## Evidence for a Spectral Interpretation of the Zeros of L-Functions (1998)

Citations: | 33 - 7 self |

### BibTeX

@TECHREPORT{Rubinstein98evidencefor,

author = {Michael Oded Rubinstein},

title = {Evidence for a Spectral Interpretation of the Zeros of L-Functions},

institution = {},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

By looking at the average behavior (n-level density) of the low lying zeros of certain families of L-functions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the non-trivial zeros in terms of the classical compact groups. This is further supported by numerical experiments for which an efficient algorithm to compute L-functions was developed and implemented. iii Acknowledgements When Mike Rubinstein woke up one morning he was shocked to discover that he was writing the acknowledgements to his thesis. After two screenplays, a 40000 word manifesto, and many fruitless attempts at making sushi, something resembling a detailed academic work has emerged for which he has people to thank. Peter Sarnak- from Chebyshev's Bias to USp(1). For being a terrific advisor and teacher. For choosing problems suited to my talents and involving me in this great project to understand the zeros of L-functions. Zeev Rudnick and Andrew Oldyzko for many disc...

### Citations

392 | Modular elliptic curves and Fermat’s last theorem
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(Show Context)
Citation Context ...nds on E, is either \Sigma1. The Hasse-Weil conjecture and also the required rate of growth on LE (s) follows from the Shimura-Taniyama-Weil conjecture, which has been proven by Wiles and Taylor [36] =-=[40]-=- for elliptic curves with square free conductor (and, apparently, has been 3.3. Computing (s) 73 extended, in recent work of Conrad, Diamond, and Taylor to N 's which are not divisable by 27). Hence, ... |

285 |
La conjecture de Weil
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Citation Context ...and associate to f(z) the Dirichlet series L f (s) := 1 X 1 a n n (k\Gamma1)=2 n \Gammas : We normalize f so that a 1 = 1. This series converges absolutely when !(s) ? 1 because, as proven by Deligne =-=[8]-=-, ja n jsoe 0 (n)n (k\Gamma1)=2 ; (3.3.25) 3.3. Computing (s) 71 where oe 0 (n) := P djn 1 (= O(n ffl ) for any ffl ? 0). L f (s) admits to an analytic continuation to all of C and satisfies the funct... |

239 | Ring-theoretic properties of certain Hecke algebras
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(Show Context)
Citation Context ... depends on E, is either \Sigma1. The Hasse-Weil conjecture and also the required rate of growth on LE (s) follows from the Shimura-Taniyama-Weil conjecture, which has been proven by Wiles and Taylor =-=[36]-=- [40] for elliptic curves with square free conductor (and, apparently, has been 3.3. Computing (s) 73 extended, in recent work of Conrad, Diamond, and Taylor to N 's which are not divisable by 27). He... |

175 |
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Citation Context ...From (3.3.10) with a = 1, we have that !ffi 1=fl ? 0, so both (nffi=Q) 1=fl and (n=(ffiQ)) 1=fl have positive ! part. Examples 1) Riemann zeta function, i(s): the necessary background can be found in =-=[37]-=- Formula (3.3.20), for i(s), is \Gammas=2 \Gamma(s=2)i(s)ffi \Gammas = \Gamma 1 s \Gamma ffi \Gamma1 1 \Gamma s + 1 X n=1 G \Gamma s=2; n 2 ffi 2 \Delta +ffi \Gamma1 1 X n=1 G \Gamma (1 \Gamma s)=2; n... |

165 |
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Citation Context ...ammaw) M G(z \Gamma M;w): Both (3.3.43) and (3.3.45) are well known and are obtained via integration by parts. The second expression (3.3.44) is less known and due to Nielsen (a proof can be found in =-=[10]-=-). It is especially well suited when fl = 1 since then, in (3.3.20), the G(\Delta; \Delta)'s have their second variable in arithmetic progression and this fact can be exploited to speed up the algorit... |

111 |
Zeros of principal L-functions and random matrix theory
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Citation Context ... (for !s sufficiently large) as an Euler product of the form L(s; ) = Y p L(s;sp ) = Y p M Y j=1 (1 \Gamma ffs(p; j)p \Gammas ) \Gamma1 : (1.0.1) Basic properties of such L-functions are described in =-=[33]-=-. The L-functions that arise in the m = 1 case are the Riemann zeta function i(s), and Dirichlet L-functions L(s; ),sa primitive character. For m = 2, the L-functions in question are associated to cus... |

