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Ram Murty, M.
- Generalization of a Theorem of Hurwitz
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be the Eisenstein series of weight k attached to the full modular group. Let z be a CM point in the upper half-plane. Then there is a transcendental number Ωz such that
G2k(z)=Ω2kz. (an algebraic number).
Moreover, Ωz can be viewed as a fundamental period of a CM elliptic curve defined over the field of algebraic numbers. More generally, given any modular form f of weight k for the full modular group, and with algebraic Fourier coefficients, we prove that f(z)πk/Ωzk is algebraic for any CM point z lying in the upper half-plane. We also prove that for any automorphism σ of Gal (̅ℚ/ℚ), (f(z)πk/Ωzk)σ=fσ(z)πk/Ωzk.
Authors
Affiliations
1 Department of Mathematics, Kyungpook National University, Daegu-702-701, KR
2 Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario- K7L3N6, CA
3 Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul-120-749, KR
1 Department of Mathematics, Kyungpook National University, Daegu-702-701, KR
2 Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario- K7L3N6, CA
3 Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul-120-749, KR
Source
Journal of the Ramanujan Mathematical Society, Vol 31, No 3 (2016), Pagination: 215-226Abstract
This paper is an exposition of several classical results formulated and unified using more modern terminology. We generalize a classical theorem of Hurwitz and prove the following: let Gk (z)=∑1/(mz+n)kbe the Eisenstein series of weight k attached to the full modular group. Let z be a CM point in the upper half-plane. Then there is a transcendental number Ωz such that
G2k(z)=Ω2kz. (an algebraic number).
Moreover, Ωz can be viewed as a fundamental period of a CM elliptic curve defined over the field of algebraic numbers. More generally, given any modular form f of weight k for the full modular group, and with algebraic Fourier coefficients, we prove that f(z)πk/Ωzk is algebraic for any CM point z lying in the upper half-plane. We also prove that for any automorphism σ of Gal (̅ℚ/ℚ), (f(z)πk/Ωzk)σ=fσ(z)πk/Ωzk.
- Generalization of an Identity of Ramanujan
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∑anTn and ∑bnTn
which are both rational functions with certain property, we then explicitly show that
∑anbnTn
is again a rational function with the same property. We use this to explain Ramanujan’s identities and also analyse Rankin-Selberg convolutions of automorphic L-functions.
Authors
Sanoli Gun
1,
M. Ram Murty
2
Affiliations
1 Institute for Mathematical Sciences, C.I.T Campus, Taramani, Chennai-600113, IN
2 Department of Mathematics, Queen’s University, Kingston, Ontario-K7L3N6, CA
1 Institute for Mathematical Sciences, C.I.T Campus, Taramani, Chennai-600113, IN
2 Department of Mathematics, Queen’s University, Kingston, Ontario-K7L3N6, CA
Source
Journal of the Ramanujan Mathematical Society, Vol 31, No 2 (2016), Pagination: 125-135Abstract
In this article, we extend two identities proved by Ramanujan involving the Riemann zeta function and the Dirichlet L-function associated to the non-trivial Dirichlet character modulo 4. More precisely, given two power series∑anTn and ∑bnTn
which are both rational functions with certain property, we then explicitly show that
∑anbnTn
is again a rational function with the same property. We use this to explain Ramanujan’s identities and also analyse Rankin-Selberg convolutions of automorphic L-functions.
