LOG-TANGENT INTEGRALS AND THE RIEMANN ZETA FUNCTION
Copyright policy )LOG-TANGENT INTEGRALS AND THE RIEMANN ZETA FUNCTION
We show that integrals involving the log-tangent function, with respect to any square-integrable function on (0,π/2), can be evaluated by the harmonic series. Consequently, several formulas and algebraic properties of the Riemann zeta function at odd positive integers are discussed. Furthermore, we show among other things, that the log-tangent integral with respect to the Hurwitz zeta function defines a meromorphic function and its values depend on the Dirichlet series ζh(s) :=∑n≥1hnn−s−8, where hn=∑nk=1(2k−1)−1.
- Mohammed V University Morocco
harmonic series, Articles, Hurwitz and Lerch zeta functions, Hurwitz zeta function, Apéry's constant, \(\zeta (s)\) and \(L(s, \chi)\), QA1-939, log-tangent integrals, Apéry’s constant, Riemann zeta function, Dirichlet series, Trigonometric and exponential sums (general theory), Mathematics
harmonic series, Articles, Hurwitz and Lerch zeta functions, Hurwitz zeta function, Apéry's constant, \(\zeta (s)\) and \(L(s, \chi)\), QA1-939, log-tangent integrals, Apéry’s constant, Riemann zeta function, Dirichlet series, Trigonometric and exponential sums (general theory), Mathematics
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).2 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Average influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!