Ayyadurai Sankaranarayanan

The average behavior of Fourier coefficients of cusp forms over sparse sequences

Let λ(n) be the nth normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈ S k (Γ). In this paper we are interested in the average behavior of λ 2 (n) over sparse sequences. By using the properties of symmetric power L-functions and their Rankin-Selberg L-functions, we are able to establish that for any e > 0, formula math. where j = 2, 3, 4.

ESTIMATES FOR THE SOLUTIONS OF CERTAIN DIOPHANTINE EQUATIONS BY RUNGE'S METHOD

We consider the two Diophantine equations ym = F(x) and G(y) = F(x) under the assumption that gcd(m, deg F) > 1 and gcd (deg G, deg F) > 1, respectively. We prove that the bounds for the denominator of the coefficients of the power series arising from the above two situations can be improved considerably and thus we establish improved upper bounds for the size of the solutions (namely for |x| and |y|). We also give explicit upper bounds for the integer solutions of equations of the form \[ F(x,y) = P_1(x) Q_2(y) - P_2(y) Q_1(x) = 0 \] under the assumption that \[ {\rm gcd} ( {\rm deg} \, P_1 - {\rm deg} \, Q_1, \, {\rm deg} \, P_2 - {\rm deg} \, Q_2 ) > 1. \]

Bernoulli numbers, convolution sums and congruences of coefficients for certain generating functions

In this paper, we study the convolution sums involving restricted divisor functions, their generalizations, their relations to Bernoulli numbers, and some interesting applications.MSC: 11B68, 11A25, 11A67, 11Y70, 33E99.

On an error term related to the Jordan totient function Jk(n)

Abstract We investigate the error terms E k (x)= ∑ n⩽x J k (n)− x k+1 (k+1)ζ(k+1) for k⩾2 , where J k (n) = n k Π p|n (1 − 1 p k ) for k ≥ 1. For k ≥ 2, we prove ∑ n⩽x E k (n)∼ x k+1 2(k+1)ζ(k+1) . Also, lim inf n→∞ E k (x) x k ⩽ D ζ(k+1) , where D = .7159 when k = 2, .6063 when k ≥ 3. On the other hand, even though lim inf n→∞ E k (x) x k ⩽− 1 2ζ(k+1) , Ek(n) > 0 for integers n sufficiently large.

ARITHMETIC SUMS SUBJECT TO LINEAR AND CONGRUENT CONDITIONS AND SOME APPLICATIONS

We investigate the explicit evaluation for the sum P (a;b;x;y)2N 4 ; ab in terms of various divisor functions (whereC(x;y)

ON THE CONVOLUTION SUMS OF CERTAIN RESTRICTED DIVISOR FUNCTIONS

We study convolution sums of certain restricted divisor functions in detail and present explicit evaluations in terms of usual divisor functions for some specific situations.

CONVOLUTION SUMS AND THEIR RELATIONS TO EISENSTEIN SERIES

In this paper, we consider several convolution sums, namely, Ai(m;n; N) (i = 1; 2; 3; 4), Bj(m;n;N) (j = 1; 2; 3), and Ck(m;n;N) (k = 1; 2; 3;:::, 12), and establish certain identities involving their nite products. Then we extend these types of product convolution identities to products involving Faulhaber sums. As an application, an identity in- volving the Weierstrass }-function, its derivative and certain linear com- bination of Eisenstein series is established.

Eisenstein series and their applications to some arithmetic identities and congruences

Utilizing the theory of elliptic curves over ℂ to the normalized lattice Λτ, its connection to the Weierstrass ℘-functions and to the Eisenstein series E4(τ) and E6(τ), we establish some arithmetic identities involving certain arithmetic functions and convolution sums of restricted divisor functions. We also prove some congruence relations involving certain divisor functions and restricted divisor functions.MSC:11A25, 11A07, 11G99.