- Zeros of Ramanujan Polynomials
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Authors
Affiliations
1 Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, CA
2 School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, Scotland EH9 3JZ, GB
1 Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, CA
2 School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, Scotland EH9 3JZ, GB
Source
Journal of the Ramanujan Mathematical Society, Vol 26, No 1 (2011), Pagination: 107-125Abstract
In this paper, we investigate the properties of Ramanujan polynomials, a family of reciprocal polynomials with real coefficients originating from Ramanujan’s work. We begin by finding their number of real zeros, establishing a bound on their sizes, and determining their limiting values. Next, we prove that all nonreal zeros of Ramanujan polynomials lie on the unit circle, and are asymptotically uniformly distributed there. Finally, for each Ramunujan polynomial, we find all its zeros that are ischolar_mains of unity.- On the Asymptotics for Invariants of Elliptic Curves Modulo p
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Authors
Affiliations
1 Department of Mathematics & Computer Science, University of Lethbridge, Lethbridge, Alberta, CA
2 Department of Mathematics & Statistics, Queen’s University, Kingston, Ontario, CA
1 Department of Mathematics & Computer Science, University of Lethbridge, Lethbridge, Alberta, CA
2 Department of Mathematics & Statistics, Queen’s University, Kingston, Ontario, CA
Source
Journal of the Ramanujan Mathematical Society, Vol 28, No 3 (2013), Pagination: 271–298Abstract
Let E be an elliptic curve defined over Q. Let E(Fp) denote the elliptic curve modulo p. It is known that there exist integers i p and f p such that E(Fp) ∼= Z/i pZ × Z/i p fp Z. We study questions related to i p and f p. In particular, for any α > 0 and k ∈ N, we prove there exist positive constants cα and ck such that for any A > 0
Σ(log ip)α = cα li(x) + O(x/(logx)A)
and
Σ Tk(ip) = ck li(x) + O(x/(log x)A)
unconditionally for CMelliptic curves, where τk (n) is the number of ways of writing n as a product of k positive integers. For a CM curve E and 0 < α < 1, we prove that there exists a constant c'α > 0 such that Σ iαp = c'α li(x) + O(x3+α/4 (log x)1-α/2)
- An Introduction to Artin L-Functions
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Authors
Affiliations
1 Department of Mathematics, Queen's University-Kingston, Ontario, K7L 3N6, CA
1 Department of Mathematics, Queen's University-Kingston, Ontario, K7L 3N6, CA
Source
Journal of the Ramanujan Mathematical Society, Vol 16, No 3 (2001), Pagination: 261-307Abstract
An Artin L -function is a generalization of the Riemann zeta function and the classical Dirichlet L -functions. Just as the Dirichlet L -functions are useful in the study of primes in arithmetic progressions, so are the Artin L -functions useful in the study of the decomposition of primes in algebraic number fields. In contrast to the classical objects, we still do not have analytic continuation of these objects in the general setting. If we did, this would have profound consequences in the study of prime number theory, especially to various forms of the effective Chebotarev density theorem, which can be viewed as the most general form of the prime number theorem.- Some Variations on the Dedekind Conjecture
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Authors
M. Ram Murty
1,
A. Raghuram
2
Affiliations
1 Department of Mathematics, Queen's University, Kingston, K7L 3N6 Ontario, CA
2 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, IN
1 Department of Mathematics, Queen's University, Kingston, K7L 3N6 Ontario, CA
2 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, IN
Source
Journal of the Ramanujan Mathematical Society, Vol 15, No 4 (2000), Pagination: 225-245Abstract
In this paper we prove a group theoretic statement about expressing certain characters of a finite solvable group as a sum of monomial characters. This is used to prove holomorphy of certain products of Artin L-functions which can be thought of as a variant of the Dedekind Conjecture. This variant is then used to improve, in the solvable case, a certain inequality due to R. Foote and K. Murty which bounds the orders of some Artin L-functions, at an arbitrary but fixed point in the complex plane, in terms of the order of a suitable quotient of Dedekind zeta functions. This improved inequality has a rather striking consequence regarding non-existence of simple zeros or simple poles in such quotients.- The Turan Sieve Method and Some of Its Applications
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Authors
Yu-Ru Liu
1,
M. Ram Murty
2
Affiliations
1 Department of Mathematics, Harvard University Cambridge, Mass., 02138, US
2 Department of Mathematics and Statistics, Queen's University, Kingston Ontario, K7L 3N6, CA
1 Department of Mathematics, Harvard University Cambridge, Mass., 02138, US
2 Department of Mathematics and Statistics, Queen's University, Kingston Ontario, K7L 3N6, CA
Source
Journal of the Ramanujan Mathematical Society, Vol 14, No 1 (1999), Pagination: 21-35Abstract
We introduce the Turan sieve method and apply it to the probabilistic Galois theory problems in both the rational number field and the function field cases. We estimate the number of polynomials of degree n and height ≤ N whose Galois group is a proper subgroup of Sn- For the rational number field case, we get an estimate of O(Nn-1/3(logN)<sup.2) and in the case of the function field over Fq, we get O(Nn-1logqN).- A Remark on a Conjecture of Chowla
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Authors
Affiliations
1 Department of Mathematics, Queen’s University, Kingston, Ontario K7L 3N6, CA
2 Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, CA
1 Department of Mathematics, Queen’s University, Kingston, Ontario K7L 3N6, CA
2 Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, CA
Source
Journal of the Ramanujan Mathematical Society, Vol 33, No 2 (2018), Pagination: 111-123Abstract
We make some remarks on a special case of a conjecture of Chowla regarding the Mobius function μ(n).References
- S. Chowla, The Riemann hypothesis and Hilbert’s tenth problem, Mathematics and its Applications, Vol. 4, Gordon and Breach Science Publishers, New York-London-Paris (1965).