92 | Elliptic Curves - Knapp - 1992 |

86 |
Notes on elliptic curves
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Citation Context ...ly (computed using the PARI number theory package), hence the notation. The ranks are r = 0; 1; 0 respectively (the first two are computed in Fermigier [11], and the last in Birch and Swinnerton-Dyer =-=[3]-=-). Furthermore, E 36 is a curve with complex multiplication. The Ramanujan L-function, Ls(s), has a Dirichlet series given by Ls(s) = 1 X n=1 (n) n 11=2 n \Gammas ; !s ? 1 where 1 X n=1 (n)q n = q 1 Y... |

81 | On the distribution of spacings between zeros of the zeta function
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Citation Context ...ee [22] for a more precise definition), has the same pair correlation (as N ! 1), seems to suggest that the relevant operator, at least for i(s), might be Hermitian. Extensive computations of Odlyzko =-=[25]-=- [26] further seem to bolster the Hermitian nature of the zeros of i(s), as might the work of Rudnick and Sarnak [33] where, under certain restrictions, the n-level correlations of i(s) and L(s; ) are... |

70 |
The pair correlation of zeros of the zeta function. In Analytic number theory (Proc
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(Show Context)
Citation Context ...e operator acting on some Hilbert space, thus (depending on the properties of the operator) forcing the zeros to lie on a line. The first evidence in favor of this approach was obtained by Montgomery =-=[23]-=- who derived (under certain restrictions) the pair correlation ((1.2.1) with n = 2) of 1 2 the zeros of i(s). Together with an observation of Freeman Dyson, who pointed out that the Gaussian Unitary E... |

49 | The Distribution of Prime Numbers - Ingham - 1990 |

42 |
Automorphic Forms and Representations, Cambridge Stud
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Citation Context ...sd (n) = \Gamma d n \Delta is Kronecker's symbol. Thensandsextend to entire functions and satisfy the functional equations (s;sd ) = (1 \Gamma s;sd )s(s;sd ) =sd (\Gamma1)s(1 \Gamma s;sd ): (see [6], =-=[4]-=-). Note that Ls(s;sd ) has a zero at s = 1=2 ifsd (\Gamma1) = \Gamma1. We collected the first few zeros of various L(s;sd )'s and Ls(s;sd )'s by looking for sign changes of the L-function (rotated so ... |

35 | Ramanujan’s Notebooks, part II - Berndt - 1989 |

31 |
An Elementary Introduction to the Langlands program
- Gelbart
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(Show Context)
Citation Context ...y evaluations of (s) 86 viii Dedicated to my sister Ronit ix Chapter 1 Introduction In this thesis, I obtain evidence for a spectral interpretation of the zeros of L-functions. The Langland's program =-=[12]-=- [24] [19] predicts that all L-functions can be written as products of i(s) and L-functions attached to automorphic cuspidal representations of GLM over Q . Such an L-function is given intially (for !... |

28 |
On the zeros of the Riemann zeta function in the critical strip
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(Show Context)
Citation Context ...ters were invented. See Edwards [9] for a historical survey of these computations. To date, the most impressive computations for i(s) have been those of Odlyzko [26] and Van de Lune, te Riele, Winter =-=[39]-=-. The latter were interested in verifying the Riemann Hypothesis and their computations showed that the first 1:5 \Delta 10 9 nontrivial zeros of i(s) fall on the critical line. Odlyzko's computations... |

25 | Numerical computations concerning the ERH - Rumely - 1993 |

24 |
Modular Forms and Dirichlet Series
- Ogg
(Show Context)
Citation Context ... 1=2; n 2 ffi 2 =q \Delta + () 1=2 iqffi 2 1 X n=1 (n)nG \Gamma (1 \Gamma s)=2 + 1=2; n 2 =(ffi 2 q) \Delta (3.3.24) Here, () is the Gauss sum () = q X m=1 (m)e 2im=q : 3) Cusp form L-functions: (see =-=[27]-=-). Let f(z) be a cusp form of weight k for SL 2 (Z), k a positive even integer: 1. f(z) is entire on H , the upper half plane. 2. f(oez) = (cz + d) k f(z), oe = / a b c d ! 2 SL 2 (Z), z 2 H . 3. lim ... |

23 |
Symmetric Square L-Functions on GL(r
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(Show Context)
Citation Context ....6. Analogous results for GLM =Q 53 The conditions= ~simplies that ffs(p; j) 2 R. The Rankin-Selberg L-function L(s; \Omega ~ ) factors as the product of the symmetric and exterior square L-functions =-=[5]-=- L(s; \Omega ~ ) = L(s; \Omega ) = L(s; ;s2 )L(s; ;s2 ) and has a simple pole at s = 1 which is carried by one of the two factors. Write the order of the pole of L(s; ;s2 ) as (ffi() + 1)=2 (so that f... |

