Motivated by work of Ramanujan, Freeman Dyson defined the rank an integer partition to be its largest part minus number parts.If N.m; n/ denotes partitions n with m, then it turns out thatWe show that if ¤ 1 is a root unity, R. I q/ essentially holomorphic weight 1=2 weak Maass form on subgroup SL 2 ‫./ޚ.‬For integers 0 Ä r < t, we use this result determine modularity generating function for N.r; tI n/, whose congruent .modt /.We extend above construct infinite family vector valued forms...

Together with his collaborators, most

Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers 5, 7 and 11. A further were shown by works Atkin, O'Brien, Newman. In this paper we prove there are infinitely many such for every prime modulus exceeding 3. addition, provide simple criterion guaranteeing truth Newman's conjecture any 3 (recall asserts hits residue class given integer M often).

Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions.Generalizing works Waldspurger, Kohnen and Zagier, we prove such also serve as "generating functions" for central values derivatives quadratic twists weight 2 modular L-functions.To obtain these results, construct differentials third kind with twisted Heegner divisor by suitably generalizing Borcherds lift forms.The connection...

In 1927 P\'olya proved that the Riemann Hypothesis is equivalent to hyperbolicity of Jensen polynomials for zeta function $\zeta(s)$ at its point symmetry. This has been degrees $d\leq 3$. We obtain an asymptotic formula central derivatives $\zeta^{(2n)}(1/2)$ accurate all orders, which allows us prove a density $1$ subset each degree. Moreover, we establish 8$. These results follow from general theorem models such by Hermite polynomials. case function, this proves GUE random matrix model...

Abstract In his “lost notebook,” Ramanujan used iterated derivatives of two theta functions to define sequences q -series $\{U_{2t}(q)\}$ and $\{V_{2t}(q)\}$ that he claimed be quasimodular. We give the first explicit proof this claim by expressing them in terms “partition Eisenstein series,” extensions classical series $E_{2k}(q),$ defined $$ \begin{align*}\lambda=(1^{m_1}, 2^{m_2},\dots, n^{m_n}) \vdash n \ \longmapsto E_{\lambda}(q):= E_2(q)^{m_1} E_4(q)^{m_2}\cdots E_{2n}(q)^{m_n}....

Abstract We study “partition Eisenstein series”, extensions of the series <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mo stretchy="false">(</m:mo> <m:mi>τ</m:mi> stretchy="false">)</m:mo> </m:math> {G_{2k}(\tau)} , defined by <m:mi>λ</m:mi> <m:mo>=</m:mo> <m:msup> <m:mn>1</m:mn> <m:mi>m</m:mi> </m:msup> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>⊢</m:mo> <m:mo>↦</m:mo>...

We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-block with defect 0, completing an investigation of many authors. The only zero alttext="p minus"> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow>...

Abstract Ramanujan’s last letter to Hardy concerns the asymptotic properties of modular forms and his ‘mock theta functions’. For mock function $f(q)$ , Ramanujan claims that as $q$ approaches an even-order $2k$ root unity, we have $$\begin{eqnarray*}f(q)- (- 1)^{k} (1- q)(1- {q}^{3} )(1- {q}^{5} )\cdots 2q+ 2{q}^{4} - \cdots )= O(1).\end{eqnarray*}$$ We prove claim a special case more general result. The implied constants in are not mysterious. They arise Zagier’s theory ‘quantum forms’....

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be prime and let alttext="upper G upper F left-parenthesis p right-parenthesis"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> stretchy="false">)</mml:mo> </mml:mrow>...

If p is prime, then let φp denote the Legendre symbol modulo and be trivial character p. As usual, n+1Fn(x)p := n+1Fn „ φp, . , p, | x « Gaussian hypergeometric series over Fp. For n > 2 non-trivial values of have been difficult to obtain. Here we take first step by obtaining a simple formula for 4F3(1)p. corollary obtain result describing distribution traces Frobenius certain families elliptic curves. We also find that 4F3(1)p satisfies surprising congruences 32 11. establish mod p2...

Eighty years ago, Ramanujan conjectured and proved some striking congruences for the partition function modulo powers of 5, 7, 11. Until recently, only a handful further such were known. Here we report that are much more widespread than was previously known, describe theoretical framework appears to explain every known Ramanujan-type congruence.

We investigate the arithmetic and combinatorial significance of values polynomials jn(x) defined by q-expansion \[\sum_{n=0}^{\infty}j_n(x)q^n:=\frac{E_4(z)^2E_6(z)}{\Delta(z)}\cdot\frac{1}{j(z)-x}.\] They allow us to provide an explicit description action Ramanujan Theta-operator on modular forms. There are a substantial number consequences for this result. obtain recursive formulas coefficients forms, infinite product exponents new p-adic class formulas.

We show that the rank generating function U ( t ; q ) for strongly unimodal sequences lies at interface of quantum modular forms and mock forms. use (-1; to obtain a form which is “dual” Zagier constructed from Kontsevich’s “strange” F ). As result, we new representation certain L -values. The series i = (- form, this fact congruences enumerative functions.

The Umbral Moonshine Conjectures assert that there are infinite-dimensional graded modules, for prescribed finite groups, whose McKay–Thompson series certain distinguished mock modular forms. Gannon has proved this the special case involving largest sporadic simple Mathieu group. Here, we establish existence of umbral moonshine modules in remaining 22 cases.