13 research outputs found
A Log-Free Zero-Density Estimate and Small Gaps in Coefficients of L-Functions
No full textLet L(s, π × π ′) be the Rankin–Selberg L-function attached to automorphic representa-tions π and π ′. Let π ̃ and π ̃ ′ denote the contragredient representations associated to π and π ′. Under the assumption of certain upper bounds for coefficients of the log-arithmic derivatives of L(s, π × π̃) and L(s, π ′ × π ̃ ′), we prove a log-free zero-density estimate for L(s, π × π ′) which generalizes a result due to Fogels in the context of Dirichlet L-functions. We then employ this log-free estimate in studying the distribu-tion of the Fourier coefficients of an automorphic representation π. As an application, we examine the nonlacunarity of the Fourier coefficients bf (p) of a modular newform f(z) =∑∞n=1 bf (n) e2πinz of weight k, level N, and character χ. More precisely, for f(z) and a prime p, set jf (p):=maxx;x>p Jf (p, x), where Jf (p, x): = #{prime q; aπ (q) = 0 for all p< q ≤ x}. We prove that jf (p) f,θ pθ for some 0< θ < 1.
An improved upper bound for the argument of the Riemann zeta-function on the critical line
No full textThis paper concerns the function S(t), the argument of the Riemann
zeta-function along the critical line. Improving on the method of Backlund,
and taking into account the refinements of Rosser and McCurley it is
proved that for sufficiently large t,
|S(t)| ≤ 0.1013 log t. Theorem 2 makes the above result explicit, viz. it enables one to select values
of a and b such that, for t>t0,
|S(t)| ≤ a + b log t
Improvements to Turing’s method
No full textThis article improves the estimate of the size of the definite integral of
S
(
t
)
S(t)
, the argument of the Riemann zeta-function. The primary application of this improvement is Turing’s Method for the Riemann zeta-function. Analogous improvements are given for the arguments of Dirichlet
L
L
-functions and of Dedekind zeta-functions.</p
Toward optimal exponent pairs
No full textWe quantify the set of known exponent pairs and develop a framework to compute the optimal exponent pair for an arbitrary objective function. Applying this methodology, we make progress on several open problems, including bounds of the Riemann zeta-function in the critical strip, estimates of the moments of and the generalised Dirichlet divisor problem.38 pages, 2 figure
Selberg's Method and the Multiplicities of the Zeroes of the Riemann Zeta-Function
No full textP(論文)departmental bulletin pape
Zeroes of partial sums of the zeta-function
No full textThis article considers the positive integers N for which ζN(s)=∑Nn=1n−s has zeroes in the half-plane R(s)>1. Building on earlier results, we show that there are no zeroes for 1⩽N⩽18 and for N=20,21,28. For all other N there are infinitely many such zeroes
On the Sum of Two Squares and At Most Two Powers of 2
No full textWe demonstrate that there are infinitely many integers that cannot be expressed as the sum of two squares of integers and up to two non-negative integer powers of 2
Linnik's approximation to Goldbach's conjecture, and other problems
No full textWe examine the problem of writing every sufficiently large even number as the sum of two primes and at most K powers of 2. We outline an approach that only just falls short of improving the current bounds on K. Finally, we improve the estimates in other Waring–Goldbach problems