12 |
Small zeros of quadratic L-functions
- Özlük, Snyder
- 1993
(Show Context)
Citation Context ...atistic for zeros of L-functions, one finds the fingerprints of the classical compact groups. For n = 1 this was done, for quadratic twists of i(s), and with certain restrictions, by Ozluk and Snyder =-=[29]-=-. A stronger result (which takes into account certain non-diagonal contributions and allows one to choose test funtions whose fourier transform is supported in (\Gamma2=M; 2=M)) which applies for i(s)... |

11 |
On computing Artin L-functions in the critical strip
- Lagartas, Odlyzko
- 1979
(Show Context)
Citation Context ...boring zeros. Yoshida [42] [41] has also used summation by parts (though in a different manner) to compute the first few zeros of certain higher degree (M = 2; 3; 4) L-functions. Lagarias and Odlyzko =-=[21]-=- have computed the low lying zeros of several Artin L-functions. More recently, Fermigier [11] computed the zeros to height T = 15, of several hundred L-functions attached to elliptic curves. Both use... |

10 |
Zéros des Fonction L de Courbes Elliptiques
- Fermigier
- 1992
(Show Context)
Citation Context ... These have conductors Q = 11; 37; 36 respectively (computed using the PARI number theory package), hence the notation. The ranks are r = 0; 1; 0 respectively (the first two are computed in Fermigier =-=[11]-=-, and the last in Birch and Swinnerton-Dyer [3]). Furthermore, E 36 is a curve with complex multiplication. The Ramanujan L-function, Ls(s), has a Dirichlet series given by Ls(s) = 1 X n=1 (n) n 11=2 ... |

9 |
Introduction to the Langlands program, Representation Theory and Automorphic Forms
- Knapp
- 1996
(Show Context)
Citation Context ...ons of (s) 86 viii Dedicated to my sister Ronit ix Chapter 1 Introduction In this thesis, I obtain evidence for a spectral interpretation of the zeros of L-functions. The Langland's program [12] [24] =-=[19]-=- predicts that all L-functions can be written as products of i(s) and L-functions attached to automorphic cuspidal representations of GLM over Q . Such an L-function is given intially (for !s sufficie... |

8 | Asymptotics and special functions, Academic Press [A subsidiary of Harcourt Brace - Olver - 1974 |

7 |
Zeros of zeta functions, their spacings, and their spectral nature
- Katz, Sarnak
(Show Context)
Citation Context ... Section 1.2). First, analogs with function field zeta functions, where there is a spectral interpretation of the zeros in terms of Frobenius on cohomology, point towards the classical compact groups =-=[16]-=-. Second, even though all the mentioned families of matrices have the same n-level correlations, there is another statistic, called n-level density, which is sensitive to the particular family. By loo... |

7 |
An asymptotic representation for the Riemann zeta function on the critical line
- Paris
- 1994
(Show Context)
Citation Context ...led to difficulties (which are overcome in this thesis) concerning computing G(z; w) with both z and w complex. Other computations of L-functions included those of Berry and Keating [2] (i(s)), Paris =-=[30]-=- (i(s)), Tollis [38] (Dedekind zeta functions), Spira [35] (Ls(s)), and Keiper [18] (Ls(s)). 1.4. Thesis results 19 0 1 2 3 4 0.0 0.5 1.0 Pair correlation for the Ramanujan L-function, N=138693 0 1 2 ... |

6 | Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachträge - Riemann - 1990 |

6 | Zeros of Dedekind zeta functions in the critical strip
- Tollis
- 1997
(Show Context)
Citation Context ...(which are overcome in this thesis) concerning computing G(z; w) with both z and w complex. Other computations of L-functions included those of Berry and Keating [2] (i(s)), Paris [30] (i(s)), Tollis =-=[38]-=- (Dedekind zeta functions), Spira [35] (Ls(s)), and Keiper [18] (Ls(s)). 1.4. Thesis results 19 0 1 2 3 4 0.0 0.5 1.0 Pair correlation for the Ramanujan L-function, N=138693 0 1 2 3 4 0.0 0.5 1.0 Pair... |

5 |
On the mean value of L(1=2; ) for real characters, Analysis 1
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- 1981
(Show Context)
Citation Context ...e number of divisors of m), so that the second bracketed term iss" 0 @X " X mX ff fi fi fi fi fi fi X d2D(X)sd (m) fi fi fi fi fi fi 2 1 A 1=2 : (2.2.13) 2.2. l.h.s. 36 Applying the methods =-=of Jutila [15], we find that the a-=-bove iss" \Gamma X "+1+ff log A X \Delta 1=2 ; for some constant A (A = 10 is admissable) which, combined with (2.2.12), shows that (2.2.11) iss" lim X!1 1 jD(X)j X "+(1+ff)=2 But,... |