- P. D. T. A. Elliott, On the correlation of multiplicative and the sum of additive arithmetic functions, Mem. Amer. Math. Soc., 112 (1994) no. 538.
- H. Li and H. Pan, Bounded gaps between primes of a special form, International Mathematics Research Notices, 23 (2015) 12345–65.
- H. Li and H. Pan, Erratum to “Bounded gaps between primes of a special form”, International Mathematics Research Notices, 21 (2016) 6732–6734.
- K. Matomaki, M. Radziwiłl and T. Tao, An averaged form of Chowla’s conjecture, Algebra Number Theory, 9 (2015) no. 9, 2167–2196.
- H. Montgomery and R. Vaughan, Multiplicative number theory I, classical theory, Cambridge Studies in Advanced Mathematics, Vol. 97, Cambridge University Press (2007).
- M. Ram Murty and J. Esmonde, Problems in algebraic number theory, Second edition, Graduate Texts in Mathematics, Springer-Verlag, New York (2005) 190.
- M. Ram Murty and A. Vatwani, Twin primes and the parity problem, Journal of Number Theory, 180 (2017) 643–659.
- Nathan Ng, The Mobius function in short intervals, Anatomy of integers, CRM Proc. Lecture Notes, Amer. Math. Soc., Providence, RI, Edited by Jean-Marie De Koninck, Andrew Granville, Florian Luca 46 (2008) 247–257.
- T. Tao, The Chowla conjecture and the Sarnak conjecture, available at https://terrytao.wordpress.com/2012/10/14/the-chowla-conjecture-and-the-sarnakconjecture/
- T. Tao, The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, Forum Math. Pi, 4 (2016) e8, 36pp.
- H. Siebert and D. Wolke, Uber einige Analoga zum Bombierischen Primzahlsatz, Math. Z., 122 (1971) no. 4, 327–341.
- Admissible Primes and Euclidean Quadratic Fields
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Authors
Affiliations
1 Department of Mthematics and Statistics, Jeffery Hall, Queen’s University, Kingston, Ontario, K7L 3N6, CA
2 Institute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai - 600113, IN
3 Chennai Mathematical Institute, SIPCOT IT Park, Siruseri, Chennai - 603103, IN
1 Department of Mthematics and Statistics, Jeffery Hall, Queen’s University, Kingston, Ontario, K7L 3N6, CA
2 Institute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai - 600113, IN
3 Chennai Mathematical Institute, SIPCOT IT Park, Siruseri, Chennai - 603103, IN
Source
Journal of the Ramanujan Mathematical Society, Vol 33, No 2 (2018), Pagination: 135-147Abstract
Let K be a real quadratic field with ring of integers ΟK . We exhibit an infinite family of real quadratic fields K, such that ΟK contains an admissible set of primes with two elements.We then study the implications of this construction to the determination of Euclidean real quadratic fields and related questions.References
- V. Buniakovsky, Sur les diviseurs numeriques invariables des fonctions rationnelles entieres, Mem Acad. Sci. St Petersburg, 6 (1857) 305–329.
- David A. Clark and M. Ram Murty, The Euclidean algorithm for Galois extensions, Journal fur die reine und angewandte Mathematik, 459 (1995) 151–162.
- David S. Dummit and Richard M. Foote, Abstract Algebra, John Wiley & Sons (2004).
- Jody Esmonde and M. Ram Murty, Problems in Algebraic Number Theory, Graduate texts in Mathematics, Springer Science and Business Media (2005).
