The Euler characteristic of the moduli space of curves.
The first moment of quadratic Dirichlet L-functions
The group structure for ζ(3)
The growth rate of the Dedekind Zeta-function on the critical line
The joint universality and the generalized strong recurrence for Dirichlet L-functions
The k-functions in multiplicative number theory. I. On complex explicit formulae
The k-functions in multiplicative number theory. II. Uniform distribution of zeta zeros
The k-functions in multiplicative number theory. III. Uniform distribution of zeta zeros; discrepancy
The k-functions in multiplicative number theory. IV. On a method of A. E. Ingham
The k-functions in multiplicative number theory. V. Changes of sign of some arithmetical error terms
The lowest zero of sections of the zeta function.
The mean square of the divisor function
Let d(n) be the divisor function. In 1916, S. Ramanujan stated without proof that , where P(y) is a cubic polynomial in y and , with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), . In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce . In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove .
The mean square of the Riemann zeta-function in the critical strip III
The mean square of the Riemann zeta-function in the critical strip II
The mean value of |L(k,χ)|² at positive rational integers k ≥ 1
Let k ≥ 1 denote any positive rational integer. We give formulae for the sums (where χ ranges over the ϕ(f)/2 odd Dirichlet characters modulo f > 2) whenever k ≥ 1 is odd, and for the sums (where χ ranges over the ϕ(f)/2 even Dirichlet characters modulo f>2) whenever k ≥ 1 is even.
The Mellin transform of powers of the zeta-function
The moments of infinite series.
The multiplicative and functional independence of Dedekind zeta functions of abelian fields
It is shown that the multiplicative independence of Dedekind zeta functions of abelian fields is equivalent to their functional independence. We also give all the possible multiplicative dependence relations for any set of Dedekind zeta functions of abelian fields.
The number-theoretic work of Paul Turán
The phase transition of the number-theoretical spin chain.