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Lower bounds for the largest eigenvalue of the gcd matrix on { 1 , 2 , ⋯ , n }

Jorma K. Merikoski (2016)

Czechoslovak Mathematical Journal

Consider the n × n matrix with ( i , j ) ’th entry gcd ( i , j ) . Its largest eigenvalue λ n and sum of entries s n satisfy λ n > s n / n . Because s n cannot be expressed algebraically as a function of n , we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that λ n > 6 π - 2 n log n for all n . If n is large enough, this follows from F. Balatoni (1969).

Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line

Thomas Christ, Justas Kalpokas (2013)

Journal de Théorie des Nombres de Bordeaux

We establish unconditional lower bounds for certain discrete moments of the Riemann zeta-function and its derivatives on the critical line. We use these discrete moments to give unconditional lower bounds for the continuous moments I k , l ( T ) = ∫ 0 T | ζ ( l ) ( 1 2 + i t ) | 2 k d t , where l is a non-negative integer and k ≥ 1 a rational number. In particular, these lower bounds are of the expected order of magnitude for I k , l ( T ) .

Lower powers of elliptic units

Stefan Bettner, Reinhard Schertz (2001)

Journal de théorie des nombres de Bordeaux

In the previous paper [Sch2] it has been shown that ray class fields over quadratic imaginary number fields can be generated by simple products of singular values of the Klein form defined below. In the present article the second named author has constructed more general products that are contained in ray class fields thereby correcting Theorem 2 of [Sch2]. An algorithm for the computation of the algebraic equations of the numbers in Theorem 1 of this paper has been implemented in a KASH program...

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