- Rajiv Gupta, M. Ram Murty and V. Kumar Murty, The Euclidean algorithm for S-integers In: Number Theory (Montreal, June 1985), CMS Conf. Proc. 7, Amer. Math. Soc. (1987) 189–201.
- Malcom Harper, Z[√14] is Euclidean, Canad. J. Math., Vol. 56 (2004) no. 1, 55–70.
- M. Harper and M. Ram Murty Euclidean rings of algebraic integers, Canadian Journal of Math., 56 (2004) no. 1, 71–76.
- C. Hooley, On Artin’s conjecture, J. Reine Agew. Math., 225 (1967) 209–220.
- Franz Lemmermeyer, The euclidean algorithm in algebraic number fields, Exposition. Math., 13 (1995) no. 5, 385–416
- Daniel A. Marcus, Number Fields, Graduate texts in mathematics, Springer-Verlag (1977).
- T. Motzkin, The Euclidean algorithm, Bull. Am. Math. Soc., 55 (1949) no. 12, 1142–1146.
- M. RamMurty and V. Kumar Murty, A variant of the Bombieri-Vinogradov theorem, in Number Theory, Proceedings of the 1985 Montreal Conference, 7 (1987) 243–272.
- M. Ram Murty and Kathleen Petersen, A Bombieri-Vinogradov theorem for all number fields, Transactions of the American Math. Society, 365 (2013) no. 9, 4987–5032.
- T. Nagell, Zur Arithmetik der Polynome, Abhandl. Math. Sem. Hamburg, 1 (1922) 179–194.
- G. Ricci, Ricerche aritmetiche sui polinomi, Rend. Circ. Mat. Palermo, 57 (1933) 433–475.
- Pierre Samuel, About Euclidean rings, Journal of Algebra, Vol. 19 Issue 2 (1971) 282–301.
- P. J. Weinberger, On Euclidean rings of algebraic integers, Proc. Symp. Pure Math., Analytic number theory, AMS, 24 (1973) 321–332 6.
- Leonardo Zapponi, Parametric solutions of Pell equations arXiv:1503.00637v1.
- The Chebotarev Density Theorem and the Pair Correlation Conjecture
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Authors
Affiliations
1 Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, CA
2 Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, CA
1 Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, CA
2 Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, CA
Source
Journal of the Ramanujan Mathematical Society, Vol 33, No 4 (2018), Pagination: 399-426Abstract
In this note, we formulate pair correlation conjectures and refine the effective version of the Chebotarev density theorem established by the first two authors. Also, we apply our result to study Artin’s primitive ischolar_main conjecture and the Lang-Trotter conjectures and obtain shaper error terms.References
- A. C. Cojocaru, Cyclicity of elliptic curves modulo p, Ph.D. Thesis, Queen’s University (2002).
- A. C. Cojocaru and C. David, Frobenius fields for elliptic curves, American Journal of Math., 130 (2008) no. 6, 1535–1560.
- A. C. Cojocaru, E. Fouvry and M. R. Murty, The square sieve and the Lang-Trotter conjecture, Canadian Journal of Math., 57, (2005) no. 6, 1155–1177.
- A. C. Cojocaru and M. R. Murty, Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem, Math. Annalen, 330 (2004) no. 3, 601–625.
- P. Deligne, Formes modulaires et representations l-adiques, Sem. Bourbaki 355, Lecture Notes in Mathematics, Springer Verlag, Heidelberg, 179 (1971) 139–172.
- R. Gupta and M. R. Murty, A remark on Artin’s conjecture, Inventiones Math., 78 (1984) 127–130.
- R. Gupta and M. R. Murty, Cyclicity and generation of points mod p on elliptic curves, Inventiones Math., 101 (1990) 225–235.
- D. R. Heath-Brown, Gaps between primes, and the pair correlation of zeros of the zeta-function, Acta Arith., 41 (1982) 85–99.
- C. Hooley, On Artin’s conjecture, J. Reine Angew. Math., 225 (1967) 209–220.
- J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem, Algebraic number fields: ζ -functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London (1977) 409–464.
