufficiently large t, then one can show that µ (σ≥) (1 σ)/2 for 0 σ≪ 1.
    [Show full text]
  • LINEAR INDEPENDENCE in the SELBERG CLASS by J.KACZOROWSKI, G.MOLTENI and A.PERELLI C
    LINEAR INDEPENDENCE IN THE SELBERG CLASS by J.KACZOROWSKI, G.MOLTENI and A.PERELLI C. R. Math. Acad. Sci. Soc. R. Can., 21(1):28–32, 1999 Abstract. We prove that distinct functions in the Selberg class S are linearly independent over the ring F of p-finite Dirichlet series. As a consequence, S has unique factorization if and only if primitive functions of S are algebraically independent over F. R´esum´e. Nous montrons que les fonctions dans la classe de Selberg S sont lin´eairement ind´ependantes sur l’anneau F des s´eries de Dirichlet p-finies. Comme cons´equence, S a une factorization unique si et seulement si les fonctions primitives de S sont alg´ebriquement ind´ependantes sur F. AMS 1991 Mathematics Subject Classification: 11 M 41 Roughly speaking, the main conjecture about the Selberg class S (see the survey paper [4] for the basic definitions, conjectures and properties of S) is that S can be essentially identified with the class of suitably normalized automorphic L-functions. Based on such conjecture, one may guess that S contains only countably many essentially different functions, although S itself is certainly not countable since it contains all shifts Fθ(s) = F (s + iθ), θ ∈ R, of any entire function F ∈ S. However, one may guess that S becomes a countable set provided shifts are not counted. This is the countability problem for S. This problem was communicated to us by Peter Sarnak, and we now understand that it was independently raised also by Ram Murty and Eduard Wirsing.
    [Show full text]
  • Selberg Conjectures and Artin L-Functions
    APPEARED IN BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 31, Number 1, July 1994, Pages 1-14 SELBERG’S CONJECTURES AND ARTIN L-FUNCTIONS M. Ram Murty 1. Introduction In its comprehensive form, an identity between an automorphic L-function and a “motivic” L-function is called a reciprocity law. The celebrated Artin reciprocity law is perhaps the fundamental example. The conjecture of Shimura-Taniyama that every elliptic curve over Q is “modular” is certainly the most intriguing reciprocity conjecture of our time. The “Himalayan peaks” that hold the secrets of these nonabelian reciprocity laws challenge humanity, and, with the visionary Langlands program, we have mapped out before us one means of ascent to those lofty peaks. The recent work of Wiles suggests that an important case (the semistable case) of the Shimura-Taniyama conjecture is on the horizon and perhaps this is another means of ascent. In either case, a long journey is predicted. To paraphrase the cartographer, it is not a journey for the faint-hearted. Indeed, there is a forest to traverse, “whose trees will not fall with a few timid blows. We have to take up the double-bitted axe and the cross-cut saw and hope that our muscles are equal to them.” At the 1989 Amalfi meeting, Selberg [S] announced a series of conjectures which looks like another approach to the summit. Alas, neither path seems the easier climb. Selberg’s conjectures concern Dirichlet series, which admit analytic contin- uations, Euler products, and functional equations. The Riemann zeta function is the simplest example of a function in the family of functions F (s) of a complex variable s satisfying the following properties: S s (i) (Dirichlet series) For Re(s) > 1, F (s)= n∞=1 an/n ,wherea1 =1,andwe will write an(F )=an for the coefficients of the Dirichlet series.
    [Show full text]
  • L-Functions of the Selberg Class S
    A CONCISE SURVEY OF THE SELBERG CLASS OF L-FUNCTIONS LI ZHENG ABSTRACT. In this survey paper, I first present some classical L-functions and its basic properties. Then I give the introduction of Selberg class of L-functions, and present some basic properties, important conjec- tures and consequences, and the relation with prime number theorem. Ever since Riemann’s revolutionary paper [1], the Riemann zeta function and its various generaliza- tions have been extensively studied by mathematicians for over a century. These functions are generally referred to as L-functions. Deep connections have been established between the properties of the L- functions and other theories (for example, prime number theory). Later in 1992, in attempt to capture the core properties of classical L-functions, Selberg gave an axiomatic characterization of what would be called general L-functions. This is paper is a concise survey for Selberg class of L-functions. 1. CLASSICAL L-FUNCTIONS In this section we will recall some common properties shared by a lot of classical L-functions. Proofs and details will be avoided; references will be provided. Also, we take the convention to write the variable s as σ Å i t. Example 1. Talking about L-functions, the first one to come to mind is of course Riemann’s ³ function, σ È which is defined, for 1, X Y ¡ ¡ ¡ ³(s) Æ n s Æ (1 ¡ p s) 1. n¸1 p It has a meromorphic continuation to the complex plane C, having a unique pole at s Æ 1. Setting ¡ ©(s) Æ ¼ s/2¡(s/2)³(s), we have the functional equation ©(s) Æ ©(1 ¡ s).
    [Show full text]
  • The Selberg Class of Zeta- and L-Functions
    MONASTIR MINI-COURSE: THE SELBERG CLASS OF ZETA- AND L-FUNCTIONS JORN¨ STEUDING ”What is a zeta-function (an L-function)? We know one when we see one! ” This quotation is attributed to M.N. Huxley and reflects the number-theoretical prob- lem to classify those generating functions which carry arithmetically relevant data. In 1989, Selberg introduced a general class of Dirichlet series having an Euler product S representation, analytic continuation and a functional equation of Riemann-type, and for- mulated some fundamental conjectures. In only twenty years this so-called Selberg class of L-functions became an important branch of research in Analytic Number Theory with im- portant arithmetical applications. It is widely expected that the Selberg class contains all aritmetically important L-functions, moreover, that it consists of exactly all automorphic L-functions. In this mini-course, consisting of six lectures, we give an introduction to this topic. Our focus is on general methods, e.g., Hecke’s approach to functional equations and analytic continuation, Tauberian theorems to deduce information about prime numbers, as well as linear and non-linear twists, a powerful tool recently invented by Kaczorowski & Perelli to investigate the structure of the Selberg class. The course is mainly based on the excellent surveys of Kaczorowski [20] and Perelli [42, 43], and the monographs [38, 54] of M.R. Murty & V.K. Murty and the author, respectively. Basic knowledge in Real and Complex Analysis is expected, and a background from Number Theory is useful, although we have added several appendices in order to present a self-contained introduction.
    [Show full text]