In this paper, we study a multidimensional divisor problem concerning the symmetric j^th power L-function. More precisely, we establish an asymptotic formula for the discrete mean square of the Fourier coefficients of the symmetric j^th power L-function in k-dimensions.
Inspired by the existence of multiple alternative definitions of continued fraction expansions for elements in ℚ_p , we study the p-adic convergence of periodic continued fractions with partial quotients in ℤ[1/p] from a geometric point of view. To this end, following a previous work by Brock, Elkies, and Jordan, we consider certain algebraic varieties whose points represent formal periodic continued fractions with period and preperiod of fixed lengths, satisfying a given quadratic equation. We then focus on the p-adically convergent loci of these varieties, describing the zero and one-dimensional cases by combining tools from algebraic geometry, arithmetic, and the theory of Pell equations and of linear recurrences.
Let p>3 be a prime. We obtain explicit congruences for ∑ _k=0^p-1w(k)( [ 2k; k ]) ^3/(-8)^k, ∑ _k=0^p-1w(k)( [ 2k; k ]) ^2( [ 3k; k ]) /(-192)^k, ∑ _k=0^p-1w(k)( [ 2k; k ]) ^2( [ 4k; 2k ]) /(-144)^k, ∑ _k=0^p-1w(k)( [ 2k; k ]) ^2( [ 4k; 2k ]) /648^k, ∑ _k=0^(p-1)/2( [ 2k; k ]) ^3/(-8)^k(k+1)^r, ∑ _k=0^p-2( [ 2k; k ]) ^2( [ 3k; k ]) /(-192)^k(k+1)^r, ∑ _k=0^p-2( [ 2k; k ]) ^2( [ 4k; 2k ]) /(-144)^k(k+1)^r, ∑ _k=0^p-2( [ 2k; k ]) ^2( [ 4k; 2k ]) /648^k(k+1)^r, ∑ _k=0^p-1k^r( [ 2k; k ]) ^3/64^k, ∑ _k=0^p-1k^r( [ 2k; k ]) ^2( [ 3k; k ]) /108^k, ∑ _k=0^p-1k^r( [ 2k; k ]) ^2( [ 4k; 2k ]) /256^k, ∑ _k=0^p-1k^r( [ 2k; k ]) ( [ 3k; k ]) ( [ 6k; 3k ]) /1728^k mod p^2 and partial results for ∑ _k=0^(p-1)/2( [ 2k; k ]) ^31/m^k(k+1)^r and ∑ _k=0^p-1( [ 2k; k ]) ^3w(k)/m^k mod p^2 , where w(k)∈{k^2,k^3, 1/2k-1,1/(2k-1)^2} , r∈{1,2,3} and m∈{1,16,-64,256,-512,4096} .