- S. Lang and H. Trotter, Frobenius distributions in GL2-extensions, Lecture Notes in Mathematics, Springer-Verlag, 504 (1976).
- F. Momose, On the l-adic representations attached to modular forms, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28, (1981) no. 1, 89–109.
- H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 227 (1971).
- M. R. Murty, On Artin’s conjecture, Journal of Number Theory, 16 (1983) 147–168.
- M. R. Murty, V. K. Murty, and N. Saradha,Modular forms and the Chebotarev density theorem, American Journal of Math., 110 (1988) 253–281.
- M. R. Murty and A. Perelli, The pair correlation of zeros of functions in the Selberg class, International Math. Res. Notices, 10 (1999) 531–545.
- M. R. Murty and A. Zaharescu, Explicit formulas for the pair correlation of zeros of functions in the Selberg class, Forum Math., 14 (2002) no. 1, 65–83.
- V. K. Murty, Explicit formulae and the Lang-Trotter conjecture, Rocky Mountain J. Math., 15(2) (1985) 535–551.
- V. K. Murty, Modular forms and the Chebotarev density theorem II, Analytic Number Theory, Ed. Y. Motohashi, Cambridge University Press (1997) 287–308.
- K. Ribet, Galois representations attached to eigenforms with Nebentypus, Lecture Notes in Mathematics, Springer-Verlag, Heidelberg, 601 (1976) 17–52,
- K. Ribet, On l-adic representations attached to modular forms II, Glasgow J. Math., 27 (1985) 185–194.
- J.-P. Serre, Resume des cours de 1977–1978, Annuaire du College de France 1978, 67–70. (See also Collected Papers, Volume III, Springer-Verlag (1985).)
- J.-P. Serre, Quelques applications du theoreme de densit´e de Chebotarev, Publ. Math. IHES, 54 (1981) 123–201.
- On a Conjecture of Bateman About r5(n)
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Authors
Arpita Kar
1,
M. Ram Murty
1
Affiliations
1 Department of Mathematics and Statistics, Queen’s University, Kingston, CA
1 Department of Mathematics and Statistics, Queen’s University, Kingston, CA
Source
Journal of the Ramanujan Mathematical Society, Vol 34, No 1 (2019), Pagination: 133-142Abstract
Let r5(n) be the number of ways of writing n as a sum of five integer squares. In his study of this function, Bateman was led to formulate a conjecture regarding the sum Σ|j|≤√n σ(n−j2) where σ(n) is the sum of positive divisors of n. We give a proof of Bateman’s conjecture in the case n is square-free and congruent to 1 (mod 4).References
- P. T. Bateman, The asymptotic formula for the number of representations of an integer as a sum of five squares, Analytic Number Theory, Vol. 1 (Allerton Park, IL, 1995), 129–139, Progr. Math., 138, Birkh¨auser Boston, Boston, MA (1996).
- P. T. Bateman and M. I. Knopp, Some new old-fashioned modular identities, Paul Erdos (1913–1996). Ramanujan J., 2 no. 1–2, (1998) 247–269.
- J. Esmonde and M. Ram Murty, Problems in algebraic number theory, Graduate Texts in Mathematics, 190. Springer-Verlag, New York, (1999).
- G. H. Hardy, On the representation of a number as the sum of any number of squares, and in particular of five, Trans. Amer. Math. Soc., 21 no. 3, (1920) 255–284.
- M. Ram Murty, Problems in analytic number theory. Second edition. Graduate Texts in Mathematics, 206. Readings in Mathematics, Springer, New York (2008).
- Murty, M. Ram and Murty V. Kumar, The mathematical legacy of Srinivasa Ramanujan. Springer, New Delhi (2013).
- C. L. Siegel, Advanced analytic number theory, second edition, Tata Institute of Fundamental Research Studies in Mathematics, 9, Tata Institute of Fundamental Research, Bombay (1980).
- C. L. Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen. Nachr. Akad. Wiss. Gˆottingen, Math.-Phys. Klasse, 10 (1969) 87–102.
- R. C. Vaughan, The Hardy Littlewood Method, Cambridge University Press (1981).
- D. Zagier, On the values at negative integers of the zeta-function of a real quadratic field, Enseignement Math. (2), 22 no. 1–2, (1976) 55–95.