Papers by Wladimir Pribitkin
a.mod c/ ad 1.mod c/ maCnd 2 i c e: Here the numbers m and n are any integers, and the modulus c ... more a.mod c/ ad 1.mod c/ maCnd 2 i c e: Here the numbers m and n are any integers, and the modulus c is a positive integer. Ac.m; n / is a \genuine " Kloosterman sum if mn ¤ 0, Ac.m; 0 / D Ac.0; m / is a Ramanujan sum if m ¤ 0, and Ac.0; 0 / is simply Euler's totient.c/. The significance of Kloosterman sums to the theory of modular forms dates back a century to an astonishingly little-known work of Poincare [10]. In 1926 Kloosterman [4] published his seminal paper regarding Ramanujan's problem of representing sufficiently large integers by quaternary quadratic forms. Since then these sums have surfaced with an almost unreasonable ubiquity throughout arithmetic. It is plain that Ac.m; n / D Ac. m; n / and hence Ac.m; n / is real. As such, it is natural to ask whether the sequence fAc.m; n/g1 cD1 is oscillatory for fixed integers m and n. That is, are there infinitely many c such that Ac.m; n /> 0 and infinitely many c such that Ac.m; n / < 0? Obviously, fAc.0; 0/g1 c...
The American Mathematical Monthly, 2001
Note 2. The forms (9) and (11) of the Law of Cosines follow immediately by adding ±2vw to both si... more Note 2. The forms (9) and (11) of the Law of Cosines follow immediately by adding ±2vw to both sides of u − v − w = −2vw cos U and then factoring the resulting left-hand side u2 − (v ± w)2 as a difference of two squares. It is interesting to note that multiplying the resulting identities 2(p − v)(p − w) = vw(1 − cos U ), 2p(p − u) = vw(1 + cos U ) and using the fact that the area of U V W is given by | U V W |= 12vw sin U yields Heron’s Formula p(p − u)(p − v)(p − w) =| U V W |2 .
Proceedings of the American Mathematical Society, 2001
The Lipschitz formula is extended to a two-variable form. While the theorem itself is of independ... more The Lipschitz formula is extended to a two-variable form. While the theorem itself is of independent interest, we justify its existence further by indicating several applications in the theory of modular forms.
Acta Arithmetica, 1999
Dedicated to the memory of my greatest teachers, my parents, Thereza de Azevedo Pribitkin and Edm... more Dedicated to the memory of my greatest teachers, my parents, Thereza de Azevedo Pribitkin and Edmund Pribitkin 1. Historical introduction. In 1989 Knopp [6] found explicit formulas for the Fourier coefficients of an arbitrary cusp form and more generally, but conditionally, of a holomorphic modular form (with a possible pole at i∞) on the full modular group, Γ (1), of weight k, 4/3 < k < 2, and multiplier system v. He assumed that there are no nontrivial cusp forms on Γ (1) of complementary weight 2−k and conjugate multiplier system v. In our initial paper we remove this assumption and capture the Fourier coefficients of an arbitrary "Niebur modular integral" on Γ (1) of weight k, 1 < k < 2. En route we also obtain expressions for the Fourier coefficients of an arbitrary cusp form on Γ (1) of weight k, 0 < k < 1. In particular we present formulas for the Fourier coefficients of η r (τ), 0 < r < 2, where η(τ) is the Dedekind eta-function. An actual formula for the Fourier coefficients of an arbitrary modular form, even in the case of the full modular group, is not always available. For forms of weight greater than two the problem was solved by Petersson [11], who introduced the (parabolic) Poincaré series. Additionally, by considering a nonanalytic version of this series, he derived the coefficients of certain forms of weight two [12]. By integrating one of these forms, Petersson [12, p. 202] was the first to find the coefficients of the absolute modular invariant J(τ). For forms of negative weight Rademacher and Zuckerman [18] discovered expressions for the coefficients by relying on the circle method. Furthermore, Rademacher [15] employed a sharpened version of this method to rediscover Petersson's formula for J(τ). We remark that both approaches
Illinois Journal of Mathematics, 2009
By employing work concerning Selberg's Kloosterman zeta-function, we carry out the decomposition ... more By employing work concerning Selberg's Kloosterman zeta-function, we carry out the decomposition of a special value of a nonanalytic Poincaré series of nonpositive even weight, with a nonsingular multiplier system, on the full modular group. The summands that emerge are connected meaningfully to each other as well as to classical expressions for Eichler integrals and modular forms.
