Related papers: Variations of Ramanujan's Euler Products

In this paper, we study some properties of Euler polynomials arising from umbral calculus. Finally, we give some interesting identities of Euler polynomials using our results. Recently, Dere and Simsek have studied umbral calculus related…

We call $R_G(a):=\sum_{q=1}^{\infty}G(q)c_q(a)$ the 'Ramanujan series', of coefficient $G:$N$\to$C, where $c_q(a)$ is the well-known Ramanujan sum. We study the convergence of this series (a preliminary step, to study Ramanujan expansions…

In a previous paper the authors elaborated notions and technique which could be applied to compute such invariants of polynomials as Euler characteristics of fibres and zeta-functions of monodromy transformations associated with a…

We define a parametric variant of generalized Euler sums and construct contour integration to give some explicit evaluations of these parametric Euler sums. In particular, we establish several explicit formulas of (Hurwitz) zeta functions,…

We consider a variant expression to regularize the Euler product representation of the zeta functions, where we mainly apply to that of the Riemann zeta function in this paper. The regularization itself is identical to that of the zeta…

In this Ph.D. thesis, written under the direction of D.B. Zagier and R.W. Bruggeman, we study the mock theta functions, that were introduced by Ramanujan. We show how they can be interpreted in the theory of (real-analytic) modular forms.…

For a class of generalized holomorphic Eisenstein series, we establish complete asymptotic expansions (Theorems~1~and~2), which together with the explicit expression of the latter remainder (Theorem~3), naturally transfer to several new…

This paper completes the proof of the Ramanujan Conjecture for holomorphic Hilbert modular forms whose weights are all congruent modulo 2. As a consequence, the Weight-Monodromy Conjecture and the zeta function conjecture of Langlands are…

The aim of this article is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of $L$-functions, namely, non-principal Dirichlet and those based on cusp…

We present generating functions for extensions of multiplicative invariants of wreath symmetric products of orbifolds presented as the quotient by the locally free action of a compact, connected Lie group in terms of orbifold sector…

We study the analytic behaviour of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power L-functions and the Sato-Tate conjectures we…

Product-to-sum identities for trigonometric functions play a fundamental role in function theory and numerous applications. In this spirit, we present convolution-to-sum identities for Mittag-Leffler type functions. Using a Laplace domain…

The study of Fourier coefficients of meromorphic modular forms dates back to Ramanujan, who, together with Hardy, studied the reciprocal of the weight 6 Eisenstein series. Ramanujan conjectured a number of further identities for other…

Let $c$ be a characteristic form of degree $k$ which is defined on a Kaehler manifold of real dimension $m>2k$. Taking the inner product with the Kaehler form $\Omega^k$ gives a scalar invariant which can be considered as a generalized…

In this paper we evaluate integrals of products of gamma functions of Ramanujan type in terms of bilateral hypergeometric series. In cases where the bilateral hypergeometric series are summable, then we evaluate these integral as beta…

In the present paper we introduce and studied two subclasses of multivalent functions denoted by $\mathcal{M}^{\lambda}_{p,n}(\gamma;\beta)$ and $\mathcal{N}^{\lambda}_{p,n}(\mu,\eta;\delta)$. Further, by giving specific values of the…

In this paper we give an attempt to extend some arithmetic properties such as multiplicativity, convolution products to the setting of operators theory. We provide a significant examples which are of interest in number theory. We also give…

This paper has twofold. The first is to establish a second main theorem for meromorphic functions on the complex disc $\Delta (R_0)\subset\mathbb C$ with finite growth index and small functions, where the counting functions are truncated to…

In the spirit of Ramanujan, we derive exponentially fast convergent series for Epstein zeta functions $ E^{\varGamma_0(N)}(z,s)$ on the Hecke congruence groups $ \varGamma_0(N),N\in\mathbb Z_{>0}$, where $z$ is an arbitrary point in the…

In this paper, the Douglas curvature of (\alpha,\beta)-metrics, a special class of Finsler metrics defined by a Riemannian metric \alpha and a 1-form \beta, is studied. These metrics with vanishing Douglas curvature in dimension n\geq3 are…