“I think it’s hard when someone who has been registered for seven or eight months in the district says they know what the people want,” Ms. Beatty said in a jab at Ms. Harper, a Columbus native who moved back to the city in December after a career in New York and Washington.

Ms. Harper is campaigning by attending as many community events as possible, and by old-fashioned neighborhood canvassing. There are some signs her strategy is working. She said she had raised money from nearly every ZIP code in her district. (Ms. Beatty, who hasn’t reported third-quarter fund-raising, had $1.3 million cash on hand as of July.)

Ms. Harper has focused on residents left out of the recent growth of Columbus — now Ohio’s most dynamic city, surpassing Cleveland — and the gentrification that has followed.

A wave of new apartment blocks, craft breweries and espresso shops radiating from downtown has pushed out poor residents, many of them black. Ms. Harper’s website calls Columbus the second most economically segregated city in the country.

In the barbershop, Mr. Copeland, the retired principal, said he was a personal friend of Ms. Beatty from “way back,” but that wouldn’t cement his vote.

“We’re in a bad state of affairs,” he said, referring to local and national events. “When we’re looking at elections and who we’re going to be sending to Congress and that sort of thing, we’re probably in a position where good friends and all that kind of stuff is going to go by the wayside.”