Research in Number Theory, 2019
The Ramanujan Journal, 2016
Marvin Knopp was the greatest. He understood that it's not just the first impression that counts.... more Marvin Knopp was the greatest. He understood that it's not just the first impression that counts. It's the thousandth one. For over half a century, he worked unflaggingly as a mathematician. He made fundamental contributions to the analytic theory of classical automorphic forms and their associated functions, as well as foundational discoveries in the broader areas of automorphic integrals and generalized modular forms. He shone in all facets of the profession-as researcher and expositor, teacher and advisor, speaker and organizer, referee and reviewer, collaborator and editor. He served wholeheartedly on the board of the Ramanujan Journal since its inception in 1996. Below we portray Professor Knopp's impact on the world of mathematics, focusing chiefly on his profound investigations, consummate devotion to his students, and passionate promotion of his peers and predecessors alike. Throughout his career Knopp studied modular forms and, more inclusively, automorphic forms. In 1958 he completed his doctoral work at the University of Illinois (Urbana) under the guidance of Paul T. Bateman. In his dissertation, "On the Construction of Certain Automorphic Forms of Nonnegative Dimension," Knopp extended a converse theorem of Hans Rademacher regarding the modular invariant J (τ) to a The author thanks Krishna Alladi, Bruce Berndt, and YoungJu Choie for their steadfast support of him, as well as their unwavering dedication to this volume. Additionally, he is grateful to his wife, Michelle Kasnikowski, who was enthralled by Marvin from the moment they met in 1993 and he burst into song, with a rendition of "Chicago," his beloved hometown.
The Ramanujan Journal, 2016
We establish a vast generalization of an observation made by Marvin Knopp half a century ago conc... more We establish a vast generalization of an observation made by Marvin Knopp half a century ago concerning the nonvanishing of Ramanujan's tau-function.
数理解析研究所講究録, Sep 1, 2012
The significance of Kloosterman sums to the theory of modular forms dates back a century to an as... more The significance of Kloosterman sums to the theory of modular forms dates back a century to an astonishingly little-known work of Poincare [10]. In 1926 Kloosterman [4] published his seminal paper regarding Ramanujan's problem of representing sufficiently large integers by quatemary quadratic forms. Since then these sums have surfaced with an almost unreasonable ubiquity throughout arithmetic.
Int J Number Theory, 2007
Developments in Mathematics, 2012
Let (a n ) n≥1 be a sequence of real numbers. Then we say that (a n ) n≥1 is oscillatory if there... more Let (a n ) n≥1 be a sequence of real numbers. Then we say that (a n ) n≥1 is oscillatory if there exist infinitely many n with a n >0 and infinitely many n with a n <0.
Proceedings of the American Mathematical Society, 2008
Under what conditions do the (possibly complex) coefficients of a general Dirichlet series exhibi... more Under what conditions do the (possibly complex) coefficients of a general Dirichlet series exhibit oscillatory behavior? In this work we invoke Laguerre’s Rule of Signs and Landau’s Theorem to provide a rather simple answer to this question. Furthermore, we explain how our result easily applies to a multitude of functions.
Developments in Mathematics, 2003
Let be a discrete subgroup of SL(2, R) with a fundamental region of finite hyperbolic volume. (Th... more Let be a discrete subgroup of SL(2, R) with a fundamental region of finite hyperbolic volume. (Then, is a finitely generated Fuchsian group of the first kind.) Let f (z) = n+κ>0 a(n)e 2πi(n+κ)z/λ , z ∈ H. be a nontrivial cusp form, with multiplier system, with respect to. Responding to a question of Geoffrey Mason, the authors present simple proofs of the following two results, under natural restrictions upon. Theorem. If the coefficients a(n) are real for all n, then the sequence {a(n)} has infinitely many changes of sign. Theorem. Either the sequence {Re a(n)} has infinitely many sign changes or Re a(n) = 0 for all n. The same holds for the sequence {Im a(n)}.
Contemporary Mathematics, 2000
Contemporary Mathematics, 2000
Ramanujan Journal, 2000
We provide a new proof of Rademacher's celebrated exact formula for the partition function. A... more We provide a new proof of Rademacher's celebrated exact formula for the partition function. Along the way we present a simple treatment of an integral which is ubiquitous in the theory of nonanalytic automorphic forms.
Ramanujan Journal, 2000
In 1921 Hamburger proved that Riemann's functional equation characterizes the Riemann zeta fu... more In 1921 Hamburger proved that Riemann's functional equation characterizes the Riemann zeta function in the space of functions representable by ordinary Dirichlet series satisfying certain regularity conditions. We consider solutions to a more general functional equation with real weight k. In the case of Hamburger's theorem, k = <img src="/fulltext-image.asp?format=htmlnonpaginated&src=W46821736L733186_html\11139_2004_Article_257538_TeX2GIFIE1.gif" border="0" alt=" $$ - \frac{1}{2}$$ " />. We show that,
Contemporary Mathematics, 2000
The Ramanujan Journal, 2007
We tweak Siegel's method to produce a rather simple proof of a new upper bound on the number of p... more We tweak Siegel's method to produce a rather simple proof of a new upper bound on the number of partitions of an integer into an exact number of parts. Our approach, which exploits the delightful dilogarithm function, extends easily to numerous other counting functions.
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Papers by Wladimir Pribitkin