For fixed large
x we give upper and lower bounds for the minimum
of
∑n≤xχ(n)/n as we minimize over all real-valued Dirichlet characters
χ.
Expanding our considerations to all multiplicative, real-valued
multiplicative functions of absolute value
≤1, the minimum equals
−0.4553···+o(1), and in this case we can classify the set of optimal functions.
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