5 |
On calculations of zeros L-functions related with Ramanujan’s discriminant function on the critical line
- Yoshida
- 1988
(Show Context)
Citation Context ...st few zeros (T = 15) to four places after the decimal point for many elliptic curve L-functions, including the L-functions associated to E 11 and E 37 . My results agree with his. For Ls(s), Yoshida =-=[41]-=- computed the zeros to height T = 100, and my data agrees to all his 12 decimal places, as well as that of Spira [35] (first 3 zeros to all places listed). Keiper [18] computed the first 5018 zeros of... |

4 |
Multiplicative number theory, second ed., Graduate Texts
- Davenport
- 1980
(Show Context)
Citation Context ...wheresd (n) = \Gamma d n \Delta is Kronecker's symbol. Thensandsextend to entire functions and satisfy the functional equations (s;sd ) = (1 \Gamma s;sd )s(s;sd ) =sd (\Gamma1)s(1 \Gamma s;sd ): (see =-=[6]-=-, [4]). Note that Ls(s;sd ) has a zero at s = 1=2 ifsd (\Gamma1) = \Gamma1. We collected the first few zeros of various L(s;sd )'s and Ls(s;sd )'s by looking for sign changes of the L-function (rotate... |

3 |
A new asymptotic representation for i( + it) and quantum spectral determinants
- Berry, Keating
- 1992
(Show Context)
Citation Context ...ement it since it led to difficulties (which are overcome in this thesis) concerning computing G(z; w) with both z and w complex. Other computations of L-functions included those of Berry and Keating =-=[2]-=- (i(s)), Paris [30] (i(s)), Tollis [38] (Dedekind zeta functions), Spira [35] (Ls(s)), and Keiper [18] (Ls(s)). 1.4. Thesis results 19 0 1 2 3 4 0.0 0.5 1.0 Pair correlation for the Ramanujan L-functi... |

3 |
Haselgrove, The evaluation of Dirichlet L-functions
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- 1961
(Show Context)
Citation Context ...w Odlyzko) agrees with Yoshida's, Spira's, and mine to only the 5th (or more) decimal place. My data also agrees with the Dirichlet L-function computations of Rumely [34] and of Davies and Haselgrove =-=[7]-=-. In the T -experiment, the zeros were checked against the predictions of the asymptotic formula for N(T ). Finally, in both the T and d experiments, the data collected agrees with our theoretical pre... |

3 |
Riemann’s zeta function, Academic Press [A subsidiary of Harcourt Brace Jovanovich
- Edwards
- 1974
(Show Context)
Citation Context ...e used the formulae Ns(T )s2T log ` T 2e ' + 11 2 NEQ (T )s2T log ` Q 1=2 T 2e ' + 1 2 to verify that all zeros had been found, though not in any rigorous fashion. In principle, using Turing's method =-=[9]-=-, this could be turned into a proof that none of 1.4. Thesis results 11 the zeros to height slightly less than T were off the critical line. (note that, because E 37 has rank 1, when comparing against... |

3 |
A motivated introduction to the Langlands program, Advances in number theory
- Murty
- 1991
(Show Context)
Citation Context ...luations of (s) 86 viii Dedicated to my sister Ronit ix Chapter 1 Introduction In this thesis, I obtain evidence for a spectral interpretation of the zeros of L-functions. The Langland's program [12] =-=[24]-=- [19] predicts that all L-functions can be written as products of i(s) and L-functions attached to automorphic cuspidal representations of GLM over Q . Such an L-function is given intially (for !s suf... |

1 |
On the zeros of the Ramanujan -Dirichlet series in the critical strip
- Keiper
- 1996
(Show Context)
Citation Context ...ree with his. For Ls(s), Yoshida [41] computed the zeros to height T = 100, and my data agrees to all his 12 decimal places, as well as that of Spira [35] (first 3 zeros to all places listed). Keiper =-=[18]-=- computed the first 5018 zeros of Ls(s), and 2028 zeros near the 20001st zero, but his data (obtained in a file from Andrew Odlyzko) agrees with Yoshida's, Spira's, and mine to only the 5th (or more) ... |

1 |
Calculation of the Ramanujan -Dirichlet series
- Spira
- 1973
(Show Context)
Citation Context ...ctions associated to E 11 and E 37 . My results agree with his. For Ls(s), Yoshida [41] computed the zeros to height T = 100, and my data agrees to all his 12 decimal places, as well as that of Spira =-=[35]-=- (first 3 zeros to all places listed). Keiper [18] computed the first 5018 zeros of Ls(s), and 2028 zeros near the 20001st zero, but his data (obtained in a file from Andrew Odlyzko) agrees with Yoshi... |

1 | Edited and with a preface by Raghavan Narasimhan - Weber, Dedekind |